Weakly Differ en Ti Able Functions

Graduate Texts in Mathematics 120Editorial Board

J.H. Ewing F.W. Gehring P.R. Halmos

Graduate Texts in Mathematics

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fO4fYY44 :11101f7

William P. Ziemer

Weakly DifferentiableFunctionsSobolev Spaces and Functions ofBounded Variation

Springer-VerlagNew York Berlin HeidelbergLondon Paris Tokyo Hong Kong

William P. ZiemerDepartment of MathematicsIndiana UniversityBloomington, IN 47405USA

Editorial BoardJ. H. EwingDepartment of

MathematicsIndiana UniversityBloomington, IN 47405USA

F. W. GehringDepartment of

MathematicsUniversity of MichiganAnn Arbor, MI 48109USA

P. R. HalmosDepartment of

MathematicsSanta Clara UniversitySanta Clara, CA 95053USA

With l illustration

Mathematics Subject Classifications (1980)• 46-E35, 26-830, 31-B15

Library of Congress Cataloging-in-Publication DataZiemer, William P.

Weakly differentiable functions: Sobolev spaces and functions ofbounded variation William P. Ziemer.

p. cm.—(Graduate texts in mathematics; 120)Bibliography: p.Includes indexISBN 0-387.97017-71 Sobolev spaces. 2 Functions of bounded variation 1. Title

II. Series.QA323 Z53 1989515' .73—dc20

89-10072

Printed on acid-free paper.

C) 1989 by Springcr-Vcrlag New York IncAll rights reserved This work may not be translated or copied in whole or in part without thewritten permission of the publisher (Springer-Verlag, 175 Fifth Avenue. New York, NY 10010,USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connec-tion with any form of information storage and retrieval, electronic adaptation, computer software,or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use of general descriptive names, trade names, trademarks, etc. to this publication, even ifthe former are not especially identified, is not to be taken as a sign that such names, as under-stood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely byanyone

Camera-ready copy prepared using LaTEX.Printed and bound by R R Donnelley & Sons, Harrisonburg, VirginiaPrinted in the United States of America.

987654321

ISBN 0-3R7-97017-7 Springer-Verlag New York Berlin HeidelbergISBN 3-540-97017-7 Springer-Verlag Berlin Heidelberg New York

To Suzanne

Preface

The term "weakly differentiable functions" in the title refers to those inte-grable functions defined on an open subset of R" whose partial derivativesin the sense of distributions are either L" functions or (signed) measureswith finite total variation. The former class of functions comprises whatis now known as Sobolev spaces, though its origin, traceable to the early1900s, predates the contributions by Sabolev. Both classes of functions,Sobolev spaces and the space of functions of bounded variation (BV func-tions), have undergone considerable development during the past 20 years.From this development a rather complete theory has emerged and thus hasprovided the main impetus for the writing of this book. Since these classesof functions play a significant role in many fields, such as approximationtheory, calculus of variations, partial differential equations, and non-linearpotential theory, it is hoped that this monograph will be of assistance to awide range of graduate students and researchers in these and perhaps otherrelated areas. Some of the material in Chapters 1 -4 has been presented ina graduate course at Indiana University during the 1987-88 academic year,and I am indebted to the students and colleagues in attendance for theirhelpful comments and suggestions.

The major thrust of this book is the analysis of pointwise behavior ofSobolev and BV functions. I have not attempted to develop Sobolev spacesof fractional order which can be described in terms of Bessel potentials,since this would require an effort beyond the scope of this book. Instead,I concentrate on the analysis of spaces of integer order which is largelyaccessible through real variable techniques, but does not totally excludethe use of Bessel potentials. Indeed, the investigation of pointwise behaviorrequires an analysis of certain exceptional sets and they can be convenientlydescribed in terms of elementary aspects of Bessel capacity.

The only prerequisite for the present volume is a standard graduatecourse in real analysis, drawing especially from Lebesgue point theory andmeasure theory. The material is organized in the following manner. Chap-ter 1 is devoted to a review of those topics in real analysis that are neededin the sequel. Included here is a brief overview of Lebesgue measure, L"spaces, Hausdorff measure, and Schwartz distributions. Also included aresections on covering theorems and Lorentz spaces--the latter being neces-sary for a treatment of Sobolev inequalities in the case of critical indices.Chapter 2 develops the basic properties of Sobolev spaces such as equiva-lent formulations of Sobolev functions and their behavior under the opera-

viii Preface

Lions of truncation, composition, and change of variables. Also included is aproof of the Sobolev inequality in its simplest form and the related Rellich-Kondrachov Compactness Theorem. Alternate proofs of the Sobolev in-equality are given, including the one which relates it to the isoperimetricinequality and provides the best constant. Limiting cases of the Sobolevinequality are discussed in the context of Lorentz spaces.

The remaining chapters are central to the book. Chapter 3 develops theanalysis of pointwise behavior of Sobolev functions. This includes a dis-cussion of the continuity properties of functions with first derivatives inD' in terms of Lehesgue points, approximate continuity, and fine conti-nuity, as well as an analysis of differentiability properties of higher orderSobolev functions by means of L"-derivatives. Here lies the foundation formore delicate results, such as the comparison of I/-derivatives and dis-tributional derivatives, and a result which provides an approximation forSobolev functions by smooth functions (in norm) that agree with the givenfunction everywhere except on sets whose complements have small capacity.

Chapter 4 develops an idea due to Norman Meyers. He observed thatthe usual indirect proof of the Poincaré inequality could be used to es-tablish a Poincaré-type inequality in an abstract setting. By appropriatelyinterpreting this inequality in various contexts, it yields virtually all knowninequalities of this genre. This general inequality contains a term which in-volves an element of the dual of a Sobolev space. For many applications,this term is taken as a measure; it therefore is of interest to know preciselythe class of measures contained in the dual of a given Sobolev space. For-tunately, the Hedberg-Wolff theorem provides a characterization of suchmeasures.

The last chapter provides an analysis of the pointwise behavior of BVfunctions in a manner that runs parallel to the development of Lebesguepoint theory for Sobolev functions in Chapter 3. While the Lebesgue pointtheory for Sobolev functions is relatively easy to penetrate, the corre-sponding development for BV functions is much more demanding. Theintricate nature of BV functions requires a more involved exposition thandoes Sobolev functions, but at the same time reveals a rich and beautifulstructure which has its foundations in geometric measure theory. After thestructure of 13V functions has been developed, Chapter 5 returns to theanalysis of Poincaré inequalities for BV functions in the spirit developedfor Sobolev functions, which includes a characterization of measures thatbelong to the dual of $V.

In order to place the text in better perspective, each chapter is con-cluded with a section on historical notes which includes references to allimportant and relatively new results. In addition to cited works, the Bib-liography contains many other references related to the material in thetext. Bibliographical references are abbreviated in square brackets, such as[DLL. Equation numbers appear in parentheses; theorems, lemmas, corollar-ies,and remarks are numbered as a.b.c where b refers to section b in chapter

Preface ix

a, and section a.b refers to section b in chapter a.I wish to thank David Adams, Robert Glassey, Tero Kilpelâinen,

Christoph Neugehauer, Edward Stredulinsky, Tevan Trent, and WilliamK. Ziemer for having critically read parts of the manuscript and suppliedmany helpful suggestions and corrections.

WILLIAM P. ZIEMER

Contents

Preface vii

1 Preliminaries 1

1.1 Notation 1Inner product of vectorsSupport of a functionBoundary of a setDistance from a point to a setCharacteristic function of a setMulti-indicesPartial derivative operatorsFunction spaces—continuous, Holder continuous,

Holder continuous derivatives1.2 Measures on R" 3

Lebesgue measurable setsLebesgue measurability of Borel setsSustin sets

1.3 Covering Theorems 7Hausdorff maximal principleGeneral covering theoremVitali covering theoremCovering lemma, with n-balls whose radii vary in

Lipschitzian wayBesicovitch covering lemmaBesicovitch differentiation theorem

1.4 Hausdorff Measure 15Equivalence of Hausdorff and Lebesgue measuresHausdorff dimension

1.5 LP-Spaces 18Integration of a function via its distribution

functionYoung's inequalityHolder's and Jensen's inequality

1.6 Regularization 21LP-spaces and regularization

xii Contents

1.7 Distributions 23Functions and measures, as distributionsPositive distributionsDistributions determined by their local behaviorConvolution of distributionsDifferentiation of distributions

1.8 Lorentz Spaces 26Non-increasing rearrangement of a functionElementary properties of rearranged functionsLorentz spacesO'Neil's inequality, for rearranged functionsEquivalence of LP-norm and (p, p)-normHardy's inequalityInclusion relations of Lorentz spaces

Exercises 37Historical Notes 39

2 Sobolev Spaces and Their Basic Properties 42

2.1 Weak Derivatives 42Sobolev spacesAbsolute continuity on linesL'-norin of difference quotientsTruncation of Sobolev functionsComposition of Sobolev functions

2.2 Change of Variables for Sobolev Functions 49Rademacher's theoremBi-Lipschitzian change of variables

2.3 Approximation of Sobolev Functions by SmoothFunctions 53

Partition of unitySmooth functions are dense in Wk ,P

2.4 Sobolev Inequalities 55Soholev's inequality

2.5 The Rellich-Kondrachov Compactness Theorem 61Extension domains

2.6 Bessel Potentials and Capacity 64Riesz and Bessel kernelsBessel potentialsBessel capacityBasic properties of Bessel capacityCapacitability of Suslin setsMinimax theorem and alternate formulation of

Bessel capacity

Contents xiii

Metric properties of Bessel capacity2.7 The Best Constant in the Sobolev Inequality 76

Co-area formulaSobolev's inequality and isoperimetric inequality

2.8 Alternate Proofs of the Fundamental Inequalities 83Hardy- Littlewood-Wiener maximal theoremSabolev's inequality for Riesz potentials

2.9 Limiting Cases of the Sobolev Inequality 88The case kp = n by infinite seriesThe best constant in the case kp = nAn L°°-bound in the limiting case

2.10 Lorentz Spaces, A Slight Improvement 96Young's inequality in the context of Lorentz spacesSobolev's inequality in Lorentz spacesThe limiting case


3 Pointwise Behavior of Sobolev Functions 112

3.1 Limits of Integral Averages of Sobolev FunctionsLimiting values of integral averages except for

capacity null set3.2 Densities of Measures3.3 Lebesgue Points for Sobolev Functions

Existence of Lebesgue points except for capacitynull set

Approximate continuityFine continuity everywhere except for capacity null set

3.4 LP-Derivatives for Sobolev FunctionsExistence of Taylor expansions D'

3.5 Properties of LP-DerivativesThe spaces Tk , tk, Tk,p , tk,P

The implication of a function being in Tk'' at allpoints of a closed set

3.6 An LP-Version of the Whitney Extension TheoremExistence of a CO° function comparable to the

distance function to a closed setThe Whitney extension theorem for functions in

Tk.P and tk.P3.7 An Observation on Differentiation3.8 Rademacher's Theorem in the LP-Context

A function in Tk.P everywhere implies it is int k 'p almost everywhere

1I2

116118

126

130

136

142145

xiv Contents

3.9 The Implications of Pointwise Differentiability 146Comparison of L"-derivatives and distributional

derivativesIf u E tkPF(x) for every z, and if the

V'-derivatives are in LP, then u E Wk,P3.10 A Lusin-Type Approximation for Sobolev Functions 153

Integral averages of Sobolev functions are uniformlyclose to their limits on the complement of setsof small capacity

Existence of smooth functions that agree with Sobolevfunctions on the complement of sets ofsmall capacity

3.11 The Main Approximation 159Existence of smooth functions that agree with

Sobolev functions on the complement of sets ofsmall capacity and are close in norm


4 Poincaré Inequalities—A Unified Approach 177

4.1 Inequalities in a General Setting 178An abstract version of the Poincaré inequality

4.2 Applications to Sobolev Spaces 182An interpolation inequality

4.3 The Dual of Wm'P(S2)

185The representation of (W' (1))'

4.4 Some Measures in (Wâ '"(11))' 188Poincaré inequalities derived from the abstract

version by identifying Lebesgue and Hausdorffmeasure with elements in (Wm'P(SZ))"

The trace of Sobolev functions on the boundary ofLipschitz domains

Poincaré inequalities involving the trace ofa Sobolev function

4.5 Poincaré Inequalities 193Inequalities involving the capacity of the set on

which a function vanishes4.6 Another Version of Poincaré's Inequality 196

An inequality involving dependence on the set onwhich the function vanishes, not merely on itscapacity

4.7 More Measures in (Wm ,P(S2))e 198Soholev's inequality for Riesz potentials involving

Contents xv

measures other than Lebesgue measureCharacterization of measures in (Wm.P(R")).

4.8 Other Inequalities Involving Measures in (WkiP)` 207Inequalities involving the restriction of Hausdorff

measure to lower dimensional manifolds4.9 The Case p = 1 209

Inequalities involving the L'-norm of the gradientExercises 214Historical Notes 217

5 Functions of Bounded Variation 220

5.1 Definitions 220Definition of BV functionsThe total variation measure IIDulI

5.2 Elementary Properties of 13V Functions 222Lower seinicontinuity of the total variation measureA condition ensuring continuity of the total -

variation measure5.3 Regularization of BV Functions 224

Regularization does not increase the 13V normApproximation of BV functions by smooth functionsCompactness in L 1 of the unit ball in BV

5.4 Sets of Finite Perimeter 228Definition of sets of finite perimeterThe perimeter of domains with smooth boundariesIsoperimetric and relative isoperimetric inequality for

sets of finite perimeter5.5 The Generalized Exterior Normal 233

A preliminary version of the Gauss-Green theoremDensity results at points of the reduced boundary

5.6 Tangential Properties of the Reduced Boundary and theMeasure-Theoretic Normal 237

Blow-up at a point of the reduced boundaryThe measure-theoretic normalThe reduced boundary is contained in the

measure-theoretic boundaryA lower bound for the density of IIDXEiiHausdorff measure restricted to the reduced boundary

is bounded above by IIDXEDD5.7 Rectifiability of the Reduced Boundary 243

Countably (n - 1)-rectifiable setsCountable (n - 1)-rectifiability of the

measure-theoretic boundary

xvi Contents

5.8 The Gauss-Green Theorem 246The equivalence of the restriction of Hausdorff

measure to the measure-theoretic boundaryand if DXEII

The Gauss-Green theorem for sets of finite perimeter5.9 Pointwise Behavior of BV Functions 249

Upper and lower approximate limitsThe Boxing inequalityThe set of approximate jump discontinuities

5.10 The Trace of a BV Function 255The bounded extension of BV functionsTrace of a BV function defined in terms of the

upper and lower approximate limits of theextended function

The integrability of the trace over themeasure-theoretic boundary

5.11 Sobolev-Type Inequalities for BV Functions 260Inequalities involving elements in (BV(SZ))•

5.12 Inequalities Involving Capacity 262Characterization of measure in (BV (SZ) )'Poincaré inequality for BV functions

5.13 Generalizations to the Case p > 1 2705.14 Trace Defined in Terms of Integral Averages 272Exercises 277Historical Notes 280

Bibliography 283

List of Symbols 297

Index 303

1

PreliminariesBeyond the topics usually found in basic real analysis, virtually all of thematerial found in this work is self-contained. In particular, most of the in-formation contained in this chapter will be well-known by the reader andtherefore no attempt has been made to make a complete and thorough pre-sentation. Rather, we merely introduce notation and develop a few conceptsthat will be needed in the sequel.

1.1 Notation

Throughout, the symbol Si will generally denote an open set in Euclideanspace R" and 0 will designate the empty set. Points in -Rn are denoted byz = (z,, ... , zn ), where z 1 E R 1 , 1 < i < n_ If x, y E R", the inner productof r and y is

x • y= Exiy,i=1

and the norm of z is1x1 = (z • z)'/2 .

If u: SZ — R 1 is a function defined on SZ, the support of u is defined by

spt u = 1 n {z: u(x) 0},

where the closure of a set S C R" is denoted by S. If S C 11, S compactand also 3 c 0, we shall write S cc SZ. The boundary of a set S is definedby

ôS=S n (Rn — S).

For E C R" and z E R", the distance from z to E is

d(z, E) = inf{Iz – yI . y E E }.

It is a simple exercise (see Exercise 1.I) to show that

Id(r, E) – d(y, E)I < Ix – yi

whenever z, y E Rn. The diameter of a set E C R" is defined by

diam(E) = sup{Ir – yI x, y E E),

2 1. Preliminaries

and the characteristic function E is denoted by xE. The symbol

B(x, r) _ {y : Ix — yi < r}

denotes the open ball with center x, radius r and

a(x, r) _ {p : tx -. yi < r}

will stand for the closed ball. We will use a(n) to denote the volume of theball of radius 1 in R. If a = (a l , ... ,an ) is an n-tuple of non-negativeintegers, a is called a multi-index and the length of a is

n

iai = E ai.1=l

If x = (x l , ... , xn ) E Rte, we will let

â = xal • x2 • • ân1 2 ' n

and a! = ar!a2! • • • a,a !. The partzal derivative operators are denoted byD, = a fax, for 1 < i < n, and the higher order derivatives by

D° =D,'.. D = axQ'...axQR •1 n

The gradient of a real-valued function u is denoted by

Du(x) = (D iu(x), ..., Dnu(x)).

If k is a non-negative integer, we will sometimes use Dku to denote thevector Dku = {Dau}â ^ =k.

We denote by C° (1) the space of continuous functions on O. More gen-erally, if k is a non-negative integer, possibly oo, let

Ck (SZ) = {u : RL, Dau E C° (St), p< lai < k },

Co (SZ) = Ck(Sa) fl {u : spt u compact, spt u C iZ),

and

Ck`(SZ) = ck(fZ) n {u : Dau has a continuous extension to SZ , o < lcxi < k}•

Since Ii is open, a function u E Ck (SZ) need not be bounded on U. However,if u is bounded and uniformly continuous on fZ, then u can be uniquelyextended to a continuous function on i We will use Ck (1; Rm) to denotethe class of functions u: f/ Rm defined on SZ whose coordinate functionsbelong to Ck (SZ). Similar notation is used for other function spaces whoseelements are vector-valued.

1.2. Measures on R71 3

if 0 < a < 1, we say that u is Hdlder continuous on SI with exponent aif there is a constant C such that

Iu(x) — u(y)I < CIx — y1', x,y E S1.

We designate by C°'° (Sl) the space of all functions u satisfying this condi-tion on a In case a = 1, the functions are called Lipschitz and the constantC is denoted by Lip(u). For functions that possess some differentiability,we let

ck k(Sl) = C°*°(S2) n { u : D/3u E C"(S?), 0 < 101 < k).

Note that Ck•°(S2) is a Banach space when provided with the norm

sup sup IDAu(x) D^ü(g)i + max sup 1,603 u(x)I.—

19I=k s,yEn Ix vI °<v31<k xEfr#y

1.2 Measures on Rn

For the definition of Lebesgue outer measure, we consider closed n-dimen-sional intervals

I = {x .a;< x ; < , i--1,...,n}

and their volumes

v(I) = ll (b,_ ai).

The Lebesgue outer measure of an arbitrary set E C R" is defined by

00AEI = inf E v(4) : E C J Ik , Ik an interval (1.2.1)

k=1 k=1

A set E is said to be Lebesgue measurable if

BAI = IA n E1+IA n (R"—E)I (1.2.2)

whenever A C R".The reader may consult a standard text on measure theory to find that

the Lebesgue measurable sets form a o- algebra, which we denote by A; thatis

(i) 0,R" E A.

(ii) If E 1 , Ea , . . . E A, then00

UE1 € A.;=i

(1.2.3)

fl Ed — lim 'Edî^00^^1

00

(1.2.6)

4 1. Preliminaries

(iii) If E E .A, then R"- E E ,4.Observe that these conditions also imply that A is also closed under count-able intersections. It follows immediately from (1.2.2) that sets of measurezero are measurable. Also recall that if E l , E2, ... are pairwise disjointmeasurable sets, then

00E Êi' .

i=1(1.2.4)

Moreover, if E1 C E2 C ... are measurable, then

00U Ei;_i = lirn'Ei Ii^0o

(1.2.5)

and if E l D E2 D ..., then

provided that lEkI < oo for some k.Up to this point, we find that Lebesgue measure possesses many of the

continuity properties that are essential for fruitful applications in analysis.However, at this stage we do not yet know whether the o-algebra, A, con-tains a sufficiently rich supply of sets to be useful. This possible objectionis met by the following result.

1,2.1. Theorem. Each closed set C C R" is Lebesgue measurable.

In view of the fact that the Borel subsets of R" form the smallest a-algebra that contains the closed sets, we have

1.2.2. Corollary. The Borel sets of R" are Lebesgue measurable.

Proof of Theorem 1.2.1. Because of the subadditivity of Lebesgue mea-sure, it suffices to show that for a closed set C C R",

A[ > IA nC'+ 'A n (R" —C)) (1.2.7)

whenever A C W'. This will follow from the following property of Lebesgueouter measure, which follows easily from (1.2.1):

'A U BI ='AÎ + IBI (1.2.8)

whenver A, B E R" with d(A,B) = inf {'x — y' : x E A, g E B} > 0. Indeed,it is sufficient to establish that 'AU B1 > IA' + I B'. For this purpose, choosee > O and let

00

A U B C U Ik wherek=I

1.2. Measures on R" 5

00

E v(4) < IAUBI + e. (1.2.9)

Because d(A, B) > 0, there exists disjoint open sets U and V such that

ACU, BCV. (1.2.10)

Clearly, the covering of AU B by {Ik} can be modified so that, for each k,

IkCUUV (1.2.11)

and that (1.2.9) still remains valid. However, (1.2.10) and (1.2.11) imply00

> v(Ik) > IAI + 1BI.^ -1

In order to prove (1.2.7), consider A C Rn with IAI < co and let Ct ={x : d(x,C) < 1/i}. Note that

d(A —C=,A n C) > 0and therefore, from (1.2.8),

lAI ? I(A — C;) u (A n C)I > IA — C;I + IA n CI. (1.2.12)

The proof of (1.2.7) will be concluded if we can show that

Urn IA—C,1—IA—CIi-.00Note that we cannot invoke (1.2.5) because it is not known that A — C; ismeasurable since A is an arbitrary set, perhaps non-measurable. Let

T; =An x. 1 <d(x,C)< 1+1

and note that since C is closed,

A— Ci(A—C)U (ÛT )

;^=1

which in turn, implies00

(1.2.13)

(1.2.14)

I A - C I IT1I< oo.==1

(1.2.16)

6 1. Preliminaries

To establish this, first observe that d(11, Ti ) > 0 if Ii — > 2. Thus, weobtain from (1.2.8) that for each positive integer m,p m

Tz^ _ ^ iTzi l <— lAl < oo,

E T2i—é=1

U T21-1i-1

<lAl<oo.

This establishes (1.2.16) and thus concludes the proof. ❑

1.2.3. Remark. Lebesgue measure and Hausdorff measure (which will beintroduced in Section 1.4) will meet most of the applications that occurin this book, although in Chapter 5, it will be necessary to consider moregeneral measures. We say that is a measure on R" if assigns a non-negative (possibly infinite) number to each subset of R" and p(0) = 0. Itis also accepted terminology to call such a set function an outer measure.Following (1.2.2), a set E is called p-measurable if

µ(A) = p(A n E)+ p(An (Rn — E))

whenever A C R" . A measure p on R" is called a Bores measure if everyBorel set is µ-measurable. A Borel measure with the properties that eachsubset of R" is contained within a Borel set of equal p measure and thatp(K) < oo for each compact set K C R" is called a Radon measure.

Many outer measures defined on Rn have the property that the Borel setsare measurable. However, it is sometimes necessary to consider a larger a-algebra of sets, namely, the Suslin sets, (often referred to as analytic sets).They have the property of remaining invariant under continuous mappingson R", a property not enjoyed by the Borel sets. The Suslin sets of R" canbe defined in the following manner. Let N denote the space of all infinitesequences of positive integers topologized by the metric

00[! 2 -ilai

— b:l j_ 1+1a,— b,)

where (a i l and {bt ) are elements of N. Let R" x N be endowed with theproduct topology. If

p:Rn xN—.R"

is the projection defined by p(x,a) = x, then a Suslin set of 11" can bedefined as the image under p of some closed subset of R" x N.

The main reason for providing the preceding review of Lebesgue measureis to compare its development with that of Hausdorff measure, which isnot as well known as Lebesgue measure but yet is extremely important ingeometric analysis and will play a significant role in the development ofthis monograph.

1.3. Covering Theorems 7

1.3 Covering Theorems

Before discussing Hausdorff measure, it will be necessary to introduce sev-eral important and useful covering theorems, the first of which is based onthe following implication of the Axiom of Choice.

Hausdorff Maximal Principle. If E is a family of sets (or a collectionof families of sets) and if {UF : F E .9 E E for any subfamily .F of Ewith the property that

FCG or GCF whenever F,G E.F,

then there exists E C E which is maximal In the sense that it is not a subsetof any other member of E.

The following notation will be used. If B is a closed ball of radius r, letB denote the closed ball concentric with B with radius 5r.

1.3.1. Theorem. Let ç be a family of closed balls with

R - sup{diamB : B E Q} < oo.

Then there is a subfamily .F C ç of pairwise disjoint elements such that

{UB: B E Ç}C{UB: B E .F} .

In fact, for each B E ç there exists B 1 E F such that B n B1 o Q andB C B l .

Proof. We determine .F as follows. For j = 1, 2, ... let

ÿj = ç : Bn B < diam B<---) ,41 21 -1

and observe that ç = U7 1 ç, . Now proceed to define Yi C çj inductivelyas follows.

Let .F1 C ÇÇ1 be an arbitrary maximal subcollection of pairwise disjointelements. Such a collection exits by the Hausdorff maximal principle. As-suming that .F1 , F2 ,... , îi-1 have been chosen, let Yi be a maximal pair-wise disjoint subcollection of

j-1çj n B : B n B' =0 whenever B' E .F: . (1.3.1)

Thus, for each B E ç3, j ? 1, there exists B1 E U;=1 .7; such that Bn B 1 o0. For if not, the family F7 consisting of B along with all elements of .Fj

S 1. Preliminaries

would be a pairwise disjoint subcollection of (1.3.1), thus contradicting themaximality of Moreover,

diam B< 1 =2R2

< 2 diem B 12

which implies that B C È. Thus,

{UB: BE ç;} C UB:B E U .^^ ,i=i

and the conclusion holds by taking

00

.F= u.F';. CIi -1

1.3.2. Definition. A collection ç of closed balls is said to cover a setE C R" finely if for each x E E and each e > 0, there exists B(x,r) E çand r < C.

1.3.3. Corollary. Let E C Rn be a set that is covered finely by g, whereÇ and ,F are as in Theorem 1.3.1. Then,

E -{UB:BE.r}c{UB:BEF-.F`}

for each finite collection .F' C

Proof. Since Rn -- {UB : B Er} is open, for each z E E - {UB : B E .F' }there exists B E ç such that z E B and B n[{UB : B E .F'} = O. FromTheorem 1.3.1, there is B 1 E .F such that B fl B1 0 0 and B 1 J B. NowB 1 it F' since B n B1 ,- 0 and therefore

xE{UB.BE .F^-^`}.

The next result addresses the question of determining an estimate forthe amount of overlap in a given family of closed balls. This will also beconsidered in Theorem 1.3.5, but in the following we consider closed ballswhose radii vary in a Lipschitzian manner. The notation Lip(h) denotesthe Lipschitz constant of the mapping h.

1.3.4. Theorem. Let SCITC R" and suppose h : U -, (0, oo) is Lipschitzwith Lip(h) < A. Let a > 0, fi > 0 with as < 1 and a/3 < 1. Suppose thecollection of closed n-balls {B(a, h(s)) : s E S} is disjointed. Let

= S n Is : B(x, ah(z)) n B(s, /3h(sp 0 0).


Then

(1 - A0)/( 1 + Aa) < h(x)/h(s) < (1 + A03)/(1 - Act) (1.3.2)

whenever s E 52 and

card(S= ) < ja + 03+1)(11)(1 + Aa)(1 - )Q) -1 r n [(1 + A0)/(1 - »J))"

where card(S.) denotes the number of elements in Sz .

Proof. If s E Ss , then clearly ix - sf < ah(x) + aka) and therefore

!h(x) - h(s)I < -- si < Aah(x) + A/3h(s),

(1 - Afi)h(s) < (1 + Aa)h(x),

(1 - Arx)h(x) < (1 + A13)h(a). (1.3.3)Now,

^x - + h(s) < ah(x) + (A + 1 )h(s)< ah(x) + (0 + 1)[(1 + Aa) /(1 - A,Q)]h(x)= ryh(x)

where ry = ar + (0+ 1)(1 + Aor)/(1 - Afl). Hence

B(s, h(s)) C B(x, ryh(x))

Since {B(s, h(s))) is a disjoint family,

L I$(8, h(s))I < IB(a, 7h(x))j.ES:

or from (1.3.3)

card(SZ)a(n)[(l+Aa)(1-A0)-1h(x))" < E a(n)h(s)" < a(n)[7h(x))". QDES=

We now consider an arbitrary collection of closed balls and find a sub-cover which is perhaps not disjoint, but whose elements have overlap whichis controlled.

1.3.5. Theorem. There is a positive number N > 1 depending only on nso that any family B of dosed balls in R" whose cardinality is no less thanN and R = sup{r : B(a, r) E B) < oo contains disjointed subfamilies B i ,82, ... , BN such that if A is the set of centers of balls in B, then

Ac U{uB:B E B{}.i-1

whenever s E Sx .

10 1. Preliminaries

Proof.Step I. Assume A is hounded.Choose B 1 = B(a l , r l ) with r 1 > 1R. Assuming we have chosen B 1 , ...,

B,_, in 8 where j > 2 choose B, inductively as follows. 1f A3 = A NU_ i Bs = 0, then the process stops and we set J = j. If A; 0, continueby choosing Eli = B(a1, r,) E B so that a, E Ai and

r2 > 4 su p{r : B(a,r) E 8 , a E Ai l. (1.3.4)

If Ai 0 for all j, then we set J = +oo. In this case lirr,? moo, r^ = 0because A is bounded and the inequalities

r; 2 ri riIai — ail>ri= —

33 + 2, for i<j,

imply that{B(a1,ri/3) : 1 < j < J} is disjointed. (1.3.5)

In case J < oo, we clearly have the inclusion

AC{UB,:1<j<J}. (1.3.6)

This is also true in case J = +oo, for otherwise there would exist B(a, r) EB with a E nl°_ 1 Al and an integer j with r, < 3r/4, contradicting thechoice of B1 .

Step II. We now prove there exists an integer M (depending only on n)such that for each k with 1 < k < J, M exceeds the number of balls B iwith 1<i< k and B1 n Bk0 0.

First note that if r; < 10rk, then

B(a„ ri /3) C B(ak, 15rk)

because if z E r; /3),

Iz—akI <Iz — ail+Ias — akI

< lOrk /3 + r; + rk

< 43rk/3 < 15rk.

Hence, there are at most (60)" balls B; with

1<i< k, B, n Bk 0, and r i <IOrk

because, for each such i,

B(as , ri/3) C B(ak , 15rk),and by (1.3.4) and (1.3.5)

IB(as, rs /3 )I = iBi I ^ ( ! )r > IB1I ^ ()" r 60n I B(ak , 15rk)I-


To complete Step II, it remains to estimate the number of points in the set

I ={i . 1 <i< k,Bt f1 Bk 0,rt > IOrk}.

For this we first find an absolute lower bound on the angle between the twovectors

a, — ak and as — ak

corresponding to i,j E I with i < j. Assuming that this angle a < 7112,2,consider the triangle

and assume for notational convenience that rk = 1, d = lai — aki. Then

10< ri < lai — akI <r ; +1 and lai — ajI > ri

because j E I, ak Bs , BJ fl Bk 0, and as V B. Also

10 <rt < d< rt +1 < 3 rt +1

because j E I, ak it Bi , BJ fl Bk 0, and (1.3.4) applies to ri.The law of cosines yields

ia i —akI 2 +d2 — iai -- a;l 2 (r,+1) 2 +d2 —r; cos a = <

21ai — akld 2rtd_ 2r,+1+d2 _ 1 1 d 1 1 4ri 1

gr i d — d + 2rid + 2ri < d + gri d + 6r; + 2rt1 1 4 1

< 10+200++ 20 <.822;

hence lcv > arccos .822 > O. Consequently, the rays determined by a s — akand a i — ak intersect the boundary of B(ak,1) at points that are separatedby a distance of at least /2(1 — cow). Since the boundary of B(ak, 1)

12 1. Preliminaries

has finite H"-l measure, the number of points in I is no more than some constant depending only on n.

Step Ill. Choice of BI! ... , BM in case A is bounded. With each positive integer j f we define an integer Aj such that Aj = j

whenever 1 <j ~ M and for j > M we define A;+l inductively as follows. From Step II there is an integer A,;+I e {I, 2, ... , M} such that

B j + 1 n {UBi : 1 < i < i, Ai = ,\ j + l} = 0.

Now deduce from (1.3.6) that the unions of the disjointed families

covers A. Step IV. The case A is unbounded. For each positive integer I, apply Step III with A replaced by Et =

A n {x : 3(t - l)R < Ixl < 31R} and 8 replaced by the subfamily Ct of 8 of balls with centers in Et. We obtain disjointed subfamilies 8f, ... ,B'M of Ct such that

14

Et c U{UB : BE Bf}· .=1

Since P n Q = 0 whenever P E 8t , Q E B'" and m > I + 2, the theorem follows with

00 00

8 U 8 2t-l 8 U 8 2i- 1 1= 1 , ... , M= 14

t~l t=1

00 00

81.1+1 = U 8~t, . .. ,8214 = U 8~L t=1 t~l

and N = 2M. o

We use this result to establish the following covering theorem which contains the classical result of Vitali involving Lebesgue measure. An interesting and novel aspect of the theorem is that the set A is not assumed to be I-£-measurable. The thrust of the proof is that the previous theorem allows 118 to obtain a disjoint 8ubfamily that provides a fixed percentage of the 1-£ measure of the original set.

1.3.6. Theorem. Let ~ be a Radon measure on R n and Juppose :F is a family of closed ball3 that covers a set A C IF finely, where ~(A) < 00.

Then there exists a countable disjoint Bubfamily 9 of F such that

Jl(A - tuB : BEg}) = o.


Proof. Choose e > 0 so that e < 1/N, where N is the constant that appearsin the previous theorem. Then .F has disjointed subfamilies B,, ... , Bnr suchthat

A C U{UB: BE B;}i-,

and therefore

A(A) < p({u(A n B) : B E Bi)).

Thus, there exists 1 < k < N such that

p({U(A n B): B E > 1/N

which implies

p(A - {UB : B E Bk)) < (1— 1/N)p(A).

Hence, there is a finite subfamily Bk, of Bk such that

p(A - {UB : B E Bk 1 }) < (1 — 1/N + -€)14(A).

Now repeat this argument by replacing A with AL = 1 - {UB : B E Bk 1 }and .F with .FE _ .F n {B : B n {UB : B E Bk,) = 0) to obtain a finitedisjointed subfamily Bk2 of .F1 such that

p(A ) - {UB : B E BO) < (1 — 1/N + C);4111).

Thus,p(A-{UB: B EBk, UBk2 })<(1-1/N+e) 2 i(A).

Continue this process to obtain the conclusion of the theorem with00

ç =U Bk,,-t

o

1.3.7. Lemma. Let p. and v be Radon measures on R". For each positivenumber a let

Ea = x: sup ^ [$ (x ' T )) > a .

r>0 l (x, )1Then, p(Ea.) > av(Ea ).

Proof. By restricting our attention to bounded subsets of Ea , we mayassume that p(Ea ), v(Ea ) < oo. Let U D Ea be an open set. For e > 0and for each z: E Ea , there exists a sequence of closed balls B(s, r 1 ) C Uwithri -'Osuchthat

µ[B(x, ri)] > (a + €)v[B(x , ri)].

14 1. Preliminaries

This produces a family .7" of closed balls that covers E„, finely. Hence, byTheorem 1.3.6, there exists a disjoint subfamily Ç that covers v almost allof E.. Consequently

(a + e)v(Ea ) < (a + e) E v(B) < E p(B) < p(U).BEç BEST

Since E and U are arbitrary, the conclusion follows. D

If f is a continuous function, then the integral average of f over a ball ofsmall radius is nearly the same as the value of f at the center of the ball.A remarkable result of real analysis states that this is true at (Lebesgue)almost all points whenever f is integrable. The following result provides aproof relative to any Radon measure. The notation

f(y) CLAWB (s,r)

denotes,IB(x, r)]-1 f (y) dµ01)•

B(z,r)

1.3.8. Theorem. Let p be a Radon measure on R" and f a locally Inte-grable function on R" with respect to p. Then

]im f (y) dp(y) = f (x)r^d B(= ir)

for p almost all x E R" .

Proof. Note that

f (y)dp(v) — f (x)I (s,r)If (y) — g(y)Idgy)

B( 2,r)

+ I9(v) — f(x)Idµ(y)B(z,r)and if g is continuous, the last term converges to Ig(x) — f(x)I as r —, 0 .

Letting L(x) denote the upper limit of the term on the left, we obtain

L(x) < supI f(v) — 9(01001) +19(x) — f(x)I .r>D B (s,r)

Hence,

{x : L(x) > a} C x : sup If(y) — g(y)Idp(y) > a/2r>ü B (s,r)

1.4. Hausdorff Measure 15

Li {x : Ig(x) - f (x)I > a/2},and therefore, by the previous lemma,

p({x : L(x) > rx}) < 2/a R 1 f - gldN + 2/a R

I f - gidu•Rn

Since dR" If — gldp can be made arbitrarily small with appropriate choiceof g, cf. Section 1.6, it follows that µ({x : L(x) > a}) - 0 for each a > O.

1.3.9. Remark. If p and y are Radon measures with p absolutely con-tinuous with respect to y, then the Radon-Nikodym theorem providesf E L 1 (R", v) such that

p(E) =E f (x) dv(x).

The results above show that the Radon-Nikodym derivative f can be takenas the derivative of p with respect to v; that is,

hm 1`[(B

((x , r)1 f(x)

r-0 v(B(x, r)J

for v almost all x E Rn.

1.4 Hausdorff MeasureThe purpose here is to define a measure on R" that will assign a reason-able notion of "length," "area" etc. to sets of appropriate dimension. Forexample, if we would like to define the notion of length for an arbitrary setE C R" , we might follow (1.2.1) and let

A(E) = inf E diam.4 ; : E C U A ; , .; -i i=i

However, if we take n = 2 and E = {(t,sin(1/t)) : 0 < t < I}, it is easilyseen that A(E) < co whereas we should have À(E) = oo. The difficulty withthis definition is that the approximating sets A ; are not forced to followthe geometry of the curve. This i8 changed in the following definition.

1.4.1. Definition. For each -y > 0, E > 0, and E C R", let00 00

H£ (E) = infE a(7)2r"diarn(A;r : E C UA;, diam A< < c .._1 ■• x

16 1. Preliminaries

Because HE (E) is non-decreasing in r, we may define the 7 dimensionalHausdorif measure of E as

H7(E) = l âHgE)• (1.4.1)

In case 7 is a positive integer, or(1) denotes the volume of the unit ballin R7 . Otherwise, a(7) can be taken as an arbitrary positive constant.The reason for requiring 41(1) to equal the volume of the unit ball in /Vwhen Y is a positive integer is to ensure that 1-1 7 (E) agrees with intuitivenotions of "-y-dimensional area" when E is a well-behaved set. For example,it can be shown that H" agrees with the usual definition of n-dimensionalarea on an n-dimensional C i submanifold of Rn+ k , k > O. More generally,if f: Rn —. Rn+k is a univalent, Lipschitz map and E C Rn a Lebesguemeasurable set, then

J = H n [f (E)]fEwhere J f is the square root of the sum of the squares of the n x n deter-minants of the Jacobian matrix. The reader may consult [F4, Section 3.2]for a thorough treatment of this subject. Here, we will merely show thatH " defined on Rn is equal to Lebesgue measure.

1.4.2. Theorem. If E C Rn , then fin (E) = 1E1.

Proof. First we show that

HE (E) < lEI for every e > O.

Consider the case where IEI = 0 and E is bounded. For each ii > 0, letU E be an open set with WI < r). Since U is open, U can be written asthe union of closed balls, each of which has diameter less than e. Theorem1.3.1 states that there is a subfamily .2 of pairwise disjoint elements suchthat

U C {UB : $ E .^ }.

Therefore,

1/1:(E) Ç HE (U) < E H£ (B.) < E 2-na(n)(diamBi)nBEEF B; EN

E 2 - "a(n)5n (cliamB;)"B,E7

=5" E1B11B,EF

< 5"IUI < 5" 7/)which proves that H"(E) = 0 since e and ri are arbitrary. The case when Eis unbounded is easily disposed of by considering E ( B(0,1), i = 1, 2, ....

1.4. Hausdorff Measure 17

Each of these sets has zero n-dimensional Hausdorff measure, and thus sodoes E.

Now suppose E is an arbitrary set with IEI < oo. Let U J E be an openset such that

lUl < IEI + >7. (1.4.2)

Appealing to Theorem 1.3.6, it is possible to find a family .7" of disjointclosed balls B1, B2 i ..., such that U°_ 1 B; C U, diatn Bi < e, i = 1, 2, ...,and

00

E— B, —0. (1.4.3)i=1

Let E' V U;_ 1 (EnBi ) and observe that E = (E— E')UE' with IE — E`l =O. Now apply (1.4.1) and (1.4.2) to conclude that

00

H£ (E') < E 2a(n)(diam Bii = 100

DBil.=100U Bi

1=1

=lUl<1El + 17 .

Because e and n are arbitrary, it follows that H"(E') < IEj. However,II" (E) < HIE —E')+H" (Ea ) with H" (E—E') = 0 because I E—E'I = O.Therefore, H"(E) < 1E1.

In order to establish the opposite inequality, we will employ the isodi-arnetric inequality which states that among all sets E C R" with a givendiameter, d, the ball with diameter d has the largest Lebesgue measure;that is,

IEI < 2 —"a(n)(diarnE)'s (1.4.4)

whenever E C R. For a proof of this fact, see [F4, p. 197]. From this thedesired inequality follows immediately, for suppose

00

E 21 (n)(diam Ej)" <117(E) + rti=1

where E C U°2 1 Ei . Applying (1.4.3) to each Ei yields

IEI < EIEl < E 2 -"a(ra)(diam Eir < IÎNE) +n,

n

1=1

18 1. Preliminaries

which implies, 1E1 < H"(E) silice e and i are arbitrary.

1.4.3. Remark. The reader can easily verify that the outer measure, H'',has many properties in common with Lebesgue outer measure. For example,(1.2.4), (1.2.5), and (1.2.6) are also valid for H'' as well as the analogof Corollary 1.2.2. However, a striking difference between the two is that'E) < oo whenever E is bounded whereas this may be false for H''(E). Oneimportant ramification of this fact is the following. A Lebesgue measurableset, E, can be characterized by the fact that for every e > 0, there existsan open set U D E such that

lU — EE < e. (1.4.5)

This regularity property cannot hold in general for /P.The fact that IP (E) may be possibly infinite for bounded sets E can be

put into better perspective by the following fact that the reader can easilyverify. For every set E, there is a non-negative number, d = d(E), suchthat

H''(E) = 0 if 7 > d

H^ (E)= oo if 7< d.

The number d(E) is called the Hausdorff dimension of E.

Finally, we make note of the following elementary but useful fact. Sup-pose f; Rk —► R k +n is a Lipschitz map with Lip(f) = M. Then for any setECRk

Hk1 f (E)] < MHk (E). (1.4.6)

In particular, sets of zero k-dimensional Hausdorff measure remain invari-ant under Lipschitz maps.

1.5 LP SpacesFor 1 < p < oo, E.,(11) will denote the space consisting of all measurablefunctions on SI that are p"-power integrable on each compact subset of SZ.LP(11) is the subspace of functions that are p th -power integrable on Si Incase the underlying measure is i rather than Lebesgue measure, we willemploy the notation L o,([; p.) and U(11, ,L) respectively. The norm onLP(S1) is given by

1/plIt llp;c^ = lulpdx (1.5.1)(LI

and in case p — oc, it is defined as

iu ll^.n = esse sup lul (1.5.2)

1.5. L" Spaces 19

Analogous definitions are used in the case of LP(f2; p) and then the normis denoted by

IItIIp,N.S-The notation f u(x) dx or sometimes simply f u dx will denote integrationwith respect to Lebesgue measure and f u dp the integral with respect tothe measure p. Strictly speaking, the elements of L"(11) are not functionsbut rather equivalence classes of functions, where two functions are saidto be equivalent if they agree everywhere on SI except possibly for a set ofmeasure zero. The choice of a particular representative will be of specialimportance later in Chapters 3 and 5 when the pointwise behavior of func-tions in the spaces W k .P(SZ) and BV (0) is discussed. Recall from Theorem1.3.8 that if u E L 1 (R" ), then for almost every zo E R", there is a numberz such that

u(y)dy --+ z as r 0+ ,e(zo,r)

where f- denotes the integral average. We define u(xo) = z, and in thisway a canonical representative of u is determined. In those situations whereno confusion can occur, the elements of LP(n) will be regarded merely asfunctions defined on O.

The following lemma is very useful and will be used frequently through-out.

1.5.1. Lemma. If u > 0 is measurable, p > 0, and Et = {x : u(x) > t},then

More generally, if µ is a measure defined on some a-algebra of Rn, u > 0is a p-measurable function, and t is the countable union of sets of finitep measure, then

00 00updµ = N(Et)dtP = p

n o j (1.5.4)

The proof of this can be obtained in at least two ways. One method is toemploy Fubini's Theorem on the product space SZ x [0, oo). Another is toobserve that (1.5.3) is immediate when u is a simple function. The generalcase then follows by approximating u from below by simple functions.

The following algebraic and functional inequalities will be frequently usedthroughout the course of this book.

Cauchy's inequality: if e > 0, a, b E R1 , then

lab! < la^^ + 2E Ib;^ (1.5.5)

00 00u(x)pdx —IEtldtP ` p tp-1lEtldt. (1.5.3)

0

20 1. Preliminaries

and more generally, Young's inequality:

IabI < IeaI + [b/e }p ,

P P'

where p > 1 and lip + 1 /p' = 1.From Young's inequality follows Holder's inequality

(1.5.6)

I uv dx < ;fi, p > 1, (1.5.7)

which holds for functions u E L'(1), t1 E LP ' (ft). In case p = 1, wetake p' = co and IIvIIp ,

;n = esse sup M. Holder's inequality can be ex-tended to the case of k functions, uh ... , uk lying respectively in spacesL1" (0), ... ,17k (SZ) where

^` 1 (1.5.8)

By an induction argument and (1.5.7) it follows that

I u l . . . ukdx 5_1101 ;0 . • • Iludlpkst- (1.5.9)

One important application of (1.5.7) is Minkowski's inequality, which statesthat (1.5.3) yields a norm on 11(11). That is,

lIu + vIIp;fl <— Iiullp;n + IIt1Ilp;1 (1.5.10)

for p > 1. Employing the notation

dx = ISZI -f u dx,fo u n

another consequence of Holder's inequality is

1/p 1/qup dx < u'dx (1.5.11)

n n

whenever 1 < p < q and SZ C R" a measurable set with ISZI < oo.We also recall Jensen's inequality whose statement involves the notion

of a convex function. A function A: R" —. R I is said to be convex if

A[(1 — t)xt + tx2J < (1 — t)A(xl) + tA(x2),

whenever x , z2 E R" and 0 < t < 1. Jensen's inequality states that if A isa convex function on R" and E C R" a bounded measurable set, then

n f(x)dx < A(f (x)1Ê E(1.5.12)

1.6. Regularization 21

whenever f E L' (E).A further consequence of Holder's inequality is

I11411pIlullr's u E Lr(o), (1.5.13)

where p < q < r, and 1/q _ A/p+(1—A)/r. In order to see this, let a = aq,= (1 — A)q and apply Holder's inequality to obtain

1/3rlulQdx = lu l° lurdx < 1 ulardZ lulf^V dZT n

(10 I

where z = p/Aq and y = r/(1 A)q.When endowed with the norm defined in (1.5.1), L 3'(11), 1 _< p < oo,

is a Banach space; that is, a complete, linear apace. If 1 < p < oo, it isalso separable. The normed dual of V'(1) consists of all bounded linearfunctionals on LP(0) and is isometric to L" (f1) provided p < oo. Hence,L" (0) is reflexive for 1 < p < oo. We recall the following fundamental resultconcerning reflexive Banach spaces, which is of considerable importance inthe case of L"(1).

1.5.2. Theorem. A Banach space is reflexive if and only if its closed urritball is weakly sequentially compact.

1.6 Regularization

Let cp be a non-negative, real-valued function in Co (R") with the propertythat

cp(x)dt = 1, spt cp C B(0, 1), (1.6.1)

An example of such a function is given by

^(^) _ Cexp[-1/(1 — 1x1 2 )1 if 1x1 < 1 (1.6.2)0 ifixl>x

where C is chosen so that fR„ cp = 1. For e > 0, the function (Mx)e —ncp(x/e) belongs to Co (R") and spt C B(0, c). (pc is called a regular-izer (or mollifier) and the convolution

ue (x) = SPE * u(x) =R cpc(x

—y)u(y)dy (1.6.3)

n

defined for functions u for which the right side of (1.6.3) liar meaning,is called the regularization (mollification) of u. Regularization has severalimportant and useful properties that are summarized in the following the-orem.

22 1. Preliminaries

1.8.1. Theorem.

(i) If u E Ltoc (R"), then for every e > 0, uE E CO°(R") and D°'(cp E *u)(Dave ) * u for each multi-index a.

(ii) nc (x) u(x) whenever r as a Lebesgue point for u. In case u iscontinuous then uc converges uniformly to u on compact subsets ofR'

(iii) If u E !J(R"), 1 < p < oo, then of E L"(Rn), Iluelip < Ilulip, andlime —.0 — ullp = O.

Proof. For the proof of (i), it suffices to consider Iarl 1, since the case ofgeneral a can be treated by induction. Let el, ... , e" be the standard basisof R" and observe that

hIle (X + hei) — ti c (x) =

0 D; çpF (x — z + te ; )u(z)dtdz

R" h

=D,^pE(x — z + te,)u(z)dzdt.4 Rn

As a function of t, the inner integral on the right is continuous, and thus(1) follows.

In case (ii) observe that

1 uE (x) — u(x)l <f wr (x — y)Iu(y) — u(x)Idy

< sup cPE—" f Iu(x) — u(y)Idy —' 013(x ,t)

as E —, 0 whenever x is a Lebesgue point for u. Clearly the convergenceis locally uniform if u is continuous because u is uniformly continuous oncompact sets.

For the proof of (iii), Holder's inequality yields

Iuc(x)I = Spc (x — y)u(y)dy

lip' 1/p< co (x — y)dy SPF(x — y)lu(y)I p dy

The first factor on the right is equal to 1 and hence, by Fubini's theorem,

lue rdx < ^PF(x — y)Iu(y)lpdydxR^ R^ R"

(Mx — y)lu(y)I pdxdyf..L.

Iu(y)Ipdy.f"

1.7. Distributions 23

Consequently,IIuE Ilp llullp• (1.6.4)

To complete the proof, for each rî > O let v E C0(Rn ) be such that

Ilu - vllP<^-

(1.6.5)

Because u has compact support, it follows from (ii) that Ilv -- ve lip < ri fore sufficiently small. Now apply (1.6.4) and (1.6.5) to the difference y -- uand obtain

Ilu -ue llp<-Ilu-v llp+llU- vE lip +IlvE tic lip <_ 317.

Hence u, u in LP(Rn) as e O. ❑

1.8.2. Remark. If u E L'(11), then uE (x) = (pc * u(x) is defined provided• E S2 and e < dist(x, OD). It is a simple matter to verify that Theorem1.6.1 remains valid in this case with obvious modification. For example, ifu E C(R) and S2' CC S2, then u E converges uniformly to u on S2' as -' O.

Also note that (iii) of Theorem 1.6.1. implies that mollification doesnot increase the norm. This is intuitively clear since the norm must takeinto account the extremities of the function and mollification, which is anaveraging operation, does not increase the extremities.

1.7 DistributionsIn this section we present a very brief review of some of the elementaryconcepts and techniques of the Schwartz theory of distributions [SCH} thatwill be needed in subsequent chapters. The notion of weak or distributionalderivative will be of special importance.

1.7.1. Definition. Let SZ C Rn be an open set. The space (S2) is theset of all (iv in Co (1l) endowed with a topology so that a sequence {}converges to an element cp in g(1) if and only if

(i) there exists a compact set K C St such that spt Sp; C K for every i,and

(ii) lirn, . D°(#0= = Dapp uniformly on K for each multi-index cr.

The definition above does not attempt to actually define the topologyon gl(11) but merely states a consequence of the rigorous definition whichrequires the concept of "generalized sequences" or "nets," a topic thatwe do not wish to pursue in this brief treatment. For our purposes, itwill suffice to consider only ordinary sequences. It turns out that £ (S2)is a topological vector space with a locally convex topology but is not

24 1. Preliminaries

a normable space. The dual space, .Y'(fl), of 2(S/) is called the space of(Schwartz) distributions and is given the weak'-topology. Thus, T; E 0(0)converges to T if and only if T,(cp) -' T(cp) for every cp E . '(fi).

We consider some important examples of distributions. Let 1.1 be a Radonmeasure on S/ and define the corresponding distribution by

T (cP) = f o(x)dµ

for all E 2(0). Clearly T is a linear functional on .Y(fl) and 1T(rp) <Iµ4(spt (P)Il from which it is easily seen that T is continuous, and thus adistribution. In this way we will make an identification of Radon measuresand the associated distributions.

Similarly, let f E Li ,(fl), p > 1, and consider the corresponding signedmeasure defined for all Borel sets E C R" by

µ(E) =E f (r)dx

and pass to the associated distribution

f (cP) = ço(x)f (x)dx.R*

In the sequel we shall often identify locally integrable functions with theircorresponding distributions without explicitly indicating the identification.

1.7.2. Remark. We recall two facts about distributions that will be ofimportance later. A distribution T on an open set SZ is said to be positive ifTOp) > 0 whenever cp > 0, cp E . (1l). A fundamental result in distributiontheory states that a positive distribution is a measure. Of course, not alldistributions are measures. For example, the distribution defined on Re by

T (cp) = f (p'(z)dx

is not a measure since it is not continuous on 0(0) when endowed withthe topology of uniform convergence on compact sets.

Another important fact is that distributions are determined by their localbehavior. By this we mean that if two distributions T and S on CI havethe property that for every x E fZ there is a neighborhood U such thatT ((p) = S((p) for all ç ,

E . (1l) supported by U, then T = S. For example,this implies that if {SZ a } is a family of open sets such that Ufl a = Si andT is a distribution on fl such that T is a measure on each fL, then T is ameasure on O. This also implies that if a distribution T vanishes on eachopen set of some family it then vanishes on the union of all elements of.F. The support of a distribution T is thus defined as the complement ofthe largest open set on which T vanishes.

1.7. Distributions 25

We now proceed to define the convolution of a distribution with a testfunction Sp E O(SZ). For this purpose, we introduce the notation ( 03(x) =Sp(—x) and r2 ço(y) = Sp(y — r). The convolution of a distribution T definedon R" with yo E (SZ) is a function of class C°° given by

T * v(x) = T(rzy3). (1.7.1)

An obvious but important observation is

T * sp(0) = T(roç) = T(0).

If the distribution T is given by a locally integrable function f then wehave

(T * (P)(z) = f f (x - y)cP(y)db

which is the usual definition for the convolution of two functions. It is easyto verify that

T*( *V)= (T * (p) * bwhenever cp, t/1 E ^.

Let T be a distribution on an open set 0. The partial derivative of T isdefined as

D=T(v) = —T (D.ço)for cP E OW). Since Dop E 0(n) it is clear that D,T is again a distribu-tion. Since the test functions W are smooth, the mixed partial derivativesare independent of the order of differentiation:

D;Dj cP = Dl Dop

and therefore the same equation holds for distributions:

D; Di T = D• D 1 T.

Consequently, for any multi-index or the corresponding derivative of T isgiven by the equation

DcT(cp) = ( -1 )' a1 T (Da (p).

Finally, we note that a distribution on SI can be multiplied by smoothfunctions. Thus, if T E 93"(f1) and f E co 0(a), then the product fT is adistribution defined by

(fT)(ço) = T(fçp), cP E r(SZ)-

The Leibniz formula is easily seen to hold in this context (see Exercise 1.5).The reader is referred to [SCHJ for a complete treatment of this topic.

26 1. Preliminaries

1.8 Lorentz Spaces

We have seen in Lemma 1.5.1 that if f E L' (le), f > 0, then its integralis completely determined by the measure of the sets {x : f (x) > t}, t E W.The non-increasing rearrangement of f , (defined below) can be identifiedwith a radial function f having the property that for all t E R V , {x : f(x) >t} is a ball centered at the origin with the same measure as {x : f (x) > t).Consequently, f and f have the same integral. Because f can be thoughtof as a function of one variable, it is often easier to employ than f . Weintroduce a class of spaces called Lorentz spaces which are more generalbut closely related to L" spaces. Their definition is based on the conceptof non-increasing rearrangement. Later in Chapter 2, we will extend basicSobolev inequalities in an L" setting to that of Lorentz spaces.

1.8.1. Definition. If f is a measurable function defined on R", let

E; ={x:If(x)1>s}, (1.8.1)

and let the distribution function of f be denoted by

af(s)=IEf I- (1.8.2)

Note that the distribution function of f is non-negative, non-increasing,and continuous from the right. With the distribution function we associatethe non-increasing rearrangement of f on (0, oo) defined by

f'(t) = inf{s > 0 : af(9) < t}. (1.8.3)

Clearly f' is non-negative and non-increasing on (0, oo). Further, if off iscontinuous and strictly decreasing, then f' is the inverse of af, that is,f' = cr f 1 . It follows immediately from the definition of f'(t) that

f . (af(s)) < s

and because ci f is continuous from the right, that

af(f + (t)) t.

These two facts lead immediately to the following propositions.

1.8.2. Proposition. f' is continuous from the right.

(1.8.4)

(1.8.5)

Proof. Clearly, f' (t) > f• (t + h) for all h > O. if f' were not continuousat t, there would exist y such that f'(t) > y > f'(t+h) for all h > O. Butthen, (1.8.5) would imply that a f (y) < a f (f' (t + h)) < t+ h for all h > O.

1.8. Lorentz Spaces 27

Thus, a f (y) < t and therefore, f'(t) < y, a contradiction.

1.8.3. Proposition. af . (s) a1(s) for all s > O.

Proof. Because f' is non-increasing, it follows from the definition of a f. (t)that

a f•(8) = sup{t > 0 . f* (t) > s}. (1.8.6)

Hence, f'(a f (8)) < .q implies a f (s) > a1• (s). For the opposite inequal-ity, note from (1.8.6) that if t > a f. (s), then f' (t) < s and consequently,a f(s) < af(f +(t)) < t, by (1.8.5). Thus, af(s) < a f. (s) and the proposi-tion is established. O

1.8.4. Proposition. Let { ff ) be a sequence of measurable functions on Rnsuch that {I f,I} is a non-decreasing sequence. If If (x)I = lirn i.œ I f=(x)( foreach r E En, then a f, and fi increase to a f and f ' respectively.

Proof. Clearly

EP C Ef and = Ef

for each s and therefore a f,(8) --> ar f(s) as s — oo. It follows from definitionof non-increasing rearrangement, that f, (t) < f ' 1 (t) < f'(t) for each tand i = 1,2, .... Let g(t) = f, (t). Since f; (t) < g(t) it follows from(1.8.5) that a11 [g(t)] < af [f: (01 < t. Therefore,

o f [g(t)1 = Urn a1► [g(t)] < t

which implies that f'(t) < g(t). But g(t) < fi(t) and therefore the proofis complete. ❑

1.8.5. Theorem. If f E L", 1 < p < oo, then

I/p 1/p^fIp = [f'(t)1 dt (1.8.7)[fœ

Proof. This follows immediately from Lemma 1.5.1 and the fact that fand f' have the same distribution function (Proposition 1.8.3). ❑

We now introduce Lorentz spaces and in order to motivate the followingdefinition, we write (1.8.7) in a more suggestive form as

0or ) 1/13

I^f Îi p = (f t1/Pf(t)lPdt/1)

28 L Preliminaries

It is sometimes more convenient to work with the average of f' thanwith f' itself. Thus, we define

f"(t) = t f " (r)dr.0

1.8.6. Definition. For 1 < p < oo and 1 < q < oo, the Lorentz spaceL(p, q) is defined as

L(p, q) = {f : f measurable on Rr, lI f II(p,4) < co}

where II f It(p,4) is defined by

(1.8.8)

[tl/vf"(t)Iq,44 1/4

II/II(p,a) = j00 71

.°sup t'/p f.. (t),c>o

It will be shown in Lemma 1.8.10 that

L(p , p) = L.

1 <p<oo,1 < g< ao

1 <p< oo, q=oo.

(1.8.9)

The norm above could be defined with f" replaced by f' in case p > 1and 1 < q < oo. This alternate definition remains equivalent to the originalone in view of Hardy's inequality (Lemma 1.8.11) and the fact that f" > f'(since f' is non-increasing). For p > 1, the space L(p, oo) is known as theMarcinkiewicz space and also as Weak LP. In case p = 1, we clearly haveL(1, oo) = V. With the help of Lemma 1.5.1, observe that

1 t o0

f'(r)dr = tf'(t) + cxf(s)dsf • ( I )

and thereforef" (t) = f' (t) + a f(s)ds. (1.8.10)

t f' (t)For our applications it will be necessary to know how the non-increasing

rearrangement behaves relative to the operation of convolution. The nexttwo lemmas address this question. Because g" is non-increasing, note thatin the following lemma, the first and second conclusions are most interestingwhen t < r and t > r, respectively.

1.8.7. Lemma. Let f and g be measurable functions on Rn where sup{ f (x):r E Rn } < a and f vanishes outside of a measurable set E with 1Ei r.Let h = f * g. Then, for € > 0,

h"(t) < ac rg"(r)


andh+'(t) < a rg'* (t).

Proof. For a > 0, define

ga(x) g(x) if Ig(x)I < a

asgng(x) if fg(x)I > a

and letga (x) = g(x) _ gs(x)•

Then, define functions h i and h2 by

h=f *g —f *ga+f *ga

—h l +h 2 .

From elementary estimates involving the convolution and Lemma 1.5.1,we obtain

0osup {hz (x) : x E R") < sup{ f (x) : x E E }Ilgl l l < a a9(s)ds (1.8.11)

a

because ga(x) = 0 whenever Ig(x)I < a. Also

sup{h l (x) : x E R" ) <_ Ilf ll1 sup{ga (x) : x E E} < ara, (1.8.12)

and

02111 <_ IIfllillgalli < a a9(s)ds. (1.8.13)a

Now set a = g`(r) in (1.8.11) and (1.8.12) and obtain

eh"(t) = t

h+ (y)dy < 016

<_ Ilhlll^ +Ilhzl600

arg'(r) + a rxg (.․ )ds9 '(r)

coo< a [r?(r) +ag (s)d8

f•(r)= a rg" (r) .

The last equality follows from (1.8.10) and thus, the first inequality of thelemma is established.

30 1. Preliminaries

To prove the second inequality, set a = g' (r) and use (1.8.12) and (1.8.13)to obtain

eth"(t) = h'(yt)dyr hi(y!)dy + h2(y)dy

0 0 0vo

t11hl lly° +h;(i) dy t ll h jll 0 + 11h2111f00

< ta rg'(t) + ac r as(s)ds9' (T)

ao< a r [ t9*(t) + as (a)ds

9 '(r)< artg" (t)

by (1.8.10). 0

1.8.8. Lemma. If h, f , and g are measurable functions such that h = f *g,then for any t > 0

00h"(t) S tf"(t)g+w(t) +f'(u)g*(r^)dr^.f

Proof. Fix t > 0.Select a doubly infinite sequence {ys } whose indices ranges from -oo to

+oo such thatYo = f* (t)

yi <— y:+1

Vim y, - oc1 -1 00

Let

f(z) = fi(z)i=t-‘00

where0 if If (z)1 < yi-t

f1(z) = f(z) - y=-l sgn f(z) if ys_1 < If(z)I S ycYs — Ye —i sgn f (z) if y. < f (z)i•

Clearly, the series converges absolutely and therefore,

h=f •g-E(00

icxD ')*g

o = E fi *g+

,--on 6_i

-h l + h2

*g

1 f' (t)


withh'*(t) < hi'(t) + h2 - (t).

To evaluate hnt) we use the second inequality of Lemma 1.8.7 withE, _ {z: If(z)I> yi _ i ) = E and a=y,-y i _ i to obtain

00

h's*(t) < E(yt - 3h-1)af(y,-1)g"(t)s-1

00

l— 9-'(t) Eckf(yti-1!(yi — yi-1)•i= 1

The series on the right is an infinite Riemann sum for the integral

af (y)dy,

and provides an arbitrarily close approximation with an appropriate choiceof the sequence { y, } . Therefore,

00h2'(t) < g*`(t)

fa f(y)dy. (1.8.14)•(i)

By the first inequality of Lemma 1.8.7,00

hi*(t) 5 E(yi - y;-1)af (yi-1)9' + (af(ys-1))1= 1

The sum on the right is an infinite Riemann sum tending (with properchoice of y ; ) to the integral,

J. f' (e)a f (y)9 * * (af (y))d11.

We shall evaluate the integral by making the substitution y = f' (u) andthen integrating by parts. In order to justify the change of variable in theintegral, consider a Riemann sum

00

Eaf(Yi-09.6(af(Y8-1))(Yi — y+-1)= t

that provides a close approximation to

af(y)9 '- (CI f (y))dy•

By adding more points to the Riemann sum if necessary, we may assumethat the left-hand end point of each interval on which cr f is constant is

32 1. Preliminaries

included among the yi. Then, the Riemann sum is not changed if each yithat is contained in the interior of an interval on which a f is constant, isdeleted. It is now an easy matter to verify that for each of the remain-ing y i there is precisely one element, ui, such that y i = f'(ui) and thata f(f'(u i )) = u i . Thus, we have

ao

Eafoh-og"(af(y._,))(vs-v,_,)i=1

00

Eui-00'(ui-1)(f'(us) - 1 `y(uti -1))

which, by adding more points if necessary, provides a close approximationto

Therefore, we have

f' (f)(t) < (If (y)g.' (af (g))dEl

o00= _ u9..(,a)df.(.n)

t00= _ug.. (u)f• (u)ir + f'(u)g'(u)duf

00< tg * ' (Or (t) + f+(u)g'(u)û. (1.8.15)f

To justify the integration by parts, let A be an arbitrarily large numberand choose ui such that t = u i < u 2 < ... < A. Observe that

Ag ..

(A)f'(A)^

tg .• (t) f■(t) _ E1=1

^

+ E f'(u)[g(u,+1)u1+j g" (u,)uc]5 -1

= ^1 rui+1g" (ui+1) {f ' (?ti+ I ) — f . (%)1

1=1

++ Ef (ua) [

„, . g' (Odd

1- 1 u ,i

Ç Eth+o-scui+otf.(u,i+i) - f'(ui)1i =1

00— ug" (u)df• (u).

t


^

+ E f'(ui)g'(ui)[ui+1 ^ ui]•

i=1

This shows that

A9 '(A)f*(A)

To establish the

a a— tg" (Of' (t) c ug" (u)d f* (u) +

f *(u)g' (u)du.t

opposite inequality, write7

ag' * Nfs (A) — tg" (Or (t) = E uir(ui)[f'(u; +1) - f 'lui)i

+ ^J

r,,

f'(ut+1)lJ*'(u;♦1)u.+1 - g'*(u,)ui]i= 1

J

= Eu, g * (ui) [f * (ui+j) - f'(ui)1;=1^

+ Ef'+') [f g'(r)dr^l

> E uig"(u+)[f'(ui+,) — f`(ui)1i-r

+ E f (ui+ i )gs (Il i+ )[u i+i — ui^•i=1

Now let A 0o to obtain the desired equality. Thus, from (1.8.15), (1.8.14),

and (1.8.10),

0o ea(t) (t) < g"(t) {tf*() +^ (y)dy + f (u)g' (u)du

f • (e) ^

r

°13< tf"(t)g"(t) + f'(u)g*(u)du.f 0

1.8.9. Lemma. Under the hypotheses of Lemma 1.8.8,co

h .. (t) < f *. (u)g" (u)du.f

Proof. We may as well assume the integral on the right is finite and thenconclude

^

lirn uf'*(u)g"(u) = 0. (1.8.16)u^vo

34 1. Preliminaries

By Lemma 1.8.8 and the fact that f` < f", we have

00h"(t) < t f «.(t)g..(t) +f'(u)g'(u)duf

DID< t f .. (t)g .. (t) + f.. (u)g'(u)du. (1.8.17)f

Note that since f' and g' are non-increasing,

â4L r a.(,u) ' ^ (u) — fa* (u)]

and

du ug" (u) = g' (u)

for almost all (in fact, all but countably many) u. Since f" and g" areabsolutely continuous, we may perform integration by parts and employ(1.8.16) and (1.8.17) to obtain

h"(t) < t f..(t)g"(t) + uf'«(u)g..(u)Ir

00+ (f(u) — f'(u))g"(u)du

t^

= (f" (u) — f'(u))g" (u)dut

00f** •• D

i t

We conclude this section by proving some lemmas that provide a com-parison between various Lorentz spaces. We begin with the following thatcompares 17 and L(p,p).

1.8.10. Lemma. If 1 < p < oc and lip + 1/p' = 1, then

Ilfllp < IIfI102,0 <_ p'Ilfllp•

Proof. Since f' < f",

IIfIIp = (f*(^)1pdt = °°(tlip f'(t))p dt < f [t'/Pf»(t)1!0 o /j

= (I1fII(p,p))p.


The second inequality follows immediately from the definition of f''(t) andthe inequality

°° xl/p x n di] 1/P too dx 1/p

[J0

x fo

f(t)dt x

ç p' o rx 1/af(4P x j

which is a consequence of the following lemma with r = p - 1. ❑

The next result is a classical estimate, known as Hardy's inequality, whichgives information related to Jensen's inequality (1.5.12). If f is a non-negative measurable function defined on the positive real numbers, let

sF(x) =--

-f f(t)dt, x > 0.

Jensen's inequality gives an estimate of the pth power of F; Hardy's in-equality gives an estimate of the weighted integral of the pth power of F.

1.8.11. Lemma (Hardy). If 1 0 and f is a non-negativemeasurable function on (0, oo), then with F defined as above,

i co F^x)Pxp -r-ldx < (=

r)cc

f (t)ptF-r-ldtl r a

Proof. By an application of Jensen's inequality (1.5.12) with the measuret (r/p )-1 dt, we obtain

s Pf(t)d = f(t)tl- (r/P)tfT/p)1dt

o

(f? P

< \ xr(1 -1/p) s

( Ef(t)Y)tP_T_1+Tdt. o

Then by Fubirri's theorem,

0o

f (t)dt s-r-ldx0 (f

Z

< (! ) P_' f 00 s 1 - (r/p) If (t)]Ptp-"* - 1 +(r/P) dt dx

P

-

1 f0011 t ptp- r -1 +(r/p)Utoo

x-1-tr/P)dx dt(12)^,

o(}1

I 1F °O ( / jt - r -I dt. D�- tpl ^f(t)t^pr o

36 1. Preliminaries

The following two lemmas provide some comparison between the spacesL(p,q) and L(p, r).

1.8.12. Lemma.

f/**( <(1) 1/9 lIfll(P,Q) ^ e 1 /e Îf I;)x /p .` p xl/P —

Proof.

m(II /II(p,Q)) 4 = [eh' f..(t)^Qdt

0

z2:[ f+*(t)14t(4/P)-1dt0

z^ [ f ** (x)]9t (9/p)' l dt

0

_ ^ [f ** (^)^ 4 xP/4

The first inequality follows by solving for f'*(x) and the second by ob-

serving that ( 2 )P

1/9 < ql/4 < e lks ❑

1.8.13. Lemma (Calderón). If 1 < p < oc and 1 < q < r < co, then

(1/q)—(1/r)II /II(p.r) Ç ( i ) III II(P,4) Ç el fe ll f 11(F.4)•

Proof.00

[f**(y)lrz (r/P )-l dz0

00= [t** (x)1Q [f** (x)jr -4x ( r/P )-1 d.Z

0I r -4

< ^[f ** (X)?^ 1/4 iIf II (p, 9) x(r/P)- l dx

o [ 'tvil 1 /p

(q)("q)-

i

= (II/II(p.q))r-4(li/IIcP,9))q,

(11116.0Y —

and the first inequality follows by taking the r th root of both sides. Thesecond follows by the same reasoning as in the previous lemma. ❑

Exercises 37

Exercises

1.1. Prove that if E C R" is an arbitrary set, then the distance functionto E is Lipschitz with constant 1. That is, if d(x) = d(x, E), thenid(x) — d(y)I < ix — yI for all x, y E R".

1.2. (a) Prove that if E is a set with HQ(E) < oo, then HA(E) = 0 forevery f3 > a.

(b) Prove that any set E C R" has a unique Hausdorff dimension.See Remark 1.4.3.

1.3. Give a proof of Lemma 1.5.1. More generally, prove the following:Let So: [0, oo] —' [0, oo] be a monotonic function which is absolutelycontinuous on every closed interval of finite length. Then, under theconditions of Lemma 1.5.1, prove that

^Sp o 11,0=f(E)co'(t)dt.

R°

1.4. Prove that C"(1-1) is a Banach space with the norm defined in Sec-tion 1.1.

1.5. Let f E Cô (Rn) and T a distribution. Verify the Leibniz formula

D'( fT)_â^

Dg f D^ —ATA<a Q^(a Q)^

where we say ,3 < a provided f3 < at for 1 < i < n.

1.6. Prove that if T is a distribution and cp E Cfl (R"), then T * (P ECô (Rn) and D(T * Sp) = (DT) * ço where D denotes any partialderivative of the first order. This may be accomplished by analyzingdifference quotients and using the fact that rh(DT) = D(rhT).

1.7. Lemma 1.8.13 shows that if 1 < p < oo and 1 < q < r < oo, then

L(p, q) C L(p,q) C L(p, r) C L(p, oo).

Give examples that show the above inclusions are strict.

1.8. As we have noted in Remark 1.4.3, the measure FP does not satisfythe regularity property analogous to (1.4.5). However, it does haveother approximation properties. Prove that if A C R" is an arbitraryset, there exists a G6-set G D A such that

HI (A) = H" ( G ).

38 1. Preliminaries

It can also be shown (although the proof is not easy) that if A is aSuslin set, then

H 7 (A) = sup{H'(K) : K C A, K compact, H'r(K) < co).

See [F4, 2.10.48].

1.9. Prove the statement that leads to (1.8.10), namely, if f E L'(R Th ),then

f '(r)dr = t f'(t) +fa f(s)d9.' (t)

Hint: Consider the graph of f and employ Lemma 1.5.1.

1.10. Another Hausdorff-type measure often used in the literature is Haus-dor f spherical measure, HS. It is defined in the same manner as Ill(see Definition 1.4.1) except that the sets A; are taken as n-balls.Clearly, 11 1'(E) < HS(E) for any set E. Prove that HS(E) = 0whenever H7 (E) = 0.

1.11. Suppose u is a function defined on an open set SZ C R" with theproperty that it is continuous almost everywhere. Prove that u ismeasurable.

1.12. Using only basic information, prove that the class of simple functionsis dense in the Lorentz space L(p, q).

1.13. Let p be a Radon measure on R". As an application of Theorem 1.3.6prove that any open set U C R" is essentially (with respect to p) thedisjoint union of n-balls. That is, prove that there is a sequence ofdisjoint n-balls B ; C U such that

[u _ û niJ =0.

1.14. Let p be a Radon measure on R". Let I be an arbitrary index setand suppose for each a E I, that Ea is an p-measurable set with theproperty that

lim g[Ea n B(x, r)1 — 1r-'0 tt[B(z , r)]

for every x E Ea . Prove that UaEIEQ is p-measurable.

1.15. From Exercise 1.1 we know that the distance function, d, to an arbi-trary set E is Lipschitz with constant 1. Looking ahead to Theorem2.2.1, we then can conclude that d is differentiable alma't everywhere.

1

t-1

Historical Notes 39

Prove that if E is a closed set and d is differentiable at a point x « E,then there exists a unique point «(x) E E nearest x. Also prove that

Dd(x) = x d (x) x()

1.16. (a) If f is a continuous function defined on R", prove that its non-increasing rearrangement f' is also continuous. Thus, continu-ous functions remain invariant under the operation of rearrange-ment.

(b) Now prove that Lipschitz functions also remain invariant underrearrangement. For this it will be necessary to use the Brunn-Minkowski inequality. It states that if E and F are nonernptysubsets of R" , then

IE + Fill" > 1Eil/n + PFI 1/"

whereE+F= {x+y:XEE,yE F}.(c) Looking ahead to Chapter 2, prove that if f E W 1 •P(R"), then

f' E W 149 (R"). Use part (b) and Theorem 2.5.1.(d) Show by an example that C 1 (R") does not remain invariant

under the operation of rearrangement.

uh(x) = h

Prove that uh --e u' in the sense of distributions.

1.18. Let fu,) be a sequence in fP (R") that converges weakly to u inLP(R"), p > 1. That is,

lim u, v dz --►uv dxi—oo R^ R "

for every v E IP' (R"). Prove that D'u, --r D'u in the sense ofdistributions for each multi-index a.

Historical Notes

1.2. The notion of measures leas two fundamental applications: one can beused for estimating the size of sets while the other can be used to defineintegrals. In his 1894 thesis, E. Borel (cf. [BO)) essentially introduced whatis now known as Lebesgue outer measure to estimate the size of sets to assisthis investigation of certain pathological functions. Lebesgue [LEI' used

1.17. Let u E C° (R 1 ). For each h 0, let uh be the function defined by

u(x + h) — u(x)

40 1. Preliminaries

measures as a device to construct his integral. Later, when more generalmeasures were studied, Radon (1913) for example, emphasized measureas a countably additive set function defined on a o-ring of sets whereasCarathéodory (1914) pursued the notion of outer measures defined on allsets.

1.3. The material in this section represents only a very small portion of theliterature devoted to differentiation theory and the related subject of cover-ing theorems. Central to this theory is the celebrated theorem of Lebesgue[LE2J which states that a locally integrable function can be representedby the limit of its integral averages over concentric balls whose radii tendto zero. Theorem 1.3.8 generalizes this result to the situation in whichLebesgue measure is replaced by a Radon measure. This result and thecovering theorems (Theorems 1.3.5 and 1.3.6) which lead to it are due toBesicovitch, [BE1), [BE2]. The proof of Theorem 1.3.5 was communicatedto the author by Robert Hardt. The original version of Theorem 1.3.6 is dueto Vitali [VI) who employed closed cubes and Lebesgue measure. Lebesgue(LE2) observed that the result is still valid if cubes are replaced by gen-eral sets that are "regular" when compared to cubes. A sequence of sets{Ek } is called regular at a point zo if zo E flk 1Ek, diam(Ek) —+ 0 andlim infk_.Q p(Ek) > 0 where p(Ek ) is defined as the infimum of the numbersICI/IEkI with G ranging over all cubes containing Ek. In particular, one isallowed to consider coverings by nested cubes or balls that are not neces-sarily concentric. However, in the case when Lebesgue measure is replacedby a Radon measure, Theorem 1.3.6 no longer remains valid if the balls inthe covering are allowed to become too non-concentric. At about the timethat Besicovitch made his contributions, A.P. Morse developed a theorywhich allowed coverings by a general class of sets rather than by concentricclosed balls. The following typifies the results obtained by Morse [MSE2):Let A C R" be a bounded set. Suppose for each z E A there is a set H(x)satisfying the following two properties: (1) there exist M > 0 independentof r and r(x) > 0 such that

B(x, r(x)) C H(x) C B(x, Mr(x));

(ii) H(x) contains the convex hull of the set {y} U B(x, r(z)) whenevery E H(x). Then a conclusion similar to that in Theorem 1.3.5 holds.

Another useful covering theorem due to Whitney [WHI states than anopen set in R" can be covered by non-overlapping cubes that becomesmaller as they approach the boundary. Theorem 1.3.5 is a similar resultwhere balls are used instead of cubes and where the requirement of disjoint-ness is replaced by an estimate of the amount of overlap. This treatmentis found in [F4, Section 3.1).

Among the many results concerning differentiation with respect to irreg-ular families is the following interesting theorem proved in [JMZ]: Suppose

Historical Notes 41

u is a measurable function defined on R" such that

f 'ul(1 +log+ Pul)n -l dx < oo.

Then, for almost every x E Rn,

lim III `1 Iu(y) - u(x)Idy = 0

where the limit is taken over all bounded open intervals I containing thepoint x. This result is false if u is assumed only to be integrable. Such ir-regular intervals are useful in applications concerning parabolic differentialequations, where it is natural to consider intervals of the form C x 10, r2J,where C is an (n - 1)-cube of side-length r.

For further information pertaining to differentiation and coverings, thereader may consult [DG], [F4, Section 2.8].

1.4. Carathéodory [CAY] was the first to introduce "Hausdorff" measure inhis work on the general theory of outer measure. He only developed linearmeasure in Rn although he indicated how k-dimensional measure could bedefined for integer values of k. k-dimensional measure for general positivevalues of k was introduced by Hausdorff [HAU] who illustrated the useof these measures by showing that the Cantor ternary set has fractionaldimension log 2/ log 3.

1.7. There are various ways of presenting the theory of distributions, butthe method employed in this section is the one that reflects the originaltheory of Schwartz [SCI-1] which is based on the duality of topological vectorspaces. The reader may wish to consult the monumental work of Gelfandand his collaborators which contains a wealth of material on "generalizedfunctions" [GE1], [GE2], [GE3], [GE4], [GEM.

1.8. Fundamental to the notion of Lorentz spaces is the classical conceptof the non-increasing rearrangement of a function which, in turn, is basedupon a notion of symmetrization which transforms a given solid in R3

into a ball with the same volume. There are a variety of symmetrizationprocedures including the one introduced by J. Steiner [ST] in 1836 whichchanges a solid into one with the same volume and at least one plane ofsymmetry. The reader may consult the works by Pôlya and Szegô [PS] orBurago and Zalgaller [BUZ] for excellent accounts of isoperimetric inequal-ities and their connection with symmetrization techniques. In 1950 G.G.Lorentz [LO1], 1LO21, first discussed the spaces that are now denoted byL(p, 1) and L(p, oo). Papers by Hunt [HU] and O'Neil [0] present interest-ing developments of Lorentz spaces. Much of this section is based on thework of O'Neil and the main results of this section, Lemmas 1.8.7-1.8.9,were first proved in [O]. The reader may consult LCA21, [CA31, 1LP], [PE]for further developments in this area.

2

Sobolev Spaces and TheirBasic PropertiesThis chapter is concerned with the fundamental properties of Sobolevspaces including the Sobolev inequality and its associated imbedding the-orems. The basic Sobolev inequality is proved in two ways, one of whichemploys the co-area formula (Section 2.7) to obtain the best constant in theinequality. This method relates the Sobolev inequality to the isoperinietricinequality.

The point-wise behavior of Sobolev functions will be discussed in Chap-ters 3 and 4 and this will entail a method of defining Sobolev functionson large sets, sets larger than the complement of sets of Lebesgue measurezero. It turns out that the appropriate null sets for this purpose are de-scribed in terms of sets of Bessel capacity zero. This capacity is introducedand developed in Section 2.6 but only to the extent needed for the analysisin Chapters 3 and 4. The theory of capacity is extensive and there is a vastliterature that relates Bessel capacity to non-linear potential theory. It isbeyond the scope of this book to give a thorough treatment of this topic.

One of the interesting aspects of Sobolev theory is the behavior of theSobolev inequality in the case of critical indices. In order to gain a betterappreciation of this phenomena, we will include a treatment in the contextof Lorentz spaces.

2.1 Weak DerivativesLet u E LL(n). For a given multi-index a, a function y E LL(11) is calledthe a th weak derivative of u if

Spvdx = (-1 )i' 1 uD° Spdx (2.1.1)f

for all W E (SZ). y is also referred to as the generalized derivative of uand we write v = D°u. Clearly, DQ u is uniquely determined up to setsof Lebesgue measure zero. We say that the ath weak derivative of u is ameasure if there exists a regular Borel (signed) measure g on SZ such that

i ^pdµ = (-1)i' 1 uD°`Spdx (2.1.2)n

2.1. Weak Derivatives 43

for all ' E Co (1/). In most applications, lal = 1 arid then we speak of uwhose partial derivatives are measures.

2.1.1. Definition. For p > 1 and k a non-negative integer, we define theSobolev space

Wk'p(1) = L"(1-Z) n {u : Dau E Lp(1), lal < k]. (2.1.3)

The space Wk•p(f) is equipped with a norm

Ilullk.p;sl = f E IDa^;lp^1 a j <k

(2.1.4)

which is clearly equivalent to

E IID°ullp,n•IQ!<k

(2.1.5)

It is an easy matter to verify that Wk ip(1) is a Banach space. The spaceWô'p(fl) is defined as the closure of Co (11) relative to the norm (2.1.4).We also introduce the space BV(0) of integrable functions whose partialderivatives are (signed measures) with finite variation; thus,

BV(S1) = L l (1Z) n f u : D'u is a measure, jD°ul(11) < oo, lal = 11.(2.1.6)

A norm on BV(11) is defined by

Ilullev(n) = Ilulll;n + E IDauI(n).lai.1

(2.1.7)

2.1.2. Remark. Observe that if u E Wk'p(11) t.) BV (SI), then u is deter-mined only up to a set of Lebesgue measure zero. We agree to call thesefunctions u continuous, bounded, etc. if there is a function û such thatTA- = u a.e. and t has these properties.

We will show that elements in Wk.P(SZ) have representatives that permitus to regard them as generalizations of absolutely continuous functionson R'. First, we prove an important result concerning the convergence ofregularizers of Sobolev functions.

2.1.3. Lemma. Suppose u E W k,P (SZ ), p > 1. Then the regularizers of u(see Section 1.6), u e , have the property that

li ô NE - u I'Ik.p ;n , = 0

44 2. Sobolev Spaces and Their Basic Properties

whenever f' CC 11. In case S2 = Rn, then lirn._.o Igue - uP[k, F = O.

Proof. Since 1' is a bounded domain, there exists ea > 0 such that c o <dist(ir, 8S1). For e < c o , differentiate under the integral sign and refer to(2.1.1) to obtain for x E re and [a[ < k,

Dace (x) = e-nf Dx ço u(y)dy

= (-1) 1' 1 a-n n

Dÿ (P a(L=1) u(y)dy

= e-n D°ù(y)dyn

_ (Da u)E (x)

for each x E f2'. The result now follows from Theorem 1.6.1(iii). C^

Since the definition of a Sobolev function requires that its distributionalderivatives belong to L", it is natural to inquire whether the function pos-sesses any classical differentiability properties. To this end, we begin byshowing that its partial derivatives exist almost everywhere. That is, inkeeping with Remark 2.1.2, we will show that there is a function u suchthat Ti = u a.e. and that the partial derivatives of Ft exist almost every-where. However, the result does not give any information concerning themost useful concept of total differential, the linear approximation of thedifference quotient. This topic will be pursued in Chapter 1

2.1.4. Theorem. Suppose u E LP(S2). Then u E W 1 "( )) , p > 1, if andonly if u has a representative i that is absolutely continuous on almostall line segments in I2 parallel to the coordinate axes and whose (classical)partial derivatives belong to LP(S2).

Proof. First, suppose u E W 1, P(12). Consider a rectangular cell in 1

R = [a1, b1] x ... x ian , b^]

all of whose side lengths are rational. We know from Lemma 2.1.3 that theregularizers of u converge to u in the W 1 () norm. Thus, writing x E Ras x = (z, x i ) where x E R" -1 and ri E [ai , bi], 1 < i < n, it follows fromFubini's Theorem that there is a sequence {Ell -' 0 such that

btTim Iuk(x,xi) — utx , xi)I ' + IDuk(x,zi) — Du(x , zi)I edx1 = ü

k--•oo a

for almost all x. Here, we denote li ck = uk. Since uk is smooth, for eachsuch x and for every q > 0, there is M > 0 such that for b E [a i , bi ],

uk (x b) y uk (x,t

b ta+)1 <_ ID t1k(2, x i )Id^ if


b, M. If {uk(i, a,)} converges as k oo, (which may be assumedwithout loss of generality), this shows that the sequence {uk} is uniformlybounded on [a ; , bd. Moreover, as a function of xi, the uk are absolutelycontinuous, uniformly with respect to k, because the Li convergence ofDuk to Du implies that for each e > 0, there is a 45 > 0 such thatfE < e whenever 11 1 (E) < b for all positive integers k.Thus, by the Arzelà-Ascoli theorem, {uk} converges uniformly on [a 1 , b11to an absolutely continuous function that agrees almost everywhere withu. This shows that u has the desired representative on R. The general casefollows from the familiar diagonalization process.

Now suppose that u has such a representative û. Then iacp also possessesthe absolute continuity properties of û , whenever cp E Co (S2). Thus, for1 < 1 < n, it follows that

f iD1 codx = — f D,rtSp dz

on almost every line segment in fl whose end-points belong to Rn -- spt (Pand is parallel to the i th coordinate axis. Fubini's Theorem thus impliesthat the weak derivative Diu has Dui as a representative. D

2.1.5. Remark. Theorem 2.1.4 can be stated in the following way. If u ELP(12), then u E W i 'P(fI) if and only if u has a representative i such that

E W 1 'P(A) for almost all hne segments A in S2 parallel to the coordinateaxes and /DUI E LP(S1). For an equivalent statement, an application ofFubini's Theorem allows us to replace almost all line segments A by almostall k-dimensional planes Ak in 1 that are parallel to the coordinate k-planes.

It is interesting to note that the proof of Theorem 2.1.4 reveals thatthe regularizers of u converge everywhere on almost all lines parallel to thecoordinate axes. Hu were not an element of W 14 (t1), but merely an elementof L' (e) , Fubini's theorem would imply that the convergence occurs onlyH'-a.e. on almost all lines. Thus, the assumption u E W t.P(1l) impliesthat the regularizers converge on a relatively large set of points. This is aninteresting facet of Sobolev functions that will be pursued later in Chapter3.

Recall that if u E L1'(R"), then IIu(x + h) — u(x)I6 —► 0 as h O. Asimilar result provides a very useful characterization of Wl.a(R").

2.1.6. Theorem. Let 1 < p < oo. Then u E W I 'P(R") if and only ifU E L1'(R") and

fj u(x+h)—u(x) I "e/P dx

= + h) — u.(z)1In


remains bounded for all h E R".

Proof. First assume u E Co ( R" ). Then

( ) — ( ) _ Ihlu x + h u x 1 Du ^+ t • ^ dt,Ih4

9-11o I I Ihl

so by Jensen's inequality (1.5.12),

u,(x + h) — u(x) ^ 'hl 1Duh - khl o

Therefore,

1 Ihl h pIIu { x + h) — u(x)IT^ Ihlp IhI o R

Du x + t Ih I dxdi,n

orIIu(x + h) — u(x)IIp <— Ihi IIDuII p .

By Lemma 2.1.3, this holds whenever u E W 1,p(R").Conversely, if e; is the ill unit basis vector, then the sequence

u(x + e, /k) — u(x) 1/k

is bounded in LP(R"). Hence, by Theorem 1.5.2, there exists a subsequence(which will be denoted by the full sequence) and u, E L"(R") such that

u(x + e, /k) — u(x) 1/k

weakly in L (R" ). Thus, for (,o E gv ,

[ u (x+ ei/k)_u(x) ] ()duopdx= lirn 1k

rk-.00f " /

P

+ t —h dt.II

u;

= GmR u(x)

kôo "

f ço(x — ei /k) — cP(x) j 1/k dx

-- — uD ; cp dx.R"

This shows thatD i u =u;

in the sense of distributions. Hence, u E W 1 "p(R"). o

2.1.7. Definition. For a measurable function u: SZ -F R 1 , let

u+ = max{u, 0}, u - = min{u, 0}.


2.1.8. Corollary. Let u E W""p(S2), p > 1. Then u+, u - E W l 'p(S2) and

Du+ = Du if u >00 if u<0

_ f 4 if u Du- Du if u < 0.

Proof. Because u has a representative that has the absolute continuityproperties stated in Theorem 2.1.4, it follows immediately that v.+, u - EW 1,p(11). The second part of the theorem is reduced to the observationthat if f is a function of one variable such that f' exists a.e., then (f + )' _

f' ' xuf>o)• D

2.1.9. Corollary. If SI is connected, u E W' (S2), p > 1, and Du = 0 a.e.on Sl, then u is constant on S2.

Proof. Appealing to Theorem 2.1.4, we see that u has a representativethat assumes a constant value on almost all line segments in Sl parallel tothe coordinate axes. ❑

2.1.10. Remark. The corollary states that elements of W1 , F(Sl) remaininvariant under the operation of truncation. One of the interesting aspectsof the theory is that this, in general, is no longer true for the space Wk .P(0).Motivated by the observation that u+ = H o u where H is defined by

H(t)= t^ oOt —< 0

we consider the composition H o u where H is a smooth function. It wasshown in [MA21 and [MA3] that it is possible to smoothly truncate non-negative functions in W 2 ' 1'. That is, if H E C°o(11. 1 ) and

sup lt2-1 H (' ) (t)l < M < oo

for j = 1, 2, then there exists C = C(p, M) such that for any non-negativev E Cp (R")

IlDaH(u)Ilp < ClID2vIl,for 1 < p < n/2 and any multi-index a with lai = 2. Here D2 v denotes thevector whose components consist of all second derivatives of u. However,it is surprising to find that this is not true for all spaces Wk ,p. Indeed, itwas established in [DA1] that if 1 1 for [ti < 1, then there exists afunction u E Wk iP(R") n C°O(R?) such that H(u) « W k ip(R"). The mostgeneral result available in the positive direction is stated in terms of Rieszpotentials, IQ * f (see Section 2.6), where f is a non-negative function in


L'. The following result is due to Dahlberg [DA2]. Let 0 < a < n and1 o

for j = 0,1, ... , a*, where a' is the smallest integer > a. If f E LP(R")and f > 0, then there exists g E IP(R") such that

H(Ic,*g)=Ia *g a.e.

and IIgIIp < CII f lip where C = C(a, p, n, M). The case of integral a wastreated in [AD4] and in this situation the result can be formulated as

I D"[H(Ia *f)1IIp CIIfIIpfor any multi-index 7 with I -II = k.

To continue our investigation of the calculus of Sobolev functions, we con-sider the problem of composition of a suitable function with u E W l .p(SZ).Before doing so, we remind the reader of the analogous problem in RealVariable theory. In general, if f and g are both absolutely continuous func-tions, then the composition, f og, need not be absolutely continuous. Recallthat a function, f, is absolutely continuous if and only if it is continuous, ofbounded variation, and has the property that If(E)1 = 0 whenever IEI = O.Thus, the consideration that prevents f o g from being absolutely continu-ous is that fog need not be of bounded variation. A result of Valle Poussin[PO] states that f o g is absolutely continuous if and only if f' o g g' isintegrable. An analogous result is valid in the context of Sobolev theory, cf.[MM1], IMM2), but we will consider only the case when the outer functionis Lipschitz.

2.1.11. Theorem. Let f : R 1 -, R` be a Lipschitz function and u EW 1,P(0), p > 1. If f o u E IP(1 ), then f o u E Wl'p(1) and for almost all

E SZ,D(f o u)(x) = flu(x)]. Du(r).

Proof. By Theorem 2,1,4, we may assume that u is absolutely continuouson almost all line segments in ft. Select a coordinate direction, say the1 th , and consider the partial derivative operator, D i . On almost all linesegments, A, in ft parallel to the it" coordinate axis, fou is clearly absolutelycontinuous because f is Lipschitz. Moreover,

Di( f o u)(x) = f'[u(x)) . Diu(x) (2.1.8)

holds at all x E A such that D,u(r) and flu(r)1 both exist. Note that ifDiu(x) = 0, then D i (f o u)(x) = 0 because

I f [u(x + heh) - f [u(x)]1 <M1' (r + h h) - u(x)111 II

2.2. Change of Variables for Sobolev Functions 49

where M is the Lipschitz constant of f and e, is the i th coordinate vector.Thus, letting N = A n {x : D iu(x) = 0), we have that (2.1.8) holds on N.Now let

P= (A- N) n{x:au(x) exists and D; te(x)# 0}

and note that P U N occupies H 1 -almost all of A. From classical consid-erations, we have that if S C P and H 1 [u(S)1 = 0, then H 1 (S) = O. Inparticular, if we let E = {y : f'(y) fails toexist}, then H 1 [u -1 (E)nPJ = 0.Since (2.1.8) holds if x E A - u - ' (E) n P and Diu(x) exists, it followstherefore that (2.1.8) holds at H 1 -almost all points of A. At all such x, wemay conclude that

AU a u)(x)I" <_ MPI D,u(x)IF. (2.1.9)

Once it is known that the set of x E 11 for which (2.1.8) holds is a mea-surable set, we may apply Fubini's Theorem to conclude that f o u sat-isfies the hypotheses of Theorem 2.1.4. This is a consequence of the factthat the functions on both sides of (2.1.8) are measurable. In particular,f' a u is measurable because f' agrees with one of its Borel measurableDini derivates almost everywhere. O

2.2 Change of Variables for Sobolev FunctionsIn addition to the basic facts considered in the previous section, it is alsouseful to know what effect a change of variables has on a Sobolev function.For this purpose, we consider a bi-Lipschitzian map

T: .

That is, for some constant M, we assume that both T and T` 1 satisfy,

IT(x) - T(y)I < Mix

IT -1 (x') - T-1 (11)1 < Mix' - y'1

for all x, y E Si,

for all x i , y' E fr. (2.2.1)

In order to proceed, we will need an important result of Rademacher whichstates that a Lipschitz map T: R" -s Rtm is differentiable at almost allpoints in R". That is, there is a set E C R" with IEi = 0 such that foreach x E Rn - E, there is a linear map dT(x): R" Mil (the differentialof T at x) with the property that

o 171 (x(x + y) - T(x) - dT(x, y)I lo

= 0. {2.2.2}^ Iui


In order to establish (2.2.2) it will he sufficient to prove the followingresult.

2.2.1. Theorem. If f : Rn — R 1 is Lipschitz, then for almost all x E R" ,

lim f(x -Fy) — f(x)—Df(x)'31 _ü.

yô

Proof. For ty E R" with H = I, and x E W', let 7(t) = f (x + tv). Since fis Lipschitz, 7 is differentiable for almost all t.

Let df (x, v) denote the directional derivative of f at x. Thus, df (x, y) =7'(0) whenever 7'(0) exists. Let

N„ = R" n {x : df (x, v) fails to exist}

Note that

N„ = x : lim sup f (x + tu) — f (x) > lirn inf f (x + iv) — f (x)}1-00

t tô t ^

and is therefore a Borel measurable set. However, for each line A whosedirection is V, we have HI (N„ fl A) = 0, because f is Lipschitz on A.Therefore, by Fubini's theorem, IN„' = O. Note that on each line A parallelto VV

df (x, v)Sp(x)dx = — f (x)dSp(x, v)dxf

for cp E Co (R"). Because Lebesgue measure remains invariant under or-thogonal transformations, it follows by Fubini's Theorem that

R df (x, v)Sp(x)dx = — f (x)dcp(x, v)dxR"

= - f (x)Dcp(x) - v dxR "

= —R

f (x)Dj çp(x) • v, dxj=1 "

-ER "

Djf(x)cP(x)•VjdxJ=1

ô(x)D f (x) - v dx.f"

lyf

1.

Because this is valid for all cp E Ca (Rn), we have that

df (x, v) = D f (x) • v , a.e. x E R11 . (2.2.3)

2.2. Change of Variables for Sobolev Functions 51

Now let u1, u2,... be a countable dense subset of Sn -1 and observe thatthere is a set E with 1E1 = 0 such that

df (x, vk) = D f (x) . vk (2.2.4)

for all x E R" — E, k --- 1, 2,....We will now show that our result holds at all points of R" — E. For this

purpose, let x E R" — E, lui = 1, t > 0 and consider the difference quotient

Q(x, v, t) - 1(x + hi) — f (x) D (x) • v.t

For v, v' E Sn -1 and t > 0 note that

IQ(z, v, t) — Q(x, v ' , t)I w If (x + tu) - f (x + tv') + (v - v') • D f (x)I

< Mlv — v'I + 1v — v'l . IDf(x)I Ç M(n + 1)1v — v i l (2.2.5)

where M is the Lipschitz constant of f. Since the sequence {v i } is dense inSn -1 , there exists an integer K such that

Iv — vkl < 2(n + 1)M for some k E {1, 2, .. ., K} (2.2.6)

whenever y E Si -1 . For x o E R" - E, we have from (2.2.4) the existenceof 6 > 0 such that

ÎQ(xo , vk, t)1 < 2 for O < t < 6, k E {1, 2, . .. , K}. (2.2.7)

SinceIQ(xo , v, t)l <_ I Q(xo, vk, t)I + IQ(xo, v, t) -- Q(xo, vk, t)I

for k E {1, 2, . . . , K}, it follows from (2.2.7), (2.2.5), and (2.2.6) that

IQ(xo, v, t)l <2 + 2 =e e

whenever Iv' = 1 and 0 < t < 6.

Recall that if L: R" , R" is a linear mapping and E C R" a measurableset, then

IL(E)i = I det Li 1E1.

It is not difficult to extend this result to more general transformations.Indeed, if T: R" -, R" is Lipschitz, we now know from Theorem 2.2.1that T has a total differential almost everywhere. Moreover, if T is alsounivalent, one can show that

t

11"[T(E)1 = L JT da for every measurable set E, (2.2.8)


where JT is the Jacobian of T. From this follows the general transformationformula

f o T J2` dx = f dx (2.2.9)T(E)

whenever f is a measurable function. We refer the reader to 1F4; 3.2.31 fora proof.

We are now in a position to discuss a bi-Lipschitzian change of coordi-nates for Sobolev functions.

2.2.2. Theorem. Let T: R" --+ R" be a bi-Lipachatzian mapping as in(2.2.1). if u E p> 1, then y = u o T E W l,p(V), V a' T -1 (f/),and

Du[T(x)] • dT (x, = Dv (x) • (2.2.10)

for a.e x E SZ and for all E R".

Proof. Let ue be a sequence of regularizers for u, defined on SZ' CC SI, (seeSection 1.6). Then ve uE o T is Lipschitz on V' = T -1 (SZ') and becausevE is differentiable almost everywhere (Theorem 2.2.1), it follows that

D i v,(x) = E D,u£[T(x))D, (x) (2.2.11)j=1

for a.e. s E V'. Here we have used the notation T = (T 1 , T2 , ... ,T") wherethe T3 are the coordinate functions of T. They too are Lipschitz. (2.2.11)holds at all points x at which the right side is meaningful, i.e., at all pointsat which T is differentiable. If M denotes the Lipschitz constant of T, wehave from (2.2.11) that

CDve (x)1 < n2 MIDuc [T(x)11 for a.e. z E V'. (2.2.12)

In view of the fact that

M —n < JT(x) < M" for a.e. x E Rn ,(2.2.12) implies that there exists a constant C = C(n, M) such that

I DvL (x)IP < CI Due [T(x)]1" . JT(x),

and thereforejDvfrdx < Cf

V' ^from (2.2.9). A quick review of the above analysis shows that in fact, wehave

I

^Dv^ - DvE ^ (Fdx < C ^Du^ - DuE^ ^pds. (2.2.13)v! n

2.3. Approximation of Sobolev Functions by Smooth Functions 53

Also,

Ive – ve 'Prix <f2

lue – ue , IPdx. (2.2.14)v'

From 2.2.11 we see that the regularizers tee converge to u in the norm ofW""F(fl') whenever 11' CC n. Thus, (2.2.13) and (2.2.14) imply that {ye }is a Cauchy sequence in W L PP (st' ), and thus converges to some elementv E W 1 4P(V') with

IIv111.P;V <_ cllulli,p,n < Îlâli^;n. (2.2.15)

Since uE (x) — u(x) for a.e. x E SI, it is clear that y is defined on V withv = u o T. Moreover, v E W `.P(V') whenever V' CC V and (2.2.15) showsthat, in fact, y E W" P(V). Finally, observe that (2.2.10) holds by lettinge—, 0in(2.2.11). O

2.3 Approximation of Sobolev Functions bySmooth Functions

From Theorem 1.6.1, we see that for each u E Wk 'P(11), there is a sequenceof Cfl (S1) functions, fu e l, such that ur u in Wk44 (SZ 1 ) for SZ' CC OE Thepurpose of the next important result is to show that a similar approximationexists on all of SZ and not merely on compact subsets of SZ.

We first require a standard result which concerns the existence of a C°°partition of unity subordinate to an open cover.

2.3.1. Lemma. Let E C Rn and let ÿ be a collection of open sets U suchthat E C {UU : U E G). Then, there exists a family F of non-negativefunctions f E Ca (Rn) such that 0 < f < 1 and

{i) for each f e F, there exists U e Ç such that spt f C U,

(ii) if K C E is compact, then spt f n K 0 for only finitely manyf E .^,

(iii)EfEF f (x) = 1 for each r E E.

Proof. Suppose first that E is compact, so that there exists a positiveinteger N such that E C U; 1 U; , U1 E Ç. Clearly, there exist compact setsE; C U, such that E C UN 1 E; . By regularizing XE, , the characteristicfunction of Ei, there exists y, E Co (U,) such that g, > 0 on E1. Let g =r+N2..1=1 g, and note that g E C°° (R") and that g > 0 on some neighborhood ofE. Consequently, it is not difficult to construct a function h E C°°(R") suchthat h > 0 everywhere and that h = g on E. Now let .7- = { f; : f; = g;/h,1 < _ < N) to obtain the desired result in case E is compact.


If E is open, let

E, = E n B(0,:) n x : dist(x, 0E) > .

Thus, Ei is compact and E = U=_ 1 E, . Let ÿi be the collection of all opensets of the form

U n {int Ei+1 - E,_2}

where U E Ç. (We take ED = E_, = 0). The elements of G, provide anopen cover for E, - int E,_ 1 and therefore possess a partition of unitywith finitely many elements. Let

00

8(x) = E s(x)i =l gEf,

and observe that only finitely many positive terms are represented and thats(x) > 0 for x E E. A partition of unity for the open set E is obtained bydefining _ f . f (x)= 0..1 for some g E .F, if ,}

If E C Rn is arbitrary, then any partition of unity for the open set{UU : U E G} provides one for E. ❑

Clearly, the set

S =Ck(Sa)n {u. ^lull^ ,p.n <oo}

is contained in WkiP(f2) and therefore, since Wk.P(12) is complete, S CW''P(1l). The next result shows that S = Wk.F(û).

2.3.2. Theorem. The space

Ilü llk,p;^ < oc}

is dense in W k .P(SZ).

Proof. Let S2, be subdornains of Sl such that f2, CC f2,+1 and u;_°_ 1 SZ, = SZ.Let F be a partition of unity of fl subordinate to the covering {52,+1 -SZ,- 1 },i = 0,1, ..., where no and SZ- r are taken as the null set. Thus, if we let f,denote the sum of the finitely many f E .7. with spt f C Sal+1 - f2,_ 1 , thenfi E Cp (fli+1 - ,li-1) and

00E f 1 on a (2.3.1)i^l

f(x) = 0 if x E.

2.4. Sobolev Inequalities 55

Choose e > 0. For u E Wk iP(S2), there exists e i > 0 such that

spt((Jiu)E,) C fi+1 - ^^-1 (2.3.2)

II(fiu }E, — ,îullkt2 < e2".With v; _ (fiu) e , (2.3.2) implies that only a finite number of the v i canfail to vanish on any given fi' CC 0, and therefore v - a°'1 vi is definedand belongs to C°° (0). For z E Di, we have

u(x) = E fl (x)u(x),; -1

Y(T) = E(fiu),, (x) by (2.3.2)j=1

and consequently,

^

Ilu - vllk,P;n, < E II(f;u)f, - f,ullk,p;‘, < E .

i=1

The conclusion follows from the Monotone Convergence theorem. ❑

The approximating space C°'°(SI) n (u : ^lul^ kn < ool admits functionsthat are not smooth across the boundary of SZ and therefore it is natural toask whether it is possible to approximate functions in Wk ,P(SZ) by a nicerspace, say

C°°(a) n {u : (IuIIk.P;st < oo}. (2.3.3)

In general, this is easily seen to be false by considering the domain n definedas an n-ball with its equatorial (n-1)-plane deleted. The function u definedby u - 1 on the top half-ball and u —I on the bottom half-ball is clearlyan element of WkiP(SZ) that cannot be closely approximated by an elementin (2.3.3). The difficulty here is that the domain lies on both sides of partof its boundary. If the domain SZ possesses the segment property, it hasbeen shown in EAR2, Theorem 3.1$] that the space (2.3.3) is then dense inW k,P(ft). A domain SZ has the segment property if for each x E OP, there isan r > 0 and a vector vs E Rn such that if y E 5In$(x, r), then y+tvs E SZfor all 0 < t < 1.

2.4 Sobolev Inequalities

One of the main objectives of this monograph is to investigate the manyinequalities that allow the L"-norm of a function to be estimated by thenorm of its partial derivatives. In this section the Sobolev inequality, which


is of fundamental importance, will be established for functions in the spaceW "(1). We will return to the topic of Sobolev-type inequalities in Chapter4.

2.4.1. Theorem. Let SZ C Rn, n > 1, be an open domain. There is aconstant C = C(n,p) such that if n > p, p > 1, and u E WW'p(1/), then

IlUllnp/(n-p);fl CÎDuI^P,^,.

If p > n and SZ bounded, then u E C(SZ) and

sup Iul < 1/PIiDullp.n.

Proof. First assume that u E Cfl (SZ) and that p = 1. Clearly, for each i,1 < i < n, z,

lu(z)I-lDiu(xi, . • - , t, • ,xn )Idt00

where t occupies the ith component of the vector in the integrand. Therefore1/n-1

+ 00

lu(x )ln/n -1dz1

0o i/(n-1)

< IDl utt , z2 7 . . . , z„ ) Idt(f

00 n 1/(n-])

I Diu#^ i(f:vadx1

- oo ^-2

00 1/(ri-1)< ID1i1(i,x2,...Zn)IdE

- oon o0 00 1/(n -1)^ ID,uldîdzl . (2.4.2)i=2 -oo - 00

Continuing this procedure and thus integrating (2.4.1) successively withrespect to each variable, we obtain

If this inequality is integrated with respéct to the first variable, z 1 , andthen Holder's inequality is applied, we obtain

lu(x)ln/n-1 < IDiuldzi . (2.4.1)n o0

i =1 foe

R"

n

lu(x )!n /(n --, )dx <

i=1 R"

t/(n -i)

I D, ujdx


and therefore, using the fact that the geometric mean is dominated by thearithmetic mean,

we have

n(rial ) l i n < —

n E aJ,J= 1

a? ?0,

)1/nalulln/("-1) <

(L"

IDiu l dxi_1 E lD,uldx

R* i=i

5. ÎlDulll.n(2.4.3)

This establishes the result in case p = 1. The result in full generality canbe obtained from (2.4.3) by replacing lui by powers of Ili!. Thus, if q > 1,

ii lug l IIn/(n-1) < fID(lur)lidxn

L.

< q luI¢—'IDuldxR^

q■/71' ll l ur^ 1 l6'11Dut!p,

by Holder's inequality. Now let q = (n - 1)p/(n - p) to obtain the desiredresult for the case 1 < p < n and u E C4 (1l). Now assume u E WW'p(a)and [et (u,} be a sequence of functions in C'°(1) converging to u stronglyin WW'' (iz). Then, with p' = np/(n - p), an application of the inequalityto ut - u3 yields

liui - uj lip. Cil !Ai - 741 111,p-Thus, u, u in L" (fl) and the desired result follows. This completes theproof in case 1 n and SI bounded, let {u,} be a sequence such that u, ECo (11) and u i — u W s "(11). The proof is thus reduced to the case whenu E C7(0). Now select x E R" and because u has compact support, notethat

Iu(x)I <a IDu(r)ldII 1 (r) (2A.4)=

where Ax is any ray whose end-point is x. Let S" -1 (x) denote the (n - 1)-sphere of radius 1 centered at r and denote by A= (B) the ray with end-pointx that passes through B, where B E Sn—L (x). By integrating (2.4.4) overS" - '(x) we obtain

lu(7)IdH"-1(e) <f ~ - (z) 5"-1(x)

f IDu(^,)Id^l (r)dHn -s(9)= (e)


= lDR (1)l rn-1 dH 1 ( r )dHn-1(e )

fi -1 W a^(8) r_ IDu(^)I1 dy

Rn lx yl

where r = Ix -- y1. Thus, for any x E Rn,

1/pw(n - 1 )1/445- IIDull') Ix - 1/1 (1-n)p' dy , (2.4.6)(f

to

where w(n - 1) = H'9Sn -1 ]. We estimate the potential on the right sideof (2.4.6) in the following way. Let B(x, R) be the ball such that 18(x, R)l =Ispt ul. Observe that for each y E spt u - B(x, R) and z E B(x, R) - spt u,we have

Ix — yl(1-11)p' < IZ — Z

I(1-71)F'

and because Isptu - B(x, R)I = IB(x, R) - spt ul, it therefore follows that

I x - y l(1-n)p' dy < I' - Y1 (1-n)p 'dy.apt u- E1(z,R) f (r,R)—spt u

Consequently,

However,

Ix — yl ( t -n)p 'dy <• ix — yl(1—n)e'dy .apt ts f (z,R)

1/p'

Ix — 01—n)P1d1J = (

7-1

Cl(n )R7) 11p'

B(x,R)

(2.4.7)

(2.4.8)

where 'y = (1 - n)p' + n and cx(n) is the volume of the unit n-ball. Buta(n)Rn = lspt ul and therefore

(a(n)R7) 1/p' = a(n - ')Inlsptul'/' -11p . (2.4.9)

The second inequality of the theorem follows from (2.4.9), (2.4.8), and(2.4.6). To show that u E C(SZ) when p > n, let (u,} E CQ (Q) be asequence converging to u in W"(). Apply the second inequality of thetheorem to the difference u = - ui and obtain that lui} is fundamental inthe sup norm on.. ❑

(2.4.5)

The first part of Theorem 2.4.1 states that the LP . norm of u can bebounded by IIuIl1,p , the Sobolev norm of u, where p• = np/(n - p). It is


possible to bound a higher LP norm of u by utilizing higher order deriva-tives of u as shown in the next theorem. Observe that the proof is slightlydifferent from that of Theorem 2.4.1 in case k = 1, p > n.

2.4.2. Theorem. Let S2 C R" be an open set. There is a constant CC(n, k,p) such that if kp < n, p > 1, and u E Wô1p(1l), then

Hub' CIIuIIk,p;i , where p* = np/(n — kp). (2.4.10)

If kp > n, then u E CM) and

sup Iul < CIKI''^p `k— ^

E (diam K)) 1a1 c } IlDaullp;K1Ql=o

—1+ (diam(K)) k

(

1k 1 )1 k

— ^ IlD k ullp,K p

where K = spt u and C = C(k,p,n).

(2.4.11)

Proof. When kp < n, the proof proceeds by induction on k. Observe thatTheorem 2.4.1 establishes the case k = 1.

Now assume for every y E Wô '3) (SZ) that

IIUII¢k-^ <- Cll^llkW t^p (2.4.12)

whereqk—Y = np/(n — kp + p)•

An application of (2.4.12) to y = Diu, 1 < j Ç n, yields

IID,u11Qt -^ <— CIID;ullk—i,p < Îlull^ , p - (2.4.13)

However, (2.4.12) holds with y replaced by u and this, combined with(2.4.13), implies

1111 11 1,9k _ 1 <- Cllu l lk,p. (2.4.14)

Since kp < n, we have qk_ i < n and therefore, Theorem 2.4.1 implies

Huh <_ Cl l ulli.gk_, (2.4.15)

where q = ngk _ 1 /(n — qk- 1 ) = np/(n — kp). (2.4.14) and (2.4.15) give thedesired conclusion.

In order to treat the case kp > n, first assume u E C(fl) and for eachy E 3) use the Taylor expansion of u to obtain, with the notation of Section1.1,

u(v) = (v) + Rz(u)

60

where

and

2. Sobo!ev Spaces and Their Basic Properties

k-1

P=(y) _ E 1i D°u(z)(y - x)aial =o Cr.

1Rz E(y) _ k ar^

- t)x + ty)dt (y - x)a.lai=k

[f0.-ok-iDo.(0

To estimate In(y)I, note that

IKI lu(y)I <_ [IP=(y)I + IR=(y)I) dx (2.4.16)

and employ Holder's inequality to obtain

IP={v)Idx <x x

k-1%---k

a^ Dan(x)(y - x)alai=o

dx

k- 1 1< IKIIIp' E (diam K) 1°1 ^ ;xa. IlDaul^p.

10.1=o=o(2.4.17)

Similarly, to estimate the remainder term, we have

f IRx(y)Idx <- (diam(K))kk E 1 o

(1 - t )k-1

x a. Klal=k• IDau((1 -- t)x + ty)Idxdt

1< (diam(K))kk E

°1 ^ (1 - t)k-1 (l - t) -nlark

a. fK`- IDau(z)Idzdt,

where K, = T,(K) and T1 (x) = (1 - t)x +ty. Note that IK,I = (1 - t)n IKI.Consequently, by Holder's inequality and kp > n, we obtain

IRs(3+)Idx < IKI L /p'(diam(K))k k ElaI-k a.

^

(1 - t)k-1(1 - t)"IID a ^llp;x(1- t)n/P'dt

< IK1 1 ^p (diam(K)) k k E 1lark ^.(k _ -Ln

p IID°ullp;x,

I„1

which, along with (2.4.16) and (2.4.17), establishes the desired inequality.If u E tiVo'' )), let (u,) be a sequence of smooth functions converging to

2.5. The Rellich-Kondrachov Compactness Theorem 61

u in W "(f2). The application of (2.4.1) to each u i thus establishes theinequality for u E Wô'p(13). To conclude that u E 0(f2), apply (2.4.11) tothe difference u i - ui and obtain that {u i } is fundamental in the sup normon S3. O

2.4.3. Remark. An important case to consider in the previous two the-orems is fi = R". In this situation, Wk.P(R") = Wô''(R") (see Exercise2.1) and therefore the results apply to Wk ,p(Rn).

Observe that for p > n, the proof of Theorem 2.4.1 as well as that ofTheorem 2.4.2 yields more than the fact that u is bounded. Indeed, u isHolder continuous, which we state as a separate result.

2.4.4. Theorem. If u E Wo'p (1 ), p > n, then u E C° ''(SZ), where ar =1 - n/p .

Proof. Assume u E Cô (f1) and select x E 1. Let B = B(x, r) be anarbitrary ball and choose z E B fl a Then,

lu(x) - tt(z)l <A.(8)

!Du(r)Ix$(r)dH'a` 1 (r)

where A1 (0) is the ray whose end-point is x and passes through the pointB, 0 E S" -1 (x). Proceeding as in (2.4.5) and (2.4.6), we obtain

1 /p'

w(n - 1)1u(x) - u(z)I _< IIDulI p Ix -- Y1(1-n)P dy13

)

But,1 /p'

ryl(l-n)Fldy = l7'la(n))1/P'rl-nip

where y and a(n) are as in (2.4.8). Since the smooth functions are densein Wô'P (13), we find that (2.4.18) holds for u E WW'p(II) and for almost allx, z. ❑

An interesting aspect of the Sobolev inequality is the limiting case kpn. This will be considered separately in Chapter 2, Section 2.4.

2.5 The Rellich—Kondrachov CompactnessTheorem

As a result of the inequalities proved in the previous section, it followsthat the Sobolev spaces Wfl'P(S3) are continuously imbedded in if (0)where p' = np/(n - kp), if kp < n. In case kp > n, the imbedding is

(2.4.18)

U,


into the space C°(st), and if kp > n + mp, it can easily be shown thatthe imbedding is into Cm (M. In this section it will be shown that theimbedding possesses a compactness property if we allow a slightly largertarget space. Specifically, we will show that the injection map from WW'p(11)into either L9 (0), q < p' , or Cm(SZ) has the property that the closure of anarbitrary closed set in Wt P (0) is compact in the range space. That is, theimage sets are precompact. We recall here that a set S in a metric spaceis said to be totally bounded if for each e > 0, there are a finite number ofpoints in S such that the union of balls of radius e with centers at thesepoints contains S.

2.5.1. Theorem. Let SZ C Rn be a bounded domain. Then, if kp < n andp > 1, WtP(11) is compactly imbedded in L 4 (0) where q < npf (n - kp). Ifkp > n + mp, Wt P(11) is compactly imbedded in Cm (S2).

Proof. Consider the first part of the theorem and let B C WtP (f) bea bounded set. We will show that B is a compact set in L4 (11). SinceCr (f2) is dense in WtP(fl), we may assume without loss of generality thatB C Co (11), For convenience, we will also assume that Ilullk,p;n < 1 for allu E B.

For e > 0, let ue be the regularization of u. That is, ti e = u * Sp, wherecp, is the regularizer (see Sectipn 1.6). If u E B, then

lue(x) I <Iu(x - y)IÊ(y)dyB (0 ,£)

< c—n sup cpllUlllE- n sup {v(y) : y E R"),

and

IDue(x)I <_ Iu(x — y)I ID(Pe(y)IdyB{ox)

€—n—i sup{ID'(y)I : y E R" } IIuIIi< e-n-1 sup{lDcp(y)I . y E Rn).

Therefore, if we let Bf = {u£ : u E B), it follows that BE is a bounded,equicontinuous subset of C ° (S2). With the help of Arzela's theorem, it fol-lows that BB is precompact in Li (SZ). Next, observe that

Iu(x) — 11 E(X)1 < Iu(x) — u(x — y)IÊ(y)dYf (0,a)1

<_ jDu o #y(t) •' '^ ' (t)IcpE(rJ)dtdyfmomo

]< IDu(x - ty)I iyf4Pf (y)dtdy

f(0,e) 0

2,5. The Rellich-Kondrachov Compactness Theorem 63

where 'y(t) = t(x -- y) + (1 - t)x = x - ty. Consequently, Fubini's theoremleads to

p1 11 (X) - u^(x)Idx <f" IDu(x - ty)I IyIcPE(y)dxdtdyJ3(0,0ff "

< ^ I IDuIdrê.C l

Thus, B is contained within an r-neighborhood of BE in L 1 (SZ). Since BE isprecompact in L 1 (fl) it is totally bounded. That is, for every r > 0, thereexist a finite number of balls in L 1 (11) of radius r whose union containsB,. Hence, B is totally bounded and therefore precompact in L 1 (W. Thisestablishes the theorem in case q = 1.

If 1 < q < rap/(rn - kp), refer to (1.5.13) to obtain

iIAIIq <_ IIuII11IuII n;/ ( n_kp)

whereA = 1 /g"(nrkp)/np.

1 - ( n - kp)/r^p

Then, by Theorem (2.4.2)

Huh < cIIuMuIIl;PA

which implies that bounded sets in Wô''(f1) are totally bounded in Lq(SZ)and therefore precompact.

The second part of the theorem follows immediately from Theorem 2.4.4and Arzela's theorem in case k = 1. The general case follows from repeatedapplications of this and Theorem 2.4.1. 0

2.5.2. Remark. The results of Sections 2.4 and 2.5 are stated in termsof functions in Wa ' p (SZ). A natural and important question is to identifythose domains S2 for which the results are valid for functions in Wk.P(II).One answer can be formulated in terms of those domains of having theproperty that there exists a bounded linear operator

L : Wk,p(H) —P Wag .p(R') (2.5.1)

such that L(u)Ii= = u for all u E Wkie(f). We say that fZ is an (k,p)-extension domain for Wk.P(i) if there exists an extension operator forWk .P(12) with 1 _< p < oo, k a aeon-negative integer. We will refer tothis definition extensively in Chapter 4, and if the context makes it clearwhat indices k and p are under consideration, for brevity we will use theterm extension domain rather than (k, p)-extension domain. Clearly, theresults of the previous two sections are valid for u E Wk ,P(f) when S2is a bounded extension domain. Indeed, by Lemma 2.3.1 there exists a


function f E Co (R") such that f ' 1 on a Thus, if u E W 1' 4'01), thenf • L(u) E W "(SZ') where i2' is some bounded domain containing spt f. Itis now an easy matter to check that the results of the previous two sectionsare valid for the space Wk1P(1) by employing Wâ111').

A fundamental result of Calderon-Stein states that every Lipschitz do-main is an extension domain. An open set fi is a Lipschitz domain if itsboundary can be locally represented as the graph of a Lipschitz function de-fined on some open ball of Rn'. This result was proved by Calderon [CA1]when I < p < n and Stein [ST) extended Calderôn's result to p = 1, co.Later, Jones (JO] introduced a class of domains that includes Lipschitzdomains, called (e, ô) domains, which he proved are extension domainsfor Sobolev functions. A domain SZ is called an (e, ô) domain if wheneverx, y E R" and ix - yi < 6, there is a rectifiable arc 7 C St joining x to yand satisfying

length 7 < F - 1 Ix — v iand

d(z, R" - SI) > fix zl ill z1 for all z on 7.ix - vi

Among the interesting results he obtained is the following: If 11 C R 2 isfinitely connected, then 0 is an extension domain if and only if it is an(e, 6) domain for some values of e, b > 0.

2.6 Bessel Potentials and CapacityIn this section we introduce the notion of capacity which is critical indescribing the appropriate class of null sets for the treatment of pointwisebehavior of Sobolev functions which will be discussed in the following chap-ter. We will not attempt a complete development of capacity and non-linearpotential theory which is closely related to the theory of Sobolev spaces,for these topics deserve a treatment that lies beyond the scope of this expo-sition. Instead, we will develop the basic properties of Bessel capacity andrefer the reader to other sources for further information, cf. [HM], [ME1],[AD6).

The Riesz kernel, IQ , 0 < a < n, is defined by

Ia (x) = 7 (a) -

llzla -n

where7r n / 2 2 a r(a/2)

7(a) = F(n/2 - a/2) .The Riesz potential of a function f is defined as the convolution

I. * f(x)=--

1 f (04 7(a) hin ix — Yin

n

2.6. Bessel Potentials and Capacity 65

The precise value of 7(ac) is not important for our purposes except for therole it plays in the Riesz composition formula:

IQ * I0 IQ+s, a >0, Q>0, a + <n

cf. [ST, p. 1181.Observe that I. * f is lower semicontinuous whenever f > O. Indeed, if

s, z, then (x; - yia -" f (y) -, Ix -- yIQ - "f (y) for all y E Rn , and lowersemicontinuity thus follows from Fatou's lemma.

The Riesz potential leads to many important applications, but for thepurpose of investigating Sobolev functions, the Bessel potential is moresuitable. For an analysis of the Bessel kernel, we refer the reader to 1ST,Chapter 5} or [DO, Part III1 and quote here without proof the facts relevantto our development.

The Bessel kernel, gQ , a > 0, is defined as that function whose Fouriertransform is

ga(x) = (27r) - "/ 2 (l + IxI2)-Q/2

where the Fourier transform is

A') = (21)-n/2 a-+z f(1/)dv. (2.6.1)

It is known that ga is a positive, integrable function which is analytic exceptat r = O. Similar to the Riesz kernel, we have

90 * gee = 90+0, a, /3 > 0. (2.6.2)

There is an intimate connection between Bessel and Riesz potentialswhich is exhibited by ga near the origin and infinity. Indeed, an analysisshows that for some C > 0,

9Q(x) N CIxl(I/2){a-n-1)e-Is) as 1x1 ~ oc.

Here, a(x) b(z) means that a(x)/b(x) is bounded above and below forall large Izj. Moreover, it can be shown that

a I x n9a(x ) = 7(a)I (IrIQ-n) + o

if 0 < a < n. Thus, it follows for some constants C 1 and C2, that

ga (x) < C1 eI x I n -a

for all x E R". Moreover, it also can be shown that

I D9Q (x)I - ^ e -c, l=I (2.6.4)

as IaJ -+O


From our point of view, one of the most interesting facts concerningBessel potentials is that they can be employed to characterize the Sobolevspaces Wk 'p(R"). This is expressed in the following theorem where weemploy the notation

L° 'p (Rn ), a > 0, 1 < p < co

to denote all functions u such that

u= 9Q*f

for some f E Lp (R") .

2.8.1. Theorem. If k is a positive integer and l < p < oo, then

Lk,P(Rn) = Wk,p(Rtz).

Moreover, if u E Lk 'p(R") with u = go * f, then

GI -1 11/11p S Ilullk,p < MG

where C = C(a, p, n).

Remark. The equivalence of the spaces Li" and Wk,p fails when p = 1 orp=oo.

It is also interesting to observe the following dissimilarity between Besseland Riesz potentials. In view of the fact that IIgo III < C, Young's inequalityfor convolutions implies

IlsQ * flip Cllfllp, 1 1 (2.6.6)

where q np f (n — ap). However, an inequality of type (2.6.6) is possiblefor only such q, cf. (Exercise 2.19), thus disallowing an inequality of type(2.6.5) for la and for every f E L".

We now introduce the notion of capacity, which we develop in terms ofthe Bessel and Riesz potentials.

2.6.2. Definition. For a > 0 and p > 1, the Bessel capacity is defined as

B ,„ p(E) = inf { flf IIp : ga * f ? 1 on E, f ? 0),

whenever E C R. In case a = 0, we take Bo ,p as Lebesgue measure. TheRiesz capacity, RC , p , is defined in a similar way, with go replaced by IQ.


Since ga (x) < I0 (x), x E R", it follows immediately from definitionsthat for 0 < cx < n, 1 < p < n, there exists a constant C = C(a, p, n) suchthat

Ram(E) < CBQ ,p (E), whenever E c R". (2.6.7)

Moreover, it can easily be shown that

110„ p(E) = 0 if and only if B0 ,p (E) = 0, (2.6.8)

(Exercise 2.5).We now give some elementary properties of capacity.

2,6.3. Lemma. For 0 < a < n and 1 < p < oo, the following hold:

(i) 13„„,p(0) = 0,

(ii) If El C E2, then 130 ,p (E 1 ) < Ba ,p (E2 ),

(iii) If E; C R' , j = 1, 2, ..., then

Ba,pû:

00

1 E. i=i

Proof. (i) and (ii) are trivial to verify. For the proof of (iii), we may assumethat E°_° 1 130 (E1 ) < oo. Since each term in the series is finite, for eache > 0 there is a non-negative function fi E IP(Rn) such that

ÿa * fi > 1 on E1, Ilfillp < Ba,p(Ei) + 2- te.

Let f(x) = sup{ f1 (a) : _ = 1, 2, ...}. Clearly, ge, * f > 1 on U;_ 1 E, andf (z)" < fi (x)F. Therefore,

Bar ,p00 00 00

Û Ei <_ llf I^ Ê E Ba,p(Ei) + F. ❑

i=t i-1 i-1

Another useful characterization of capacity is t:.e following:

Ba ,p (E) = inf{ inf gQ * f = {sup inf ya * f (x)} -1)J xEE f xEE

where f E IP(R"), f > 0 and11 flip _< 1 (Exercise 2.4).Although Lemma 2.6.3 states that Ba ,p is an outer measure, it is fruitless

to attempt a development in the context of measure theory because it canbe shown that there is no adequate supply of measurable sets. Rather, wewill establish other properties that show that the appropriate context for


B. is the theory of capacity, as developed by Brelot, Choquet, [BRIT[CH).

2.8.4. Lemma. If IL} is a sequence in I7(R") such that — f II -' 0as i -, oo, p > I, then there is a subsequence {f, } such that

9o* fi,(x) 9a*f(x)

for Ba ,p-q.e. x E Rn.

(We employ the time-honored convention of stating that a conditionholds BQ ,p -q.e., an abbreviation for B a ,p-quasi everywhere, if it holds at allpoints except possibly for a set of /3 0 ,p-capacity zero.)

Proof. It follows easily from the definition of Ba ,p capacity that if f EIF(R" ), then Igo * f(x)1 < oo for BQ ,p-q. e. x E Rn . Thus, for E > 0,

Ba,p(ix : fga * fi(x) ` 9a * f (x)I e} = BQ,p({x : I9a * (fi -- f)(x)I E})

-' IIfi — fllp.Consequently, there exists a subsequence {A}} and a sequence of sets Esuch that

Igo * fi, (l) - 9a * f(Z)I 1, x E R" - E^,

withBa,p(Ej) < e2 -3 .

Hence, ga * f;? -► ga * f uniformly on R" - U? _ 1 E3 , where Ba,,p (U °_ 1 E) ) _<e. Now a standard diagonaiization process yields the conclusion. D

2.8.5. Lemma. If { f; } is a sequence in L" (R71 ), p > 1, such that fi -e fweakly in L"(R" ), then

lim inf 9Q * fi (x) < ga * f (x) < lim sup 9a * f i ( (2.6.10)

for 13„,p-q.e. x E R". If :n addition, it is assumed that each f > 0, then

9a * f (x) < lim inf ga * fi (x) for a E R" (2.6.1 1)tiôo

and

for Ba ,p -q.e. x E Rn.

go * f (x) = Lim inf 9a * fi (x)i.00(2.6. I2)

Proof. Under the assumption that f; — f weakly in V'(Rn), by theBanach-Saks theorem there exists a subsequence of IL} (which will be


denoted by the full sequence) such that

g i - i-1 E ff3 = i

converges strongly in I'(R") to f . Lemma 2.6.4 thus yields a subsequenceof {g; } (denoted by the full sequence) such that

9Q * f(x) = lirn go, * gi(x)fôo

for Ba, p-q.e. z E Rn. However, for each z E R73 ,

lim inf ga * fc(x) < lim go * Mx),s-.00 - i -.00

which establishes the first inequality in (2.6.10). The second part of (2.6.10)follows from the first by replacing f; and f by --L and -f respectively.

In the complement of any ball, B, containing the origin, IIg0 IP p . ; R'_B <co, by (2.6.3). Thus, (2.6.11) follows from the weak convergence of f; to f.(2.6.12) follows from (2.6.11) and (2.6.10). D

2.6.6. Lemma. For every set E C Rn

B0 , p (E) = inf. { Ba ,p (U) : U D E,U open).

Proof. Since go is continuous away from the origin, the proof of the lowersemicontinuity of ga *f when f ? 0 is similar to that for the Riesz potentialgiven at the beginning of this section. The lemma follows immediately fromthis observation.

The lemma states that Ba , p is outer regular. To obtain inner regularityon a large class of sets, we will require the following continuity propertiesof B0 ,p .

2.6.7. Theorem. If {E; } is a sequence of subsets of Rn, then

Bar ,p ^lim inf E; } < lim inf Ba ,p (E;). (2.6.13)—• 00 —*CO

If E10 E2C ..., then

Ba,p u E; = lim BQ .p (E;). (2.6.14)-î_-.1.

If IC I D K2 D ... are compact sets, then(00

c=p K; = lirn

i__.00Ba,p(K;). (2.6.15)

^

70 2. Sobolev Spaces and Their Basic Propel. tm..

Proof. For the proof of (2.6.14) assume that the limit is finite and let f,be a non-negative function in L" (Rh) such that ga * fi > 1 on Ei with

Ilfillp < BQ.p(Et) + 1/i. (2.6.16)

Since Il fi if p is a bounded sequence of real numbers, Theorem 1.5.2 assertsthe existence of f E LP(Rn) and a subsequence of {AI} that convergesweakly to f. Hence, (2.6.12) implies that there exists a set B C E = U°__ L Eiwith Ba ,p (E — B) = 0 such that ga *f > 1 on B. Therefore,

Ba,p(E) = Ba,p(B) < Ilf llp<limif llf1llp< lim Ba ,p ( Ei ),i—.00

from (2.6.16). If^ o0

Ai= u n Ek,J= i k=^

then {A,} is an ascending sequence of sets whose union equals lirn inf E,.Therefore, since Ai C Ei for i > 1, (2.6.14) implies (2.6.13) because

Ba,p

00

inf E,} = Ba ,p (U Ai)iôoi =f

= Jim Ba p (A,)iâo

< lim inf Ba ,p (E,).i—oo

Finally, it {K;} is a descending sequence of compact sets, Lemma 2.6.6provides an open set U D fls_ 1 K i such that

Ba,p(U) < Ba,p + f

for an arbitrarily chosen e > O. However, K, C U for all sufficiently large àand consequently Ba , p (K,) < Ba ,p (U). (2.6.15) is now immediate and theproof is complete, y ❑

(2.6.14) states that Bail) is left-continuous on arbitrary sets whereas(2.6.15) states that Ba ,p is right continuous on compact sets. The impor-tance of these two facts is seen in a fundamental result of Choquet [CH,Theorem 11 which we state without proof.

2.6.8. Theorem. Let C be a non-negative set function defined on the Bordsets in Rn with the following properties:

II9Q * r.IÎ p' = sup{L.

supR^

9a *T/• f dZ: f> o, 1IfI1p Ç 1

l J1g^ *f du: f?o, VG ^


(i) C(0) = 0,

(ii) If B1 C B2 are Borel sets, then C(B 1 ) < C(B2 ),

(iii) If (13; } is a sequence of Borel sets, then C (U°° 1 BI ) < Er° y C(B; ),

(iv) C is left continuous on arbitrary sets and right continuous on compactsets.

Then, for any Suslin set A C Rn ,

sup{C(K) : K C A, K compact} = inf{C(U) : U D A, U open}.

Any set A for which the conclusion of the theorem applies is called C-capacitable. In view of Lemma 2.6.3 and Theorem 2.6.8, the following isimmediate.

2.6.9. Corollary. All Suslin sets are Ba ,p-capacitable.

The usefulness of Theorem 2.6.8 and its attending corollary is quite clear,for it reduces many questions concerning capacity to the analysis of itsbehavior on compact sets.

We now introduce what will eventually result in an equivalent formula-tion of Bessel capacity.

2.6.10. Definition. For 1 < p < oo, and E C Rn a Suslin set, let M(E)denote the class of Radon measures thon Rn such that p(R" — E) = O. Wedefine

bQ,p(E) = sup{p(Rn) }

(2.6.17)where the supremum is taken over all A E M(E) such that

Ilga * 1.4p' < 1. (16.18)

Clearly,b,„,p (E) = (inf{Ilga * Î^p'}) (2.6.19)

where the infimum is taken over all u E M(E) with v(R") = 1. We havethat

and thus obtain

ba ,F (E) -1 = (inîsuJ 9a * f dy (2.6.20)


where y E M(E), v(R") = 1, and f > 0 with IIf IIp < 1.Recall from (2.6.9) that if E C R", then

13.,p (E) = (sup inf 9. * f (x) } _pf xEE

where f E LP(R"), f > 0 and II fli p < 1. By considering measures concen-trated at points, this is easily seen to be

BQ ,p (E) -l ip — sup inf ga * f dv (2.6.21)f y

where f and v are the same as in (2.6.20).We would like to conclude that there is equality between (2.6.20) and

(2.6.21). For this purpose, assume E C R" is a compact set and let

F( f, v) = f gQ * f dv (2.6.22)

where f E L"(R"), f > 0, II f lIp < 1 and v E M(E), v(R") = 1. ClearlyF is linear in each variable and is lower semicontinuous in v relative toweak convergence. Since the spaces in which f and v vary are compact wemay apply the following minimax theorem, which we state without proof,to obtain our conclusion, EFA1.

Minimax Theorem. Let X be a compact Hausdorff space and Y an ar-bitrary set. Let F be a real-valued function on X x Y such that, for everyy E Y, F(x, y) is lower semicontinuous on X. If F is convex on X andconcave on Y, then

inf sup F(x, y) = sup inf F(x, yy).sEX yEY yEY zEX

We thus obtain the following result.

2.6.11. Lemma. If K C Rn is compact, then

[ba,p(K)? = Ba,p(K). (2.6.23)

Our next task is to extend (2.6.23) to a more general class of sets. Forthis purpose, observe that if E C Rn is a Suslin set, then

bQ ,p (E) = sup{ba , p (K) : K C E, K compact). (2.6.24)

To see this, for each Suslin set E, let p E M(E) with IIg. * µll p' < 1. IfK C E is compact, then v — uI K has the property that v E .A4 (E) withIlga *vllp ,

< 1. Since p is a regular measure, we have

p(E) = sup{p(K) K c E,K compact},


and therefore

bk,p (E) = sup{bk,p (K) : K C E, K compact}. (2.6.25)

From (2.6.25), (2.6.23), and Corollary 2.6.9 we conclude the following.

2.6.12. Theorem. If E C R" is a Suslin sct, then

[byi.p (E)jp = Bck,e(E)•

Thus far, we have developed the set-theoretic properties of Ba ,p . We nowwill investigate its metric properties.

2.8.13. Theorem. For p > 1, ap < n, there exists a constant C =C(a, p, n) such that

C-l rn-"P < BQ,P[B(x , r)] < Cr"'"

whenever z E Rn and 0 < r < 1/2.

Proof. Without loss of generality, we will prove the theorem only for B(0, r)and write B(r) = B(0, r). Let f E I1(R"), f > 0, have the property that

gQ * f > 1 on B(2). (2.6.26)

By a change of variable, this implies

gQ r y f ^M)r -ndy^ 1Rn

(2.6.27)

for x E B(2r). From (2.6.3) and (2.6.4), there exists C = C(a, p, n) suchthat

Clx - yla-n e- 'Ix- E/! < ga (z - 3/) < Ctx -- yIQ - ne-12-

v1 ,

and therefore

90 (L---■--y) < GIx — yIQ-nrn-Qe-1t-U1r 1 r

< Clx - y la-n rn-ck

e-21x

-vi (7- 1/2)< C2 rn

— Qgc,(z — y) (r < 1/2).

Consequently, from (2.6.27),

C2 g0 (x - y) f ( 7-1 )r r-ady > 1 for z E B(2r), (r < 1/2).R

74

However,

Hence,

2. Sobolev Spaces and Their Basic Properties

JR. [C3r-cif ()}'dy = C2Prn-°'p 11f11p.

Ba,P[B(2r)i < C2prn-ap ll f IIp, r < 1/2,for every f E LP (R") satisfying (2.6.26). Thus,

$a,P[B(2r)] < C2Pr' -aPBa,p[B( 2 )) , r < 1 / 2,

from which the conclusion follows.For the proof of the first inequality of the theorem, let f E LP(R' ),

f > 0, be such that g.* f > 1 on B(r). Then

I B(r)I fB

s f dx < IB(r)I l/4 119a * f1Iq'(r)where q = p` = np/(n —ap). It follows from (2.6.3) that ga < CIa . Becausethere is no danger of a circular argument, we employ the Sobolev inequalityfor Riesz potentials (Theorem 2.8.4) to obtain

rn-aP < CIA'Taking the infimum over all such f establishes the desired inequality. ❑

The case ap > n requires special treatment.

2.6.14. Theorem. If p > 1, ap _ n and 0 < r < I, there exists C =C(n, r) such that

C` 1 (logr -1 ) 1-P < B.,P[B(x , r)) < C(logr -11 ) L_P

whenever O<r <ir< I and x R' 1 .

Proof. As in the proof of the previous theorem, it suffices to consider onlythe case z = O. Let be a Radon measure such that µ[R" — B(r)) = 0 and119a * < 1, where we write B(r) = B(x,r). Because of the similaritybetween the Riesz and Bessel kernels discussed at the beginning of thissection, there exists a constant C independent of r such that

(IQ * µ)Pax<C (9a*µ) 1 dx<C.f (1) R"

If ly1 < r and 1x1 ? r, then Ix — y1 < (xi +therefore

C>_r(Ia*µ)Pdx—<^x^< 1 fÎ1^1 (1, P'

Ix - yl a-"dtA(y) dx

> C1[p(Rn))p ,

1x1—n dxf<,=,«

= C1[u(R°)lps [logr -i ^.

1Y1 < 1x1 + r < 21x1 and


Thus, by Theorem 2.6.12, it follows that

B°,p[B(r)] < C(logr -1 ) 1-p.

To establish the opposite inequality, let a r denote the restriction ofLebesgue measure to B(r). Since ga < I° , we have

ga * Ar(x) < C f ix - yla`ndy. (2.6.28)B(r)

If 1x1 < r/r, NI < r, then Ix - yi < cr where c(r) = 1 + 1/i. That is,B(r) C B(x, cr). Therefore,

lx - 1,1`2-"dy - vl°-ndyf (r) B(s,cr)

< C(r)r°

which, by (2.6.28), implies

9a * Ar (x) < C(f)r° if lx' < r/F. (2.6.29)

If l yj < r and r/f < Izi < 1, then Ex - yi > IxI - ly i > lx ^ - r > e(F)1x1,where now c(f) = 1 - F. Hence,

9a * Ar (x) < C Î x - y iandy < C1 rn fxIa-n if r/t < Ix' < 1. (2.6.30)B(r)

If lx1 > 1, then (2.6.3) yields

go * Ar(x) <- Crn e -Izt . (2.6.31)

Thus, (2.6.29), (2.6.30), and (2.6.31) yield

1i * A. lip, < Crn (log r-2)1/p

Appealing again to Theorem 2.6.12, we establish the desired result. ❑

2.6.15. Remark. in case ap > n, it is not difficult to show that there is aconstant C = C(ar, p, n) such that

Ba ,p (E) > C

whenever E 0. See Exercise 2.6.Because Ba ,p [B(x, r)) .:r T r"." one would expect that Bessel capacity

and Hausdorff measure are related. This is indeed the case as seen by thefollowing theorem that we state without proof, [ME1), [HM). See Exercises2.15 and 2.16.

2.6.16. Theorem. If p > 1 and ap < n, then B0 (H) = 0 if H"- °p(E) <oo. Conversely, if 13.,p (E) = 0, then H" -aP+E (E) = 0 for every e > O.


2.7 The Best Constant in the Sobolev Inequality

There is a fundamental relationship between the classical isoperimetric in-equality for subsets of Euclidean space and the Sobolev inequality in thecase p = 1. Indeed, it was shown in [FF] that the former implies the latterand, as we shall see in Remark 2.7.5 below, the converse is easily seen tohold.

We will give a method that gives the best constant in the Sobolev In-equality (Theorem 2.4.1), by employing an argument that depends criticallyon a suitable interpretation of the total variation for functions of severalvariables. This is presented in Theorem 2.7.1 and equality (2.7.1) if referredto as the co-area formula. Tbis is a very useful tool in analysis that hasseen many applications. We will give a proof for only smooth functions butthis will be sufficient for our purposes.

2.7.1. Theorem. Let u E Co (R"). Then

I„ +oo[Duldx =Hn -1 [u -1 (t) n S^]dt.

f 00(2.7.1)

Before giving the proof of this theorem, let us first consider some ofits interpretations. In case n = 1, the integrand on the right-hand sideinvolves Hausdorff 0-dimensional measure, H ° . H° (E) is merely the numberof points (including oo) in E and thus, the integrand on the right side of(2.7.1) gives the number of points m the set u -1 (t) n O. This is equivalentto the number of times the graph of u, when considered as a subset ofR2 = {(x, y)}, intersects the line y = t. In this case (2.7.1) becomes

L u'1 f N(y)dy (2.7.2)

where N(y) denotes the number of points in u -1 (y) n Il. (2.7.2) is knownas the Banach Indicatrix formula, [SK, p. 284

The Morse-Sand Theorem [MSE1], [SA], states that a real-valued func-tion u of class Cn defined on Rn has the property that H 1 [u(N)] = 0 whereN = {x : Du(x) = 0}. For example, if we consider a function u E Co (R2),an application of the Implicit Function theorem implies that u -1 (t) n S2 isa 1-dimensional class C 2 manifold for a.e. t. In this case, H 1 (u -1 (t) r1 S2]is the length of the curve obtained by intersecting the graph of u in Ra bythe hyperplane z = t. Thus, the variation of u, fn IDuodx, is obtained byintegrating the length of the curves, f ri u- ' (t), with respect to t.

The co-area formula is known to be valid for Lipschitzian functions. (Wewill see in Chapter 5, that another version is valid for BV functions.) Theproof in its complete generality requires a delicate argument from geometricmeasure theory that will not be given here. The main obstacle in the proof

2.7. The Best Constant in the Sobolev Inequality 77

is to show that if u is Lipschitz, then

H"-1[u-1(^) n N]dt = 0f'where N = {x : D(x) = 0}. Once this has been established, the remainderof the proof follows from standard arguments. Because our result assumesthat u E C", we avoid this difficulty by appealing to the Morse-Sard the-orem referred to above. In preparation for the proof, we first require thefollowing lemma.

2.7.2. Lemma. If U C R" is a bounded, open set with C2 boundary, then

sup : cp E C (R";R"), supp] <_ 1 = H" 1 [8U].{ jdiv dxi

Proof. By the Gauss-Green theorem,

divSpdx !c r^(x)dH" I (x)j,(x).U

where v is the unit exterior normal. Hence

sup div (09 dx :çP E CQ (R"; R"), suP iWi < 1 < Hit-1 (8U).u

To prove the opposite inequality, note that v is a C 1 vector field of unitlength defined on ôU and so may be extended to a C' vector field Vdefined on R" such that IV (x)I < 1 for all x E R", cf. Theorem 3.6.2. Iftke Cô (R") and 10l <1, then with ço=tkV,we have

J div Sp dx =t/J(y)dH" - 1 (y)fu

so that

sup div cp : cp E Co (R"; R"), sup 11,01 < 1u

> supt, dH"- 1 : t € Co (R"), sup I I < 1 = Hn -1 (8U). Oou

Proof of Theorem 2.7.1. We first consider linear maps L: R" - R 1 .Then there exists an orthogonal transformation f : R" -+ R" and a non-singular transformation g such that f (N1 ) = R 1 , f (N) _ R"'', (N =ker L) and

1

L —gopo f

f„,- H"-1[En L-1(z)]dz. (2.7.4)


where p: R" -. R 1 is the projection. For each y E R 1 , p -1 (y) is a hy-perplane that is a translate of the subspace p -1 (0). The inverse imagesp-1 (y) decompose Rn into parallel (n - 1)-dimensional slices and an easyapplication of Fubini's theorem yields

IEI = R Hn-1[E np 1 (y)]dye

whenever E is a measurable subset of R". Therefore

(2.7.3)

If(E)1 — IEI =R

H"-1[E n p -1 (y)]dy1

-fRI H"-

1 [ f(E) n p- 1 (v)Ïdy

-Hn-1 (E n f -1 (p-1 (y))1dy•fe

Now use the change of variables z = g(y) and observe that the last integralabove becomes

Ig'IIEI -R

Hn -1 [E n f -1 (p- 1 (g -1 (z)))Jdz^

But WI = IDLI and thus (2.7.4) establishes Theorem 2.7.1 for linear maps.We now proceed to prove the result for general u as stated in the theorem.

Let N = {x : Du(x) = 0} and for each t E R 1 , let

Et-R"n{x: u(x) >t}

and define a function fe : R" -+ R 1 by

Thus,

ft = )(Et, if

-XR" -E, ift >0t < 0.

u(x) =ft (x)dt, x E R".R 1

Now consider a test function Sp E Cô (Rn - N), such that sup IV, < 1.Then, by Fubini's theorem,

u(x)co(x)dx =ft (x)Sp(x)dtdxf" R" R I

= fe(x)cp(x)dxdt.f , R"(2.7.5)


Now (2.7.5) remains valid if cp is replaced by any one of its first partialderivatives. Since Du # 0 in the open set R" - N, the Implicit Functiontheorem implies that u -1 (t) n (R" -N) is an (n - 1)-manifold of class Cn.In addition, since spt cp C R" - N, it follows from the Divergence theoremthat

div tip dx = yo(r) • v(x)dHn - ' (x).Ee JôE e )fl(R" -N)

Therefore, if cp is now taken as cp E Cô (R" - N; Rn ) with sup kpl < 1, wehave

- Du gp dx = u div cp dx = div tic)R" R " R' E

tic) dE t

= cp(x) v(x)d1-1n -1 (x)dtRt fR" -N)nBEt

H" - 'RR" — N) n u -1 (t)ldtJR'

fin - 1 [u- E (t)]dt. (2.7.6)f ^

however, the sup of (2.7.6) over all such cp equals

Du^dx =R" -N R"

In order to prove the opposite inequality, let Lk: R" -e R 1 be piecewiselinear maps such that

lireR ILk - uldx = 0 (2.7.7)

k-1°°o "

and^l iyrn

R n IDLk id

Rx = ^ 1Duldx. (2.7.8)

LetE^ — R" n {z : Lk (x) > t ?,

Xé = XE •From (2.7.7) it follows that there is a countable set S C R 1 such that

lim lX t — Xé ldx = 0 (2.7.9)k'oo f awhenever t ft S. By the Morse -Sard theorem and the Implicit Functiontheorem, we have that u -1 (t) is a closed manifold of class C" for all t ER 1 - T where 1-1 1 (T) = O. Redefine the set S to also include T. Thus, for


t g S, and e > 0, refer to Lemma 2.7.2 to find (P E C7( Rn ; Rn) such thatIV, < 1 and

lin-i[u-1(t)1 — divcpdx < 2 .l

E: (2.7.10)

Let M = fjr, jdiv coldx and choose

iXt

ko such that for k > ko ,

€— X: idx < .

R"2M

For k > ko ,

div cP dx — div çP dxf t E t < M iXt — Xi 1dz < 2 .R "

(2.7.11)

Therefore, from (2.7.10) and (2.7.11)

H"-1[u-1(t)] <L d1 dx+t

< div 5pdx+ eE k

— (P• v dHn -1 + e8E=

< fIn-t[Lk-1 (t)1 + e.

Thus, for t ft S,

lin -1 [tr i (t)] < lip inf fin -1 [L i

Fatou's lemma, (2.7.8), and (2.7.4) imply

lin -1 {u -1 (t}]dt < lirn inf Hn-'[Lk'(t)]dtf ^ - k' Ri

<lim inf. IDLk^dx- k ^ oo R"

_'Duldx.R "

Theorem 2.7.1 is a special case of a more general version developed byFederer [F1] which we state without proof.

2.7.3. Theorem. If X and Y are separable Riemannian manifolds of class1 with

dim X = m > k = dim Y

CI


and f : X --+ Y is a Lipschitzian map, then

I, ^ f(x)dHm(=) = n f-1 (31))dH k (g)Y

whenever A C X is an Hm-measurable set. Moreover, if g is an Hu'integrable function on X, then

g(x).If (x)dHm (x) = g(x)dHm-k(a)dHk (31).J f-'(!I)

Here, .1 f (x) denotes the square root of the sum of squares of the deter-minant of the k x k minors of Jacobian matrix of f at z.

The proof of Theorem 2.7.1 above is patterned after the one by Flemingand Rishel [FR] which establishes a similar result for BV functions. Theirresult will be presented in Chapter 5.

We now give another proof of Theorem 2.4.1 that yields the best constantin the case p 1.

2.7.4. Theorem. If u E Co (R"), then

IIUIIn /{n-1) < n- 1 ^(n)- 1 ^nIIDÎI.

Proof. For t > Q, let

A t = {x : Iu(x)I > = {x : Iu(x)I = t}

and let u t be the function obtained from u by truncation at heights t and—t. If

1(0 = -1),then clearly

I^t +^ I <_ Iutl + hXa.f (t + h) < f(t) + htA1l(n-Min (2.7.12)

for h > Q. It follows from the Morse-Sard theorem that for a.e. t > 0, Bt isan (n — 1)-dimensional manifold of class oo and therefore, an applicationof the classical isoperimetric inequality yields

IAtI(n-1 )/n < n -la ( n)- 1 /nHn-1(Bt ) . (2.7.13)

It follows from (2.7.12) that f is an absolutely continuous function with

f'(t) < IAt+(n -1)/n


for a.e. t. Therefore, with the aid of (2.7.13), it follows that

(n-1)/nIuln/(n-1)dx = f ( Do ) -- Ï (o)

00

= f`(t)dt0

< n -1 a(n) -l in Nn-1[13t1dt.f0 00

The co-area formula, Theorem 2.7.1, shows that the last integral equals

l DuldX,R ^

thus establishing the theorem.

From the inequality

IIuIIn/(n-1) < nla(n)-l/nlIDull

one can deduce the inequality

IIuIÏP• < np(n - 1)/(n - p) II Du IIP

(2.7.14)

(2.7.15)

by replacing u in (2.7.14) by uQ where q = p(n - 1)/(n -- p). Then

(n-1)/n^nP/tn-P)d^ < n'a(n) -1/nq lu1 4- i lDulda

(P- 1 )/v5_ n-la(n) -E/n q unP/(n-P)dx IIDuIIP

by Holder's inequality.Of course, one cannot expect the constant in (2,7.15) to be optimal.

Indeed, Talenti [TA) has shown that the best constant C(n, p) is

C(n,p) =7r-1/2n-1/2 P - 1 1- ( 1 /p) r(1 + (n/2))r(n)

1/n

n - P L r(n/pr(I^ n- (n/p))where 1 < p < n.

2.7.5. Remark. The proof of Theorem 2.7.4 reveals that the classicalisoperimetric inequality implies the validity of the Sobolev inequality whenp = 1. It is not difficult to see that the converse is also true.

To that end let K C Rn be a compact set with smooth boundary. LetdK(z) denote the distance from x to K,

dK(x) = inf{lx - y) . y E K }.

2.8. Alternate Proofs of the Fundamental Inequalities 83

It is well-known and easy to verify that dK(x) is a Lipschitz function withLipschitz constant 1. (See Exercise 1.1.) Moreover, Rademacher's theorem(Theorem 2.2.1) implies that dK is totally differentiable at almost everypoint x with IDdk (x) f = 1 for a.e. x E f?. For each h > 0, let

Fh(x) = 1 - min[dK(x), h} • h -1

and observe that Fh is a Lipschitz function such that

(1) Fh(x) =Iif x E K

(ii) Fh(x) 0 if dK(x) ? h

(iii) IDFh(z)i < h- ' for a.e. x E Rn.

By standard smoothing techniques, Theorem 2.7.4 is valid for Fh becauseFh is Lipschitz. Therefore

(11(1)(n-1)(n) < n-'a(n)-1fn t{x :0 < dK(x) < NI h

Since lDdK (x)I = 1 for a.e. x E R", the co-area formula for Lipschitz maps,Theorem 2.7.3, implies that

I{x : 0 < dK(x) < h}I 1 IDdKIdzh h

{ o<a,<<hfI h

Hr [dxi (t))dt0

= Hn - ' [di' (th ))

where 0 < t f, < h. Because K is smoothly bounded, it follows that

Hn -1[dxl(th)) -0 H" -1 (8K) as h -' 0

and thus, the isoperimetric inequality is established.Of course, by appealing to some of the more powerful methods in geo-

metric measure theory, the argument above could be employed to cover thecase where the compact set K is a Lipschitz domain. By appealing to theproperties of Minkowski content, cf. [F4, Section 3.2.391, it can be shownthat the above proof still remains valid.

2.8 Alternate Proofs of the FundamentalInequalities

In this section another proof of the Sobolev inequality (2.4.10) is givenwhich is based on the Hardy-Littlewood-Wiener maximal theorem. This

or nlAt f ç 75 Ill I11 whenever t E R I . (2.8.2)


approach will be used in Section 2.9, where the inequality will be treatedin the case of critical indices, kp = n.

We begin by proving the Hardy-Littlewood-Wiener maximal theorem.

2.8.1. Definition. Let f be a locally integrable function defined on R'a.The maximal function of f, M(f), is defined by

M(f )(x) = supIf(y)Idy : r > 0 .B (r,r)

2.8.2. Theorem. If f E Lp (Rn ), 1 t}. Prom Definition 2.8.1it follows that for each x E A t , there exists a ball with center x E A t , suchthat

s= Ifl dy > t. (2.8.1)iBx

If we let F be the family of n-balls defined by F _ {El. : x E At}, then The-orem 1.3.1 provides the existence of a disjoint subfamily {B1, B2, ... , Bk, .. .}such that

00

EIBkI?5 -nlAtlk=1

and therefore, from (2.8.1),DO

IIfII I ? fur. 1 Bklfldy > t E l$kI ? t5IAtI ,

k=1

We now assume that 1 t/20 if If(x)1 < t/2.

Then, for all x,If(x)1 t} C {M(ft )(x) > t/2}.

2.8. Alternate Proofs of the Fundamental Inequalities 85

Applying (2.8.2) with f replaced by ft yields

'Ad <_ III R(ft)(r) > t/2}I <_ 2 •t5R If:ldy = 2

5Îf ldy.n t ffill>t/2)

Now, from Lemma 1.5.1, and (2.8.3),00

=AdtL MfY" o

00< p2 • 5" 9-2 I f(d^ dt

o flifi>02}

p2p • 5" tv-2 If ldx dto

f{ifi>e}

00= p2P • 5" tP-2p({1f Î > t})dt

0

where p is a measure defined by p(E) = fE !Mr for every Borel set E.Thus, appealing again to Lemma 1.5.1, we have

(Mf)pdx = p2p

•

5n oo

u(iifl > dtp- i

p -1 o t })

p2p.5" f 1= p — 1 JR» If lp du

p2p . 5"__ 1 Ip RRSince p > 1. This establishes the theorem. ❑

For 0 < a < n, we recall from Section 2.6 the definition of the Rieszpotential of f of order a:

L* f(x) = ici (x)= 1 f(y)du ^`(^) hin — yin —a

.

The following lemma is the final ingredient necessary to establish theSobolev inequality for Riesz potentials.

2.8.3. Lemma. If 0 < a < n, 0 > 0, and 6 > 0, then there is a constantC = C(n) such that for each z E R",

(i) If(y)Idu < C6°M(f)(z)B(:.6) Ix — yin-" —

(2.8.3)

00_ p tp-lIAeIdt0

fRn


(ii) If(y)Idy < C6-13M(f)(x).1" -B(z,6) I x 11I

Proof. Only (i) will be proved since the proof of (ii) is similar. For r E Rnand a > 0, let the annulus be denoted by

b ô ë b..4 x, 2k ' 2k + i = B x,

2k - B x, 2k +1

and note that

If(^)Idy I (z,6) Ix — yln -a =If(y)Id^

^ À(= , ^ Ix yIn - ak-o ;^ ^)vo 6 a-n

< 2k^.1k_p

e(T.*)Iflix

ao ( i )a—n ^( i L(Z,f1dT

=a(n) ^k0 1

^ C6a M(f)(z)+

where a(n) denotes the volume of the unit n-ball. This proves (i). ❑

We now will see that the Sobolev inequality for Riesz potentials is aneasy consequence of the above results.

d =

2.8.4. Theorem. Let a > 0, 1 0, Holder's inequality implies that

If(ll)I1/p'

$_ ( , fixR x 5 I yin-a dy < w (n - 1 )IIf IIp (16 °13 rn-1-p'(n-Q) dé

where r = Ix - gel. The integral on the right is dominated by ë" -(n fp ) sinceap < n, and therefore, by Lemma 2.8.3(i),

Ila(f)(x)I < C [6 M(f)(x) + ta-(Tt ) IIfIIp] • (2.8.4)

I f we choose

( M(f)(x)\-Din,II/11p )

np= n ap s

2.8. Alternate Proofs of the Fundamental Inequalitics 87

then (2.8.4) becomes

Ifa(f)(x)I<_ C[M(f)(x)r (°P/n) Ilf llp /n

or,Ifa(f)(x)IP. <— CIm(f)(x)]PIIIIIpaP /n )P • .

An application of Theorem 2.8.2 now yields the desired conclusion. Q

2.8.5. Remark. If we are willing to settle for a slightly weaker result inTheorem 2.8.4, an easy proof is available that also provides an estimate ofthe constant C that appears on the right-hand side of the inequality. Thus,if I/ is a domain with finite measure, f E LP(Q), and p < q < p', we canobtain a bound on IIII(f )IIq by a method that essentially depends only onHolder's inequality.

For this purpose, let T = 1 — (p — q) and note that because q < p',

I x - yIa-n € Lr(n) (2.8.5)

for each fixed x E R. As in the proof of (2.4.7), if I 13(x, R)I IS11, then

I^ _ Î(a-ri}rd^ f k — y ,(a-n)r dy = WR - 1)R(a-tt)r +n(l A B (s,R) (cY -n)r+ n

w(n -1)IS/I'''=' Rot - n)r + nia(n)7 -

Oa, r, 3-0

where 7 = Rot — n)r)/n + 1. For each fixed x, observe that

Ix - YI (a-n) If M I = ( Ix - 1/1 (a-n)r If (OP)

1/4

•(Ix - yI {a-n)r/PI 1 - If(Y)Ipawhere b = i - 1 . Because A + i + b = 1, we may apply Holder's inequalityP q P qto the three factors on the right side of (2.8.7) to obtain

Ifa(f)(x)I < I x - yt(°',n'rIf(v)iPdu(f

n n

1/p` 6

• Ix - bI(a-o)rdEl If(y)^pdv •

Therefore, by Fubini's theorem and (2.8.6),

(raf )Qdx n ct[f ix - Î (a -n)r If (y)IPdzdy

•C(a, 6,n)p, • IIIIIp 6<- c(ct, 6, n) • li f IT - C(a, 6, n)^, • 11/11:96 -

(2.8.6)

(2.8.7)

1/q

fa


Thus,

IIIQf1f4 < C(a, 15, H)('Iq)+(Ìp') I1flfp< C(a, ë, 0 ) 1/r fIf lfp-

2.8.6. Remark. It is an easy matter to see that Theorem 2.8.4 providesanother proof of Theorem 2.4.2. Indeed, if u E Co (R" ), recall from (2.4.5)that for every x E R",

fu(x)I 5 C(n)Il(IDuI)- (2.8.8)

In fact, if we employ the Riesz composition formula which states that

* Ip = fQ+A, a + Li < n,

an application of (2.8.8) to the derivatives of u gives the estimate

Iu(x)I < C(n , k)Ik(IDk ui)•

From Theorem 2.8.4 we have

fllk(IDkuI)IIp. < CflDkulipif kp < n. Thus,

Ilullp•<_ CIIDkuIIp <_ Cllullk.pwhich is the conclusion in (2.4.10) when SZ = R".

Of course, one could also employ Theorem 2.6.1 which states that eachu E Wk IP(R") can be represented as u = gk * f for some f E U' (R" ), whereIIf IIp Ilullk,p;R° . Then, in view of the fact that gk < CIk, (2.4.10) followsfrom Theorem 2.8.4.

2.9 Limiting Cases of the Sobolev InequalityIn previous sections all Sobolev-type inequalities were established underthe restriction kp n. We now treat the case kp = n in the context ofRiesz potentials and since the Riesz kernel I , is defined for all positivenumbers a, we will therefore replace the integer k by a.

When ap = n, one might hope that to * f is bounded because p* --' o0as ap -' n. However, while boundedness is trivially true when n = 1, itis false when n > 1. As an example, consider u(x) = I log lxII' -2

/ ("- L ) ;

clearly u E W 1 ""(B(O,r)) for r < 1, but u L°'°(B(0,r)). Although anL estimate cannot, in general, be obtained it is possible to obtain resultsthat provide a good substitute. Our first result below offers exponentialintegrability as a substitute for boundedness. We begin with a simple and

2.9. Limiting Cases of the Sabolev Inequality 89

elegant proof of this fact which follows easily from the estimate discussedin Remark 2.8.5.

2.9.1. Theorem. Let f E L"(ft), p > 1, and define

9=In/p* f.Then there are constants C1 and C2 depending only on p and n such that

P

st exp

Cl Ilf IIP dx < Ca .

Proof. Let p < q < oo and recall from Remark 2.8.5 the estimate

Il'af Ilq < C (Q, o, 1l)" Ilf llp ,

where T=1—( 1 1'

(2.9.1)

(2.9.2)

C(a' S' (1) [(a — n)r + nja(n) 7 '

and(a — n)r7 1

= +n

In the present situation, cp = n, and therefore

TIP(oc — n)r+n— pq +(P— g)Thus, we can write

C(a, ô, S-2) = K l l SZ l 7(

pq +(p — q) ) < KIS2I7qnp

where K1 and K are constants that depend only on p and n. Thus, since74/r = I, from (2.9.2) we have

IglQdx < Cq/r Ili llpi2

(qK)q/riot llflip = (qK) 1 + ( q/P' ) • IIcI Ilfllp.Now replacement of q by p'q (which requires that q > p — 1) yields

IgIP i4dx < (F'gK) 1 +gInl Ilfl1p'g.

In preparation for an expression involving an infinite series, substitute aninteger k, k > p — 1, for q to obtain

, k1^ i9Î A dx < p'K

kk ' ISZI

(K p' kL k ^llfllP (k 1). CP

cd(n — 1)Ift17


for any constant C > O. Consequently,

00 p' Te oo k , k

f ^ 1 f191 {{ dz < p'KI^ I ^ k Kp

k,ko k^ Cllflkp k —ko (k - 1 ) +. C131 )

where ko = [p]. The series on the right converges if CP' > elf); and thusthe result follows from (1.5.12) when applied to the terms involving k < ke,and the monotone convergence theorem.

By appealing to a different method, we will give another proof of expo-nential integrability that gives a slightly stronger result than the one justobtained.

2.9.2. Theorem. Let f E L"(R" ), spt f c B where B is a ball of radius R,and let p = n jot > 1. Then, for any e > O, there zs a constant C = C(c, n, p)such that

Tn/p ( f)(X ) pIlf llp

e ] dxC. n

exp^ un-1

(2.9.3)

Proof. Clearly, we may assume that 11f Ilp = 1. Then,

Ia(f)(s) - f(v)I x — y1a -ndy + f(y)lx — E/la-ndy

f (2,6) B— B(2.6)

where x E B and 0 < 6 < R. By Lemma 2.8.3(i), the first integral on theright is dominated by C6kM(f)(z). By Holder's inequality and the factthat 11f IIp = 1, the second integral on the right can be estimated as follows;if r = Ix - vi, then

R 1111;

f (âl)Iz — glla-"dy < w(n —1)ffB—B(x,6) 6

= [w(n - 1) log(R/6)1 10 ' .

ThusIIQ(f)(x)I C6°M( f )(z) + (w(n - 1) log(R/6)) 1

/n'.

If we chooseP = rnin(EC -1 [M( f )(r)] -1 , R° ),

then we have,

II„(f)(x)I < c +[w(n - 1) log + (^ -1/°`C` /° M(f)(x) 1 /°`)] 1/p a

or(Ia(f)(x) - e)

+D' w(n - 1 )n-1 log + (R"e -"C"M(f)(x) p )

2.9. Limiting Cases of the Sobolev Inequality 91

since ap = n. Because II f II p = 1, the conclusion now follows immediatelyfrom Theorem 2.8.2. D

2.9.3. Remark. Inequality (2.9.3) clearly implies that if A < n/w(n - 1),then there is a constant C = C(Q, n, p) such that

exp [°111fUlipXx)r dx < C (2.9.4)

thus recovering inequality (2.9.1).Although it is of independent mathematical interest to determine the

best possible constants in inequalities, in some applications the sharpnessof the constant can play a critical role.

The sharpness of the Sobolev imbedding theorem in the case of criticalindices has had many different approaches. For example, in [HMTI, it wasshown that the space W01 10) could not be embedded in the Orlicz spaceL,p (1l) where çp(t) = exp(Itln/(n -1) - 1). On the other hand, with thisSobolev space, it was shown by Moser IMOS1 that (2.9.4) remains valid for/3 = n/w(n - 1); that is, a can be taken to be zero in (2.9.3). Recently,Adams [AD8I has shown that (2.9.4) is valid for f3 = n/w(n - 1) with norestriction on a.

Theorems 2.9.1 and 2.9.2 give one version of a substitute for boundednessin the case ap = n. We now present a second version which was developedby Brezis and Wainger [BWI.

For this, recall the definition of the Bessel kernel, ga , introduced in (2.6.1)by means of its Fourier transform;

90 (x) = 27r -n/2 (1 + Ix 2)-a/2,

Also, recall that the space of Bessel potentials, L°•p(R"), is defined as allfunctions u such that u = ga * f where f E L 13 (R"). The norm in this spaceis defined as IIuIIa,p = II flip . Also, referring to Theorem 2.6.1, we have inthe case a is a positive integer, that this norm is equivalent to the Sobolevnorm of u.

For the development of the next result, we will assume that the readeris familiar with the fundamental properties of the Fourier transform.

2.9.4. Theorem. Let u E Lt.¢(R") with lq > n, 1 < q < co and letop = n, 1 < p < 00. If Ilulin.p < 1, then

IIu II^ <_ C [1 + log l /p1 (1 + IIuIIt ^^)^ • (2.9.5)

Proof. Because Co (R") is dense in 1,1 .9 (R") relative to its norm andalso in the topology induced by uniform convergence on compact sets, it issufficient to establish (2.9.5) for u E Co (R").


Let cp, n E C°°(R") be functions with spt tp compact, cp (27r)-n f2on some neighborhood of the origin, and Sp + iJ = (27r)_n/ 2 on R". Sinceu E Ca (R"), u may be written in terms of the inverse Fourier transforma$

u(s) = 1 =" .11û(092(âi1 R)dy + e'z

4u(^)rl(^/R)dl!

= ui(x) + u2(x), (2.9.6)

where R> 2 is a positive constant to be determined later.The proof will be divided into two parts. In Part 1, the following inequal-

ity will be established,^

Ilulll^ <_ C(log R) `^p

while in Part 2, it will be shown that

IIuaII^ < CR-44 II14

for some 5 > O. The conclusion of the theorem will then follow by taking

R max( 2 ,IIUII46 ).

Proof of Part 1. We proceed to estimate u 1 as follows:

u1(x) _ (21r)--n J2 e i= .1 (27r)"/ 2 0. IfI 2 )a,2 (Y1) (p(y1R) dy+ 11,I2)o/2

= f *xR(x)

wheref(y) = (27r)n/ 2 (1 + m 2 ) 0/2u(y)

and

KR (b) — ( 1 ç(y/R) + I1+I 2 )012

.

Note that u = go * f (see Section 2.6) and therefore

(2.9.7)

Ilflip — Ilulla,p <_ 1 .

Consequently, in order to establish Part 1 it will be sufficient to show that

II^R4Ip< ç C(iog R) l/p 1 , R > 2.

We now define a function L such that L = cp. Note that L is a rapidlydecreasing function and thus, in particular, L E L 1 (R") fl C°°(R"). Let

LR(x) = R"L(Rx).


From (2.9.7) we have that

( 2,r) -71/2 KR(y) = LR(y) • 9a(li)

and(27)-n/2KR = LR * 9a

Let B(R) be the ball of radius R centered at the origin. Define two functionsGâ and Gâ by

Gp(s) = 90 (x)XB(R- 1 ) (x)

Ga (x) = 9a(x) — Ga (z)'Then

(27 )-n/2KR(s) = LR * Gâ(s) + LR * G^(x).

An application of Young's inequality yields

HLR * GâIID' <_ IILRIIpf • IIGQII1 (2.9.8)

and it is easily verified that

IILRIfn, ; CrIp

while from (2.6.3) and ap = n, it follows that

IIGQ 111 <_ CR - n/P.

Similarly, from (2.6.3) we see that

9a(x) Ç C1lxI°`- ne -Czlsk

and therefore

ÎrGkirp' < C ^,IxI^^-nlp' dx1 /R<Isj<1

1/P

+ C11<lx1<co

e—C' !TIP' dx

< C (log R)'/p' + C< C(log R)IIp'

since R > 2. Hence,

IILR * GkIIp' < C(logR)1/721 (2.9.9)

because

)lip'

IILRII'1= IILIIi = C < oo.


Thus, from (2.9.8), and (2.9.9) we have

IIKRIIP' < C(log R) 1 /P', (2.9.10)

thereby establishing Part 1.

Proof of Part 2. We write u 2 as follows:

u2(x) = (27 ) -n/2 eiXil û(11)(27r) n/2 ( 1 + Iy12)t/ ^ I I)"2 chi

(1 + lyl)

g * KR(x) (2.9.11)

(2.9.12)

and'4(v) = ü(v)(270n / 2 ( 1 + Iu1 2 ) e/2 .

By assumption, u E Lt.g(Rn ), and therefore it follows from definition thatg E LQ(R") with u = ye * g. In order to establish Part 2, it suffices to showthat

IIKRI^Q, < CR -6 for some ë > O.

First, consider the case q = 1. Since eq > n by assumption, we havel > n. Now write

IKR(x)I =If ei" g(y/Rd

<Irl(y/R)Idb

(1 +10 2 ) 1/ 2 1 — IyIRecall that t vanishes in some neighborhood of the origin, say for all y suchthat 'y1 < ER. Thus, for all x,

I KR(x )l I 77(yIR)1dyR"—B(Q,eR) M

00< C r"-^-edr

E R< CR"

since I > n. Thus, Part 2 is established if q = 1.Now consider q > 1, so that q' < oo and without loss of generality, let

l G n. Since cp + q = (27r) -"/2 on Rn, (2.9.12) can be written as

(21r) ni2 KR(y) — 140(YiR ) I2

(1+1y17)t 2 ( 1 + ICI )Thus, we have

whereKR(y) _

^?(y/R) (1 + 11/I2) e/2

KR(x) = ge(x) — ge * LR(x) (2.9.13)


whereLR(x) = RT L(Rx), L(y) = 99 (y)•

We can rewrite (2.9.13) as

KR(x) = f [Mx) - 9i(x - y)1LR(y)dy (2.9.14)

because(270 —n/2 = cp(0) = (2 7 )—n/2 e — io.yL( y)dy•

Rn

To estimate f 19t (x -- y) - gt (z) ri dz we write

f i9t(x — y) - 9t(x)^Q^dx =

I9t(x - y) - 9c(x)r" dxn 1ri521v1

+I9t(x- y) -gt(x)f9^dxIrl>?I41

=---I1+I2.

Now

Î l < C,

1I )x1521v1 frI <2IyI

<C 19t(x-y)J (Ix +1gt(x)1Qdxfix- til< 3 1Yr j=15 2 1y/

< CIyjit-n)q'+n,

by (2.6.3). To estimate I2 , note that gt is smooth away from the origin,and therefore we may write Igt(x - y) — 9t(x)1 < ID gt (z)1 - iyl where z =t(x - y) + (1 - t)x = x - ty for some t E [0, 11. Since, Izt ? 1/(2x1) when1x1 > 2114 we have, with the help of (2.6.4),

I9t(x - y) - 9t(x)IQ'dx < C P-clz11x1(t-n-" Iyllrdx

< IvV"e -c lzllxl (t- n -r)Q'dxR "

< CIy14 .

Consequently, combining the estimates for I I and I2 , we have

II9t(x - y) — 9t(x)I1q' < CIyIS (2.9.15)

where ë = [(t + 1 - n)q' + nJ/q' > O. Referring to (2.9.14), we estimate[IKRIIq' with the aid of Minkowski's inequality and (2.9.15) as follows:

1 /q' [ /g'

IKR(x)IQ'dx <_ 1 I91(x — y) — gt(x )1 a'dx ILR(y)Idy


Cf ILR(9)t 19I 6 dy

= CR" 6

L(Ry)I 1R91 a d9

< CR -6 .

The integral f IL(z)I Izl 6dz is finite because L = cp and thus L is rapidlydecreasing. The proof of Part 2 is now complete and the combination ofParts 1 and 2 completes the proof of the theorem. ❑

2.10 Lorentz Spaces, A Slight Improvement

In this section we turn to the subject of Lorentz spaces which was intro-duced in Chapter 1, Section 8. We will show that the Sobolev inequalityfor Riesz potentials (Theorem 2.8.4) as well as the development in Chapter2, Section 9, can be improved by considering Lorentz spaces instead of LPspaces.

We begin by proving a result that is similar to Young's inequality forconvolut ions.

2.10.1. Theorem. If h = f * g, where

f € L(P1, 4i), g E L(P2, 42),

then h E L(r, a) where

and 1 + 1 >1,Pi P2

1 1 1—+--1= -P1 P2 r

and s > 1 is any number such that

1 1 1— + ____ > ,41 42 s

Moreover,II h II( r ,d) < 3TIIf IIc7)„g,)119 11(p2,q2 ).

Proof. Let us suppose that q 1 , q2 , s are all different from oc. Then, byLemma 1.8.9,

(1 I hÎ(r, ․ )) ° = o

00 OD

(-0

°° s

f*Z ^^r OD

:(t)g" (t)dtF drx l i

_ ^

[-d--1-r- y f” 1 g" ^

d^d f1.1(2.10.1)— o

o u u u .

2.10. Lorentz Spaces, A Slight Improvement 97

The last equality is by the change of variables x = 1/y, t 1/u. Now useHardy's inequality (Lemma 1.8.11) to obtain,

°° f 1 y ( 1) .. 1 du ' dy00 Y t /r f"g (;) ‘7 y

< r'oo [y '_('/r) f(h/y)g(h/y) I '

dY.o Y 2

y '

00' r

(xt+(1/r) {..(^1^..^x`1^sd2

o J 1 x '

by letting y = 1/x.Since s/q l + 5/q2 > 1, we may find positive numbers m t , m2 such that

1 11 and 1 < s 1 sm1 m2 m1 qt m2 q2

Therefore q i Ç sm li qz G sm 2 . An application of Hdlder's inequality withindices rn li na g , yields

(IIhII(r,a))a < rsop [x!/Pi f•*(x)Ta jT1/pzg*s(x)]a

dxU

xl/m' J ll xi/m2

/ i< re

oo

[x1/Pi f■slx)Jsm^

dx m

0 J z

d. [Zoo i

(x/pz 4 ..(x )^s.n 2 _

II= r" ( IIf Il(Pi ,am►))'(II9II(P,,.mz) )s.

Thus, by Lemma 1.8.13

IIhII(r,a) < rllf Ilepr ,amAgIIp,,amz)< e 1 /e e l /erllf II(p l , g , ) Ilsll(p,,0 )<_ 3r. IIf 11(p, .g1)IIgIl(p2,42)•

Similar reasoning leads to the desired result in case one or more of (it,42, 8 are oo. ❑

As an application of Theorem 2.10.1, consider the kernel 4,(x) = 1xia--nwhich is a constant multiple of the Riesz kernel that was introduced inChapter 2, Section 6. For simplicity of notation in this discussion, we omitthe constant 7(a) -1 that appears in the definition of /„,(x). Observe thatthe distribution function of l a is given by

ara (t) = I{x : Ixla-n > t}I= (Ix : IxI < t1/(a-n)}1

= a(n)tre /(Q-n)


and because I is the inverse of the distribution function, we have 1V() =( t ){a—n)/n. It follows immediately from definition thatWC)n)

I:* (t) = n t(a—n)/n )

and therefore that I. E L(n/(n - at), oo). If we form the convolution la * fwhere f E LP = L(p,p), then Theorem 2.10.1 states that

la * f E L(q,p)

where1 1 n- a 1 oc-_- + - 1 =---.q P n p n

Moreover, it follows from Lemma 1.8.13 that L(q,p) C L(q, q) and thus wehave an improvement of Theorem 2.8.4 which allows us to conclude onlythat fa * f E L4 . As a consequence of Theorem 2.10.1, we have the followingresult that is analogous to Theorem 2.8.4.

2.10.2. Theorem. If f E L(p,q) and 0 < a < n/p, then

Ia * f E L(r,q)

and

Ilia * f ll(r, g ) Illall(n/(n—a),00) IIf II(p,g)=GII fII (p,g)

where1 1 ar - p n .

We now consider the limiting case of 1 /p i +1/p2 = 1 in Theorem 2.10.1.In preparation for this, we first need the following lemma.

2.10.3. Lemma. Lei cp be a measurable function defined on (0, 1) suchthat tcp(t) E Lp(0, 1; dt/t), p > 1. Then,

1II( 1 + I log ti) -1

: 40(8)616{0,1,d1/0<_

p—1)— 1 IIt9(t)IILp(o,1;dt/t)•

Proof. By standard limit procedures, we may assume without loss of gen-erality that w E L 1 (0,1) is non-negative and bounded. Let

1 ^ 1 (1 + I logtl)!p cp(s)ds p

o (J 1 i


Since

fi (j ' co(s)ds) d(1= 1 -- log t)—p+'

integration by parts yields

j1

= cp(s)ds (1 — log t}P t

and Holder's inequality implies1

I < p p l II( 1 +I logtl) -' t ^P( 8 )d81I Ln ô ,i;dt t)1I t^(t)11^^(o,i;dt/t) ,

from which the conclusion follows. ❑

2.10.4. Remark. Before proving the next theorem, let us recall the fol-lowing elementary proof concerning convolutions. If f,g E L 1 (R") we mayconclude that

R" j" I f (r — g)g(y)I dxdy = R" Ig(g)1 - If(x , y )ldidy R"

=R" 1001 lIfllidy

= IIIlli llglli < oo•

Thus, the mapping y — f (x — y)g(y) E L 1 (R") for almost all x E R" andf * g E L 1 (Rn).

In the event that one of the functions, say f, is assumed only to be anelement of L(p, q), p > 1, q > 1, while g E L 1 (R"), then the convolutionneed not belong to L 1 (R"), but it will at least be defined. To see this, let

1 if f (z) > 1fi (x) = If (x) if — 1 < f (x) < 1

—1 if f(x) < —1

and let 12 = f — f 1 . Then f 1 * g is defined because f ' is bounded. Wewill now show that 12 E L' (R") thus implying that 12 * g is defined andtherefore, similarly for f * g. In order to see that 12 = L' (R") let

f2(x) = x) i If (x)I > 10f(iff j f (x) f < 1.

Clearly ah (a) = ar f (1) if 0 < s < 1 and ci f2 (s) = a f (8) if a > 1. Conse-quently

f2 (af(1)) = inf{.g : a-2 (s) < a1(1)} _ O.

Thus, since A. is non-increasing, f2 (t) vanishes for all t > a f (1) and itis easy to see that r (t) = JO) for all t < a f(1). We may assume


that q > p for, if q < p, then Lemmas 1.8.13 and 1.8.10 imply thatf E L(p, q) C 14,p). Consequently, by Young's inequality for convolu-tions, j * g is defined. In fact, f * g E L". With q > p and a = a f(1), wehave

°o f#.(t)q °° f.(t)g f`(t)Q 00 > fl

o -ti (g/p) (Ît > 0tl- (Q/p)dt > fa t1-(g/p) ta

> a(g(P)-1 f*(t}Qdt0

a

= a (9/p )-2 12 (t)Q dt0

00= a (Q /p )-1 (t)gdt

0

where i?, E [0, a]. Since j2 (t) vanishes for t > a whereas f2 (t) > 1 for t < a,it therefore follows that

f j2(t)dt < 0 /(t)dt

thus showing that /2 is integrable because /2 and f2 have the same distri-bution function. Therefore 12 is integrable.

2.10.5. Lemma. Let 1 < p < oo, 1 < qi < oo, 1 < qz < oo be suchthat 1/q i + 1/q2 < 1 and set 1/r = 1/q1 + 1/q2. Assume j E L(p, q 1 ) andg E (p',q2) fl L 1 (1V) and let u = f * g. Then

.. ( t) T dt t/T

I_ i + I log II j t — cIIfIlcp.gs) • (11911(pi.o) 19111)

where C depends only on p, q 1 , and q2.

Proof. Note from the preceeding remark, that u is defined. For simplicitywe set II f II(p,q,) and III! = Ilgli(pi . )+IIgll,. Also, for notational conveniencein this discussion, we will insert a factor of (q/p) 1 /4 in the definition of theIII II (p,q) ; thus,

1/q o0 (t)P o

1 !q9 sup tl/p f #* (t),--)P t >o

1 <p<oo,0<q <oo

1 <p<oo,q=oo.

We distinguish two cases:

(i) r<co

001


(ii) r = oo (i.e. q 1 = q2 = oo).

(i) The case r Coo. Recall from Lemma 1.8.8 that for every t > 0,00

u` • (t) < t f"(t)g"(t) +f'(s)g'(s)ds.f

Clearly, the following inequalities hold for every a > 0'

r(s) < f"(s) <_ s /P Cif II(p,g i)>

g'(s) < s'"(s) < szl/p, Ifsl1(P',g2)?

9 • (8) s g•` ( s ) <_ 9119f1 1.

For t < 1, it follows from (2.10.2), (2.10.3), and (2.10.4) that,

1 1 1u'`(t) <t t %p t1/p^ CCf1i( p,as )IIgI1( p)g2 ) + fi(s)g•(s)ds

e

(2.10.2)

(2.10.3)

(2.10.4)

(2.10.5)

00+f"(s)g`(s)ds. (2.10.6)f

From (2.10.3) and (2.10.5) we have that

f `(s)g•(s)d8 < P11Ifil(pm) - II9II1)-

This in conjunction with (2.10.6) yields1

u'•(t) < P1If 11 II9Il + f * (s)9 M (s)ds. (2.10.7)r

By Lemma 2.10.3, (2.10.7), and (2.10.4) we have

11( 1 + 1 log t1) - 1 u" • (t)IIL' (o,l;dt/t) <_ 01/11 11911 + cliff•(t)s'(t)IIL-(0.1;dt/t)cliff! 11911 + CiIt 1 /P f+(t)t 1 /P'

9 6 ( t)IILr(0,r ;dt/t)C11111 11911 + C111 II(P,q,)119II(p',92).

(ii) The case r o). By (2.10.7), (2.10.3), and (2.10.4) we have, for t < 1,

u" ( 1 ) <_ p11111 11911 + 1 log tl IIII1(P,00)Il9ll(p',00)and therefore

11(1 + I log tI) - ' u•' (t)IiL—(0,1) <_ Cilf i1 il9il

2.10.6. Theorem. Let 1 0.

(ii) if r < oo, there exists positive numbers C = C(p, q 1 , 42) and M =M(I(2I) such that

ec lui rg dx < M

for every f and g with Ilf II (p,gi) I and 11911(^ .q, ) + II9II1 <_ 1 .

Before proceeding with the proof, let us see how this result extends theanalogous one established in Theorem 2.9.1. To make the comparison, takeone of the functions in the above statement of the theorem, say f, asthe Riesz kernel, In/p . As we have seen from the discussion preceedingTheorem 2.10.2, f E L(p', oo). The other function g is assumed to bean element of LP(11) where S-2 is a bounded set. Thus, by Lemma 1.8.10,g E L(p, p) fl L 1 (Rn ). In this context, q 1 = oo and q3 = p > 1 thus provingthat this result extends Theorem 2.9.1.

Proof. Consider part (1) first. Because u' (t) is non-increasing we have that

Iu'(r)I'' t (1 - log s } - r d9 1and Lemma 2.10.5 implies that 1(t) -, 0 as t -' 0 where 1(t) denotes thesecond integral. Note that there exists a constant K = K(r) such that

l u'(t) I*' < K(1 + I logtl)I(t)I/(r-1) 0 < t < 1. (2.10.8)

With Q C Rn any bounded measurable set, we have with the help ofLemma 1.5.1,

i 41exp(CIu(x)I)r ^ f

=l eXp(Clu

, (t)I^T 'dt = to exptClu'(t)I) r dt

0

IQI+ exp(CIu'(t)I)T / dtJ o

(2.10.9)

where 0 < to < IQI. Because u* is non-increasing, it is only necessary toshow that the first integral is finite. For this purpose, choose t o < 1 so thatCr' KI(to ) 1 /(r -1 ) < 1. Then, from (2.10.8),

exp(Clu`(t)I) r < (e/t)Q

where a = C'' i KI{to } 1 /(r -1 ) Thus, part (i) of the theorem is established.

J

Exercises 103

For the proof of (ii), Lemma 2.10.5 and the fact that u0 (t) is non-increasing allow us to conclude that for 0 < t <1,

ile(t)I r j ( 1 _log )

- Gill IV • 11911 *

where Il f II = Ill 11(p,Q,) and 11911 =1191642) + bill. Therefore,

Iu'Mir' < K(1 — log t)IIfPIr'11911r#-Similar to part (i), the proof of (ii) now follows from (2.10.9) by choosingKC<1. ❑

Exercises

2.1. Prove that W' (R") = W:'p(R").

2.2. If f and g are integrable functions defined on R" such that

fcPdx = f gcodx

for every function cp € Co (R"), prove that f = g almost everywhereon R".

2.3. Prove the following extension of the Rellich-Kondrachov compactnesstheorem. If St is a domain having the extension property, then

Wk+rn,p(1) Wk,q(0)

is a compact imbedding if mp < n, 1non-negative integer.

<q<np/(n—tnp) and ma

2.4. Verify the following equivalent formulation of Bessel capacity:

Ba.p(E) _ z

inf EE

9a * f(z)} -P

{ suPf Q

EE

where f E IP(R'a), f > 0, and IIf 11p ? 1

* f(2)

... p

2.5. Prove that the Riesz and Bessels capacities have the same null sets;that is, RQ , p (E) = 0 if and only if BQ , p (E) = 0 for every set E C R„_


2.6. Show that there is a constant C = C(a, p, n) such that

Ba,p(E) > C

provided cep > n and E is non-empty.

2.7. As an extension of Corollary 2.1.9, prove that if 1 is connected andu E Wk ,P(11) has the property that Dau = 0 almost everywhere onn, for all lal = k, then u is a polynomial of degree at most k - 1,

2.8. Let 1 < p, kp < n. If K C R" is a compact set, let

yk,p(K) = ; u > 1 on a neighborhood of K,

u E Cfl (R"))

With the aid of Theorem 2.6.1, prove that there exists a constantC = C(p, n) such that

C-'Bk, (K) < l'k,p( K ) < CBk,p(K) .

2.9. Show that for each compact set K C R",

inf IDurdx . u > 1 on K, u E Co (R") = 4.Rn

2.10. Prove that there exists a sequence of piecewise linear maps

Lk: R" -. R I

satisfying (2.7.7) and (2.7.8). See the discussion in Exercise 5.2.

2.11. Suppose that u: R" -0 R' is Liptichitz with pull ov► < co. Defineau : R ' -' R' by a u (t) = f {x : u(x) > t}1. Since au is non-increasing,it is differentiable almost everywhere.

(a) Prove that for almost all t,

-otu(t) >u

IDur l dH"-' .(t)

(b) Prove that equality holds in (a) if

[ix . Du(x)=0}I=0.

2.12. Theorem 2.8.4 gives the potential theoretic version of Theorem 2.4.2,but observe that the latter is true for p = 1 whereas the former isfalse in this case. To see this, choose f ; > 0 with fir, f,dx - 1 and

Exercises 105

spt fi C B(0, 1/i), Prove that Ia * f; la uniformly on I?" — B(0, r)for every r > 0. Thus conclude that

(lQ * fi)"/(n- ^ ) dx > (fa* fi )"/ ( n -Q)

f n f(0,r)

The right side tends to

( fa) n /(n -a)dx = C I:C} -ndx.t (O,r) B(o,r)

ButI x rdx — oo as r 0.

B(0,r)

2.13. Show that. Theorem 2.8.4 is false when ap = n. For this consider

o( 1 +E)/n

f (x) = ^zI ^1og 51) !x^ < 1

o 'xi >

where e > O. Then f E L" since ap = n but /0 * f (0) = oo whenevera(1 + e)/n < 1.

2.14. Prove the following extension of Theorem 2.8.2. Suppose Ill log(2 +if i) is integrable over the unit ball B. Then M f E L i (B). To provethis, note that (with the notation of Theorem 2.8.2)

1 0oM < 1B} +Mdx < ^B^ + fAtldt +

A 1^

Now use (2.8.3) and Exercise 1.3.

2.15. There is a variety of methods available to treat Theorem 2,6.16. Hereis one that shows that B 1 ,p (K) = 0 if Nn -p < oo, 1 0(ii) spt u C U

(iii) KcV C {x:u(x)=1)

(iv) fir, ÎDui dx < C.


To prove Step 2, first observe that HS -P(K) < oo (see Exercise1.10). Since K is compact, there exists a finite sequence of open halls{B(ri )}, i = 1, 2, ... , m, such that

m rn

KC B(r i ) C B(2r,) C Ui = l i=1

and00

L a(n - p)ri -p < HS -P(K) + 1.i= i

Let V = Um 1 B(r i ) and define u; to be that piecewise linear functionsuch that ui=1o ❑ B(r,),u i =Oon R"- B(2r;). Let

u=max{u,.a=1,2,..., m}

to establish Step 2.STEP 3. For each positive integer k, let

Uk— x: d(x,K)< 1 .k

Employ Step 2 to find corresponding uk such that

IDukI dx < C

for k =1,2, ....STEP 4. Use Theorem 2.5.1 to find a subsequence {uk} and u EWo'P(R") such that uk -' u weakly in Wo'p(Rn) and uk -i u stronglyin LP. Hence, conclude that u = 1 almost everywhere on K and thatu=OonR" -K.STEP 5. Conclude from Theorem 2.1.4 that IKI = 0 and thereforethat u=0.STEP 6. Use the Banach-Saks theorem to find a subsequence {uk)such that

1 '

tlj = E ukk =1

converges strongly to u in WW' p (R"). Thus, IIDv;IIp - 0 as j --• oo.But

71,P(K) ÇR IDvJI pdZ"

for each j = 1, 2, ....

Exercises 107

2.16. In this problem we sketch a proof of the fact that y i , p (K) = 0 implies11"-p+e(K) _ 0 whenever c > 0. The proof requires some elementaryresults found in subsequent chapters.

STEP 1. For each positive integer i, there exists u, E Co (Rn) suchthat u; > t on a neighborhood of K and

ID dx < 2'I .

R"

Let v = Ei ° 1 u, and conclude that v E W L 7 (R" ). Also Mote thatK C interior {x : v(x) > k) whenever k > 1. Therefore, for r E K,

lim inf v(x, r) = o0r-.0

where

v(x, r) = v(y)dy.B(s,r)

STEP 2. For all x E K and e>0,

lim sup rp - T ! IDviPdy = oo.r-.0 .l B(x,r)

If this were not true, there would exist k < co such that

rp -n -E IDv1Pdy < k

for all small r > 0. For all such r, it follows from a classical versionof the Poincaré inequality (Theorem 4.2.2) that

Iv(U) — v(t,r)Ipdy < Crp- n 1DvIpdy < Cr`.113(.,r) B(s,r)

Thus conclude that

17(x, r/2) — v(x, r)j < CrEIP

for all small r > 0. Therefore, the sequence {1(z, 1/2 3 )} has a finitelimit, contradicting the conclusion of Step 1.

STEP 3. Use Lemma 3.2.1 to reach the desired conclusion.

2.17. At the end of Section 2.3 we refer to [AR2) for the result that C°° (SZ)is dense in W k,P(SZ) provided 0 possesses the segment property. Provethis result directly if the boundary of 0 can be locally represented asthe graph of a Lipschitz function.

2.18. Show that C°. 1 (Q) = W''°°(l) whenever 11 is a domain in R.


2.19. This problem addresses the issue raised in (2.6.6). If f E LP (R") andap < n, then Theorem 2.8.4 states that

Ilia *file <_ Cilfllpwhere q = p'. A simple homogeneity argument shows that in orderfor this inequality to hold for all f E D' , it is necessary for q = p`.For 6 > 0, let Tb f (x) = f (6x). Then

llâ * (T6f)II4 ^^"^pllf^lPand

* ( Tbf )— ra Td (ta* f )'

Hence,llia * (Tbf)IIg - a -ab- n /g }4ia * f Ilgi

thus requiring1 1 aQ P n

Historical Notes2.1. It is customary to refer to the spaces of weakly differentiable func-tions as Sobolev spaces, although various notions of weak differentiabilitywere used before Sobolev's work, [SO2]; see also [S01], [S03]. Beppo Leviin 1906 and Tonelli [TO] both used the class of functions that are abso-lutely continuous on almost all lines parallel to the coordinate axes, theproperty that essentially characterizes Sobolev functions (Theorem 2.1.4).Along with Sobolev, Catkin [CA] and Morrey [MOI] developed many ofthe properties of Sobolev functions that are used today. Although manyauthors contributed to the theory of Sobolev spaces, special note should bemade of the efforts of Aronszajn and Smith, [ARS1], [ARS2J, who made adetailed study of the pointwise behavior of Sobolev functions through theirinvestigations of Bessel potentials.

2.2. Theorem 2.2.1 was originally proved by Rademacher [RA]. The proofthat is given is attributed to C.B. Morrey [MO1, Theorem 3.1.6]; the proofwe give appears in [SI. In our development, Rademacher's theorem wasused to show that Sobolev functions remain invariant under compositionwith bi-Lipschitzian transformations. However, it is possible to obtain astronger result by using different techniques as shown in [Z3]. SupposeT : R" -i R" is a bi-measurable homeomorphism with the property thatit and its inverse are in Wio'' (R", R" ), p > n - 1. If u E W1 f (R") wherep` = p[p - (n - 1)] -1 , then u o T E W101 : (R"). With this it is possible toshow that if u E Wiô: (R") and T is a K-quasiconformal mapping, thenuoTE W1 (R").

Historical Notes 109

2.3. Theorem 2.3.2 is due to Meyers and Serrin, [MSE].

2.4. Theorem 2.4.1 is the classical Sobolev inequality [SO1], [SO2], whichwas also developed by Gagliardo [GA1], Morrey IMO1], and Nirenberg[NI2]. The proof of Theorem 2.4.1 for the case p < n is due to Nirenberg[NI2).

2.5. Theorem 2.5.1 originated in a paper by Rellich [RE] in the case p =2 and by Kondrachov [KN] in the general case. Generally, compactnesstheorems are of importance in analysis, but this one is of fundamentalimportance, especially in the calculus of variations and partial differentialequations. There are variations of the Rellich-Kondrachov result that yielda slightly stronger conclusion. For example, we have the following resultdue to Frehse [FRE]: Let 1 C R" be a bounded domain and supposeu ; E W' P(c1), 1 < p < n, is a bounded sequence of functions with theproperty that for each i = 1, 2, ...,

IDu i IF -2 Du; dx < be Wilco

for all cp E W 1 ,'(1l) n L°°(1). Then there exist u E W "P(Il) and a subse-quence such that u ; -, u strongly in W"'(0), whenever q < p.

2.6. Potential theory is an area of mathematics whose origins can be tracedto the 18th century when Lagrange in 1773 noted that gravitational forcesderive from a function. This function was labeled a potential function byGreen in 1828 and simply a potential by Gauss in 1840. In 1782 Laplaceshowed that in a mass free region, this function satisfies what is now knownas Laplace's equation. The fundamental principles of this theory were de-veloped during the 19th century through the efforts of Gauss, Dirichlet,Riemann, Schwarz, Poincaré, Kellogg, and many others, and they consti-tute today classical potential theory. Much of the theory is directed to theunderstanding of boundary value problems for the Laplace operator and itslinear counterparts. With the work of H. Cartan [CAR1], [CAR2], in theearly 1940s, began an important new phase in the development of potentialtheory with an approach based on a Hilbert-space structure of sets of mea-sures of finite energy. Later, J. Deny [DE] enriched the theory further withthe concepts and techniques of distributions. At about the same time, po-tential theory and a general theory of capacities were being developed fromthe point of view of an abstract structure based on a set of fundamentalaxioms. Among those who made many contributions in this direction wereBrelot [BRT], Choquet [CH], Deny, Hervé, Ninorniya, and Ohtsuka. Theabstract theory of capacities is compatible with the recent development ofcapacities associated with non-linear potential theory which, among otherapplications, is used to study questions related to non-linear partial differ-ential equations. The first comprehensive treatment of non-linear potentialtheory and its associated Bessel capacity was developed by Meyers [ME1],


Havin and Maz'ya NM], and Resetnjak [RES). Most of the material inSection 2.6 has been adopted from IME1).

2.7. The co-area formula as stated in Theorem 2.7.3 was proved by Fed-erer in [F11. In the case m 2, k = 1, Kronrod [KR] used the right side of(2.7.1) to define the variation of a function of two variables. Fleming andRishel [FR] established a version of Theorem 2.7.1 for 13V functions. An-other version resembling the statement in Theorem 2.7.3 for BV functionsappears in [F4, Section 4.5.9].

The proof of the best possible constant in the Sobolev inequality (The-orem 2.7.4) is due to Fleming and Federer [FF]. Their result can be statedas follows:

Ilulln/(n-1)where the supremum is taken over all u E Co (Rn). Talenti [TAI extendedthis result to the case p > 1 by determining the constant C(n, p) definedby

C(n,p) = sup IlDullpIIuIIp.

He showed that

1—(1/P)1/n

C(n,p) = r -1/2n -1/9 P - 1 1,( 1 + pr(n)

n-p

{ r()r(1+n- p)

He also showed that if the supremum is taken over all functions whichdecay rapidly at infinity, the function u that attains the supremum in thedefinition of C(n, p) is of the form

u(x) = (a+blzyPf ( P-where a and b are positive constants. This leads to the following obser-vation: in view of the form of the extremal function, it follows that if S2is a bounded domain and if u E W "(1) has compact support, then byextending u to be zero outside of St, we have

IIuIIp• < C(n , P)IIDuIIP.

Brezis and Lieb [BL] provide a lower bound for the difference of the twosides of this inequality for p = 2. They show that there is a constant C(S1, n)such that

C(O, n)IIull,00) +1114 -< C(n,p)IIDullâwhere q = n/(n - 2) and IIuII(q,00) denotes the weak Lq-norm of u (seeDefinition 1.8.6).2.8. The maximal theorem 2.8.2 was initially proved by Hardy-Littlewood[HLI for n = 1 and for arbitrary n by Wiener [WI]. The proofs of Theorem2.8.4 and its preceding lemma are due to Hedberg [HE1].

na(n) = sup flDull

1))1—nfP


2.9. The proof of exponential integrability in Theorem 2.9.1 is taken fromIGTI while the improved version that appears in Theorem 2.9.2 was provedby Hedberg [11E1]. The question concerning sharpness of this inequality hasan interesting history. Trudinger (TR1 proved (2.9.1) for Sobolev functionsin W k .P, nip = k, with the power p' replaced by n'. However, when nip > 1,Strichartz [STR1 noted that Trudinger's result could be improved with theappearance of the larger power p'. The reason why Trudinger's proof didnot obtain the optimal power is that the case of k > 1 was reduced to thecase of k = 1 by using the result that if u E Wk ") , k > 2, kp = n, thenu E W". However, in this reduction argument, some information is lostbecause if u E Wk •e then u is actually in a better space than W l in. In fact,by appealing to Theorem 2.10.3, we find that the first derivatives are in theLorentz space L(n, p) C L". This motivated Brezis and Wainger to pursuethe matter further in [BM where Theorem 2.9.4 and other interestingresults are proved. The sharpness of the Sobolev imbedding theorem in thecase of critical indices was also considered in [HMT1, where it was shownthat the space W(l) could not be imbedded in the Orlicz space L,(1)where çp(t) = exp(Iiln/(R -1 ) — 1).

The other question of sharpness of the inequality pertains to the constant/3 that appears in (2.9.4). It was shown in [MOST that (2.9.4) remains validfor fi = n/w(n — 1) in the case of Sobolev functions that vanish on theboundary of a domain. The optimal result has recently been proved byAdams [ADS] where (2.9.4) has been established for Q = n/w(n — 1) andall a > O.

2.10. Most of the material in this section was developed by Brezis andWainger [BWI although Theorems 2.10.1 and 2.10.2 and due to O'Neil [O1.

3

Pointwise Behavior ofSobolev FonctionsIn this chapter the pointwise behavior of Sobolev functions is investigated.Since the definition of a function u E Wk.P(S2) requires that the kth-orderdistributional derivatives of u belong to (0), it is therefore natural toinquire whether the function u possesses some type of regularity (smooth-ness) in the classical sense. The main purpose of this chapter is to showthat this question can be answered in the affirmative if interpreted ap-propriately. Although it is evident that Sobolev functions do not possesssmoothness properties in the usual classical sense, it will be shown that ifu E W k 'P(Rn), then u has derivatives of order k when computed in themetric induced by the LP-norm. That is, it will be shown for all pointsx in the complement of some exceptional set, there is a polynomial Px ofdegree k such that the L"-norm of the integral average of the remainderlu -- Ps I over a ball B(x, r) is o(rk ). Of course, if u were of class Ck , thenthe LP-norm could be replaced by the sup norm.

We will also investigate to what extent the converse of this statement istrue. To this end, it will be shown that if u has derivatives of order k in the17-sense at all points in an open set SI, and if the derivatives are in IP(Sl),then u E Wk1P(0). This is analogous to the classical fact that if a functionu defined on a bounded interval is differentiable at each point and if u' isintegrable, then u is absolutely continuous. In order to further pursue thequestion of regularity, it will be established that u can be approximated in astrong sense by functions of class CI, I < k. The approximants will have theproperty that they are close to u in the Sobolev norm and that they agreepointwise with u on large sets. That is, the sets on which they do not agreewill have small capacity, thus establishing a Lusin-type approximation forSobolev functions.

3.1 Limits of Integral Averages ofSobolev Functions

In this and the next two sections, it will be shown that a Sobolev functionu E Wk iP(Sï) can be defined everywhere, except for a set of capacity zero,in terms of its integral averages. This result is analogous to the one that

3.1. Limits of Integral Averages of Soholev Functions 113

holds for integrable functions, namely, if u E L 1 , then

(u(y) - u(x)(dy -, 0 as r -, 0B(a,r)

for almost all x E Rn. Since our result deals with Sobolev functions, theproof obviously will require knowledge of the behavior of the partial deriva-tives of u. The development we present here is neither the most efficient norelegant. These qualities have been sacrificed in order give a presentationthat is essentially self-contained and clearly demonstrates the critical roleplayed by the gradient of u in order to establish the main result, Theorem3.3.3. Later, in Section 3.10, we will return to the subject of Lebesgue pointsand prove a result (Theorem 3.10.2) that extends Theorem 3.3.3. Its proofwill employ the representation of Sobolev functions as Bessel potentials(Theorem 2.6.1) and the Hardy-Littlewood maximal theorem (Theorem2.8.2).

In this first section, it will be shown that the limit of integral averagesof Sobolev functions exist at all points except possibly for a set of capacityzero. We begin by proving a lemma that relates the integral average of uover two concentric balls in terms of the integral of the gradient.

3.1.1. Lemma. Let u E WLp[B(xo,r)1, p > 1, where xo E R" and r > 0.Let 0 < 6 < r. Then

r — "1

u(i1)dy—b—"U(11)6111= nr

—n

B(so.r) L06) fB (zo ,r)[Du(11)' (11- xo)1d11

- -1 6

-n [Du() • (y - zo)1dyn fB( 0 ,6) '

1

n J (so,r)— B(so,a)

Proof. Define p on R 1 by

(3.1.1)

{ oTh_pt 6µ(t) = t—n — p-n b < t< r

0 t > r.

Define a vector field V by V(y) = µ(1y - xoI)(FI - z0). Since u is thestrong limit of smooth functions defined on B(zo, r) (Theorem 2.3.2), anapplication of the Gauss-Green theorem implies

u(v)div v(v)dv = -Du(g) • (y - ro)p(Iy = xoI)dy. (3.1.2)J (sp,r) B (xD,r)

11/ — zo1"[Du(Y) • (y - xo)1d1l.

An easy calculation of div V establishes equation (3.1.1). ❑

114 3. Pointwise Behavior of Sobolev Functions

3.1.2. Lemma. Let t be a positive real number such that Ip < n, p > 1,and let u E W l 'p(Rn). Then

Iv — X1 1-71 11 (014 5 R Iv -- x1 1-71+1 IDu(y) Idy (3.1.3)JR .n n

for all x E Rn.

Proof. (i) We suppose first of all that u vanishes outside a hounded set.Let z E IV and for each positive integer j, define a C °° vector field Vl onRn by

1 (1/2)(t -n)V, (y) = -[ + I Y _xI2] (y x)•

Since Iul E W i rP(R°) (by Corollary 2.1.8), lul is therefore the strong limitof smooth functions with compact support. Therefore, by the Gauss-Greentheorem,

jRn div V;(y)lu(y)Idy JR n= rV(y)' D(IuI)(3►)^dy.

Moreover, since ID(ful)I = IDul a.e.,

divV; (y)lu(y)Idy <R V,(y)IDu(y)Idy. (3.1.4)

Rn n

By calculating the divergence on the left-hand side of (3.1.4) one obtains

1 (1/2)(t-n--2) nL J + ly — z1 2

[ fly — 1 1 2 + -jlu(y)Idyn

(1/2)(t—n)< 13 + ly — x 2 I 1y — lI IDu(y)Idy•

RR

The inequality (3.1.3) now follows, in this case, when j oc.(ii) The general case. Let ri be a C°° function on R, such that 0 < n < 1,

rt(t) = 1 when t < 1 and 17(t) = 0 when t > 2. Define

ui(Y) = u(y)i,(.l' -1 IyI)

for y E R''. By applying (i) to u 3 and then letting j --+ oo, one can verify(3.1.3) in the general case. D

3.1.3. Lemma. Let I be a positive real number and k a positive integersuch that (k + t —1)p < n. Then there exists a constant C = C(n, k, t) suchthat

Rn I y ^ xl n I u(y)I dy <_ G ^ f ^ lu xIt- n +k IDa U(y)Idyâ^ =k

115

ID

3.1. Limits of Integral Averages of Sobolev Functions

for all x C R' and all u E Wk•p(Rn ).

This follows from Lemma 3.1.2 by mathematical induction.

We are now in a position to prove the main theorem of this sectionconcerning the existence of integral averages of Sobolev functions.

3.1.4. Theorem. Let k be a positive integer such that kp < n, p > 1, let 0be a non-empty open subset of Rn and let u E Wk 'p(SZ). Then there existsa subset E of SZ, such that

Bk ,P(E ) = 0

andlira u(y)dy

6-.0+ B(r,6)(3.1.5)

exists for all x E ( - E.

Proof. (1) We suppose first of all that SI = R' . Define

g(y) _ E I rru(y)1

(3.1.6)101=k

for y E R". Then g E LP(RR ). Let E be the set of all those points x of Rn

for which(Ik * g)(x) = co. (3.1.7)

Then, from the definition of Riesz capacity (Definition 2.6.2),

Rk.p(E) = 0,

and therefore from (2.6.7),

Bk,p(E) = O.

Consider x E R'ti — E. By (3.1.1),

u(y)dy — n u(y)dy = n [Du(y) • (y — x)1dyf (x,l} ^- f (x,6) f (s, i )

-- 1 a-n [Du(3I) •• (t! — x))dya(=,6)

— 1 ly — xrn [Du(y) - (y — x)]dy•nf<ly—sl<1

When k = 1, it follows from (3.1.6) and (3.1.7) that

Iv — x1 1-"iDu(y)idy < 00 .B (s,l)

(3.1.8)

(3.1.9)


When k > 1, it follows from Lemma 3.1.3 with t = 1 and k — 1 substitutedfor k, that

ffinIv — slt-'ID,u(y)Idy <— C ly — XI k_n [D9Diu(y)1Idy , n lal=k-1

which, by (3.1.6) and (3.1.7), is finite. Thus (3.1.9) still holds when k > 1.By (3.1.9)

Urn ly — ri -n (Du(y) ' (y —6-.0+ 6<iy-zl<1

exists. It also follows from (3.1.9) that

al ô Iy — xl l- "IDu(y)Idy = 0,

e (:,6 )

(3.1.10)

henceb f6 [Du(y) • (y — x)jdy --y 0 (3.1.11)

(x,6)

as b -' 0+. It now follows from (3.1.8), (3.1.10), and (3.1.11) that the limitin (3.1.5) exists.

(ii) The general case. Let II be an open set of R" . There exists an in-creasing sequence {cpj ) of non-negative CO° functions on R", with compactsupports, and spt (pi C 1 for all j such that the interiors of the sets

{r : x E R" and Spy (z) = 1}

tend to S2 as j —► oo. Define

u3 (x) = 0 (x) u(x) Ex41 ft.

By applying (i) to each of the functions u T , one can easily prove the theoremin this case. ❑

3.2 Densities of Measures

Here some basic results concerning the densities of arbitrary measures areestablished that will be used later in the development of Lebesgue pointsfor Sobolev functions.

3.2.1. Lemma. Let p > 0 be a Radon measure on R". Let 0 < A < coand 0 < ci < n. Suppose far an arbitrary Bo re l set A C R" that

B z, rlire sup (z r)) >r^0 T

3.2. Densities of Measures 117

for each x E A. Then there is a constant C = C(a, n) such that

p(A) > C\H' (A).

Proof. Assume p(A) < oo and choose e > O. Let U D A be an open setwith p(U) < oo. Let Ç be the family of all dosed balls B(x, r) C U suchthat

x E A, 0 < r < e/2, p[B(â' 01> A.r

Clearly, Ç covers A finely and thus, by Corollary 1.3.3, there is a disjointsubfamily .F C Ç such that

AC[U{B:BE.F'}]U[U{B:BEf - P}]

whenever .F' is a finite subfamily of Y. Thus, by Definition 1.4.1,

H5a(A) < C ( c2y^ C5' E 6(B) 1

BEY' BEY-Y• 2

where 5(B) denotes the diameter of the ball B. Since .F C Ç and .F isdisjoint, we have

E ( 5 V <cA_1 E p(B)BEY BEY

< Ca - `p(U) < oo.

Sincec E ( o(2B) 1

BEY-Y•

can be made arbitrarily small with an appropriate choice of .F', we conclude

HsF (A) < CA -3 p(U).

Since i is a Radon measure, we have that µ(A) = inf(p(U) : U D A, Uopen). Thus, letting a --+ 0, we obtain the desired result. ❑

3.2.2. Lemma. Let > 0 be a Radon measure on R' 1 that is absolutelycontinuous with respect to Lebesgue measure. Let

A = R" fl x : lim sup p[B(x, r)] > 0 .r—o r

Then, Ha(A) = 0 whenever 0 < a < n.

Proof. The result is obvious for i = 0, so choose 0 < a < n. For eachpositive integer i let

Ai = R" fl x : lx, < i, lirn sup p[B(x'r)] > i-i }

r—.0 r°


and conclude from the preceding lemma that

µ(Ai) > Ci- 'H°'(A i ). (3.2.1)

Since A i is bounded, p(Ai) < co. Therefore Ha (Ai) < oo from (3.2.1).Since cY < n, H" (Ai) = 0 and therefore IAi(= 0 from Theorem 1.4.2. Theabsolute continuity of µ implies µ(A i ) = 0 and consequently H°' (A; ) = 0from (3.2.1). But A = U°_ I A i , and the result follows. O

3.2.3. Corollary. Suppose u E LP(Rn ), 1 0 ,r—, 0 B (x,r)

then H°(E) = O.

Proof. This follows directly from Lemma 3.2.2 by defining a measure p as

p(A) --1u1 pdx. nA

3.3 Lebesgue Points for Sobolev F unctions

We will now prove the principal result of the first three sections (Theorem3.3.3) which is concerned with the existence of Lebesgue points for Sobolevfunctions. We will show that if u E WI P(R"), then

lim iu(y) - u(x)rdx = 0r•-•0

for Bk,p-q.e. x E R". This is stronger than the conclusion reached in Theo-rem 3.1.4, which only asserts the existence of the limit of integral averages.However, in case u E W''P(Rn), the existence of the limit of integral av-erages implies the one above concerning Lebesgue points. In this case, wecan use the fact that u - p4 E WIôp(R") for each real number p and thenapply Theorem 3.1.4 to conclude that

^^ô My) — pidy

exists for B 1 ,p-q.e. x E R". Of course, the exceptional set here depends on p.The object of Exercise 3.1 is to complete this argument. This approach failsto work if u E W k iP(Rn ) since it is not true in general that lui E Wk'P(Rn),

B (s,r)

3.3. Lebesgue Points for Sobolev Functions 119

cf. Remark 2.1.10.

3.3.1. Lemma. Let k be a non-negative integer and A, p real numbers suchthat p>1,kp< n, and k<11 <n/p. If

u E Wk 'p(Rn ), (3.3.1)

thenba--k u(31)dy --► 0

B(x,a)

as b 0+, for all x E Rn except for a set E with Ba,p (E) = O.

(3.3.2)

Theorem 3.1.4 states that the integral averages converge to a finite valueat all points in the complement of a B k ,p-null set. This lemma offers a slightvariation in that the integral averages when multiplied by the factor 6A- kconverge to 0 on a larger set, the complement of a BA null set. At somepoints of this larger set, the integral averages may converge to infinity, butat a rate no faster than 5 k-A .

Proof of Lemma 3.3.1. (1) Suppose k = O. It follows from Corollary 3.2.3that

bpa^^ Ita(y)Ipdy -6 0 (3.3.3)e(s,6)

as b 0+, for all x E Rn except for a set E with Hn -ap(E) = 0. Fromthe definition of Hausdorff measure, for e > 0 there is a countable numberof sets {E,} such that E C UEi and E(diam E i )n-aa < e. Each Ei iscontained in a ball Bi of radius r i where r ; = diarn Ei. Therefore, with theaid of Theorem 2.6.13,

Ba,p(E) < E BÀ,p(E,) < CE r, -ap < Ce.ti=1 i=1

Since a is arbitrary, we have that BA, p(E) = 0.Now consider x E Rn - E. From Holder's inequality, there is a constant

C = C(n,p) such that

15A —n u(y)dy <— C bap-nf (s,6) B(s,6)

l fp

Iu(y)I°dy . (3.3.4)

(3.3.2) now follows from (3.3.3) and (3.3.4), and (i) is established.(ii) Now suppose k > O. Let E be the set of all x for which

I3/ — MIA —n E Ib°u(1!)I dy = 00•Iai =k

(3.3.5)

fs(x-,1)Since JI - k - n < 0, it follows that

(3.3.10)Iu — x I IDu(y) Idv < oc.

120 3. Pointwisc Behavior of Sobolev F'nnctions

Then R,,p (E) = 0 and therefore Ba,p (E) = 0 by (2.6.8).Consider z E Rn — E. When k = 1, it follows from (3.3.5) that

fn(x,i)

When k> 1, we replace t byÀ- k+ 1 and k by k-- 1 in Lemma 3.1.3 andagain derive (3.3.6) from (3.3.5). For z E fin — E, we now show that

ba -k Iv _ xl l-nlDu(y)Idy -■ 0,f<ly-xj<1

as b j 0. Let r E (0, 1) be arbitrary. Clearly

45.x-k I v , xl l-nlDu(y)Idy --► 0f<jg-s(<1

as 10. When 0 < b< rwe have

(3.3.7)

(3.3.8)

aJl -k xl l-ÎDu(y)Idv Ç ^31 - xI^ -k+I -n lDu(y)Idy.

6<^y-x^<r 13(x ,r)(3.3.9)

It follows from (3.3.6) that the right-hand side of (3.3.9) approaches zeroas r j 0. (33.7) now follows from (3.3.8) and (3.3.9).

Clearly,

ly - xla -k+ l -niDu(y)Idy < oc. (3.3.6)

6A-k-n I^— xl I Du(p)idy <Iv — xlk+ln lDu(v)Idy ,

B(x,6) f (x ,6)

so that by (3.3.6),

ba-k-n Iv - zI I Du(y)Idy 0 (3.3.11)f(x.6)

as b j 0. By putting r = 1 in Lemma 3.1.1, one can obtain (3.3.2) fromLemma 3.1.1, (3.3.10), (3.3.11), and (3.3.7). ❑

3.3.2. Theorem. Let 1, k be integers such that k > 1, 0 < k < l and1p < n, p > 1. Let u E WkiP(Rn) and for each z E Rn and r > 0 put

Then

ux ,r u(y)dy.fB(x,r)

u(j) -- u= .r l pdv --► 04 (m,r)r(t-k)p (3.3.12)


as r . 0, for all x E R" except for a set E with Bt,p (E) = O.

Proof. We proceed by induction on k. Suppose to begin with that k O.It follows from Corollary 3.2.3 that

rtp—"

lu(y)Ipdy -' 0 (3.3.13)B(x,r)

for all x E R", except for a set E' with H"— 'P(E') = 0 and therefore

B,,p (E') = 0. (3.3.14)

We now have, for x E R" — E',

1/p 1/pr(1- (n /P )) û(y) - uz ,r lp d^f < r(t-("/p)) û(y)^pd^1

B(x,r )[L(X)

But by Lemma 3.3.1,

^- r(t- (n/P)) Iux.r 11/p

dy . (3.3.15)[L(,)

rtux , r -► 0 (3.3.16)

as r . 0, for all z E R" except for a set E" with Bcp (E") = 0. (3.3.12) nowfollows from (3.3.13), (3.3.15), and (3.3.16) in the case k = O.

Now suppose that k > 0 and that the theorem has been proved for allfunctions of Wk -1 ip(R"). Let u E Wk 1P(Rn). By the Poincar6 inequality,which we shall prove in a more general setting later in Chapter 4 (forexample, see Theorem 4.4.2),

r(t-k)p-n ju(y) f ux,rIPdy < Cirif-(k-1AP_" IDu(y)Ipdv,J(x,r) f (z.r)

(3.3.17)for all z E R", where C depends only on n. By the induction assumption,there exists a set F', with

Bt,p ( F' ) = 0

andr,i1-- (k -1)1v --n IDiu(y) - (D+u)z,rlPdy -. 0

B(aC,r)

asr10, forall xE R" -F'. But

(3.3.18)

(3.3.19)

1 /p

1D:u(N)^pdy <[L(X.r) [fB,?)

1/3104(y) - (Dtu)=,rl Pdtl1/p


1/p

+ l(Diu)x,r l [j dv} , (3.3.20)6(x,r)

and by Lemma 3.3.1

re-(k--1)(Diu)s,r —• 0

(3.3.21)

as r J. 0, for all x E R", except for a set F" with

Bcp (F") = O.

(3.3.12) now follows from (3.3.17), (3.3.19), (3.3.20), and (3.3.21). Thiscompletes the proof. ❑

3.3.3. Theorem. Let k be a positive integer such that kp < n, let 1-2 be anopen set of R" and let u E W k ip (S2) . Then

s(s,r) lu(y) — u(x)Ipdy --, o

(3.3.22)

as r j 0, for all z E S2, except for a set E with Bk,p (E) = 0.

Proof. (i) When SZ = R", (3.3.22) follows from Theorem 3.3.2 and Theorem3.1.4.

(ii) When SZ is arbitrary, the theorem can be derived from (i) as in theproof of Theorem 3.1.4. ❑

3.3.4. Corollary. Let k be a positive integer such that kp < n, let it be anopen set of Rn and let u E W k.P(SZ). Then

lim lu(y)Ipdy exists and = lu(x)I' (3.3.23)r-.0+ g(x,r)

for all x E Si, except for a set E with Bk ,p (E) = O.

3.3.5. Remark. Theorem 3.3.3 states that on the average, the oscillationof u at x is approximately equal to u(x) at Bk ,p-q.e. z E Si This canalso be stated in terms of the classical concept of approximate continuity,which will be used extensively in Chapter 5. A function u is said to beapproximately continuous at xo if there exists a measurable set A suchthat

t/3(xo, r) n AI =1 ô B ( x o r

— 1 (3.3.24), ) I

and u is continuous at zo relative to A. It is not difficult to show that ifu has a Lebesgue point at x o then u is approximately continuous at zo.A proof of this is given in Remark 4.4.5. Thus, in particular, Theorem3.3.3 implies that u E W 1,p(R") is approximately continuous at /31,p-q.e.x R".


Approximate continuity is a concept from measure theory. A similar con-cept taken from potential theory is fine continuity and is defined in termsof thin sets. A set A C Rn is said to be thin at x0 relative to the capacityBk,p if

1 [

Bic (A n B(xo, r)11 1/(P-13 dr— <oo.o Bk,pjB(xo, r)1 r

(3.3.25)

A function u is finely continuous at x0 if there exists a set A that is thinat x0 and

lim u(x) = u(xo ).Z --+t oxvA

It follows from standard arguments in potential theory that A can be takenas a measurable set. In the case of the capacity, B1,2, which is equivalentto Newtonian capacity, these definitions are in agreement with those foundin classical potential theory. In view of the fact that

IAI GIBk.P(A)1 71/(n-kP)

for any set A C Rn, it follows that (3.3.25) implies

11m IB(xo, r) n (Rn - A)I ' 1,r---'d IB(xd, r)I

and therefore fine continuity implies approximate continuity.We now will show that the approximate continuity property of Sobolev

functions can be replaced by fine continuity. First, we need the followinglemma.

Lemma. If {A 1 } is a sequence of sets each of which is thin at x 0 , thenthere exists a sequence of real numbers {r 1 } such that

00

U Ai n B(xQ , r i )i-1

is thin at x0 .

Proof. Because A ; is thin at x0, it follows that there exists a sequence{ri } -, 0 such that

Bk,p[A11 fl B(xo, r0)I 0 as i oo.Bk,PI

r B(x0, ri )]

We may assume the r, to have been chosen so that

T' Bk,PIAi n B(zo , r)) 1/(P-') dr _ ?,d Bk,y[B

—

(xo, r)^ r < 2(^+t


Then,

' [Bk.p [Ai n B(xo, r) n B(xo , 1 1/(P-1) drBk,F [B(xo, r))

$k.p[A+ n B(xo, r) n B(xo, r^ )jfo r,

Bk,p[B(xo, r)1i [Bk,p [A, fl B(xo , r) n B(xo , r ; )1 1/(p-1) dr

+ r, Bk,p[B(xa, r)] r1 1 1 '/(p-1) dr

< 2 -(i+l) + Bk .p[A, n B(xo , r;)['/(p -1)[ CrTh_"P

]rr,

< + Bk ,p [A ; n B(xo , r,)1'/ (p -1)G1 1 1 Bkm[B(Z0, ri )jl/(P- 1)

< 2' for rs sufficiently small.

Since capacity is countably subadditive, the result easily follows. ❑

For ease of exposition, we now restrict our attention to u E W 1 ' 13 (R").Again, we see the important role played by the growth of the gradient inorder to obtain some regularity at a given point.

Theorem. Let xo E R", p > 1, and suppose u E W' '(R") has the propertythat

"(p-1) dr

rp—" IDuI dx < oc.B(xo,r) r

[irn u(y)dy = u(xo ).r—.o J(zo ,r)

Then u is finely continuous at x o .

Proof. For each e > 0, let

A(xo , e) = Rn n {x : I u(x) — 4l(x0 )I > E}.

For r > 0, letyr (x) = pr (x) [u(x) - FL(2r)]

where (so, is a smooth function such that ipr - 1 on B(xo , r), apt 4p r CB(xo , 2r), < Cr -1 and where

T../(2r) = u dx.B(x0,2r)

I1/(p-1) dr

r

1

J: '

Suppose also that


Because of the assumption û(2r)sufficiently small r,

u(x0) as r -, 0, note that for all

vr (x) > e/2 for x E A(xo , e) fl B(ro , r).

Therefore, by appealing to Exercise 2.8, which allows B1 1p to be expressedin terms of a variational integral, there exists C = C(p, n) such that

B1.1,1A(xo , e) B(xo, r)) < C(2e -1 )p^Dvr^pd^

B(so,2r)

< C(2e -1 )F ^Du^pdz

B(so,2r)

+ (C2e- 1 r-1)plu - û(2r)1 pdx. (3.3.26)

B(so,2r)

An application of Poincaré's inequality (cf. Theorem 4.4.2) yields

lu - û(2r)Ipdx < Cr" IDujpdxf(xo,2r)

f(zo,2r)

and therefore (3.3.26) can be written as

B1.p(A(ro, E) n B(xo, r)1 < Ce-1r-n f Duf pdx,rn-p13(z0,2r)

which directly implies that A(xo, e) is thin at z 0 . Now let e3 be a sequencetending to 0. By the preceding lemma, there is a decreasing sequence 5 -. 0such that

00

A = U EA(xo , ei) n B(xo> r2 )j^-1

is thin at x0 . Clearlylim u(x) = u(X0)s—oso

sER" —A

and the theorem is established. ❑

It can be shown that1 1/(p-1) dr

rp-n IDuIpdx — < co (3.3.27)0 B(sa,r) r

for B 1 ,p-q.e. xp E R" cf. 1ME31. Therefore, with Theorem 3.3.3, we obtainthe following.

Corollary. If u E W l "p(R") then u finely continuous al all points exceptfor a set of B lip capacity zero.


Observe that Corollary 3.2.3 implies

lim sup rp—"I DuI pdx = 0 (3.3.28)r -•0 B (xo,r)

for H" —p-a.e. x0 E R. Although (3.3.27) implies (3.3.28) for each z0, theexceptional set for the former is larger than that for the latter.

3.4 II-Derivatives for Sobolev Functions

In the previous three sections, the continuity properties of Sobolev func-tions were explored through an investigation of Lebesgue points and finecontinuity. We now proceed to analyze their differentiability properties. Webegin by proving that Sobolev functions can be expanded in a finite Tay-lor series such that for all points in the complement of an exceptional set,the integral average of the remainder term tends to 0, (Theorem 3.4.2). Inkeeping with the spirit of this subject, it will be seen that the exceptionalset has zero capacity. Obse rve that Theorem 3.3.3 provides the first stepin this direction if we interpret the associated polynomial as one of degree0 and the remainder at x as lu(y) — u(x)I.

When k, m are integers such that 0 _< m < k, (k — rn)p < n andu E Wl u 1 (Rn), it follows from Theorem 3.1.4 that there exists a subset Eof R" such that

B,c_,,,,p(E) = 0 (3.4.1)

11m D°û(y)dy (3.4.2)r^0+ B(z,r)

exists for all x E R" — E and for each multi-index ar with 0 < led < rn.Thus, for all such x, we are able to define the Taylor polynomial em) inthe usual way:

Pâm)(!1) = L ! Dau(x)(y — x)a•° +Q) m

(3.4.3)

(Recall the notation introduced in Section 1.1.) Observe that when 7R is aCm function on RP, Taylor's theorem can be expressed in the form

au(U) = Rzm-1) (10 + rn E ° (1 — t }

m--2

iaE=m a. • D"u[(1 — t)x + ty]dti (y — x)". (3.4.4)

and

3.4.1. Theorem. Let 1 < i < k and suppose (k — rn)p < n. Let u E

3.4. L'-Derivatives for Sobolev Functions 127

Wk"(R") and E be the set described to (3.4.1) and (3.4.2). Then

1/plu(v) — Pzm) (v)IPdy < Tm E

{L(z,r) iai =m

1(1 _ t )m-1

C! a

and[rn B(z,tr)

1/p

I D°u(y) - D aru(x) IPdy dt (3.4.5)

-

1/p 1

( < T mu(y) p=m-t)(b)^pd^1 t m[L(z ) r) ^ ^ -

a! o ( l Iai=m

1/p

• t'„ ID°u(y)Ipdv dt,B (z.tr)

for all x ER" except for a set E'iE with Bk -„a ,p (E')= O.

(3.4.6)

Proof. (i) Suppose first of all that u is a Cm function on fr. Let r E R",r > 0 and put B _ B(x, r). Let cp be a function of L" (B) with Ijcp^j< < 1where p' is the conjugate of p. By (3.4.3) and (3.4.4),

i [ 1411) - R1m)(y)14:0)dy - E 77:1 (1 - t)m -1

la+=m

{D°' u((1 - t)x + ty) - Dau(x)}(y - x)°cp(y)dy dt.[113

Hence, by Holder's inequality,

^f (1 _ t )m -1IJ ru(y) — R1m ) (0 1 cP(04- E< r„` ^

a .iaj=m

1/p

• IDau((1 - t)x + ty) - Dau(x)IPdy dt.B

By making the substitution z = x + t(y - x) in the right-hand side andthen taking the supremum over all cp, one obtains (3.4.5).

The inequality (3.4.6) can be derived similarly.(ii) Now let u be an arbitrary function of W k,P(R„ ). By Theorem 3.3.3,

there exists a set E' D E, with Bk_„,,p (E') = 0 such that

lim ID4u(y) - Dau(x)IPdy = 0 (3.4.7)4-' 0+ B(z,6)

IDauf(y)ipdy = ifE ço,(y — z)Dr4i(z)dzp

dy, E R"


when 0< Pal <m and x E R"—E'.Consider r E R" — E'. There exists a constant M (depending on s), such

thatIDau(y)Ipdy <_ M (3.4.8)

B(z,6)

for all lai = m and all b > O. Let {y,,} be a sequence of regularizers asdiscussed in Section 1.6. Thus, cp, E Cfl (Rn),

apt cPc C B(0, e) and

cPE(x)dx = 1,Rn

(3.4.9)

sup w,(x) < CC" (3.4.10)'ER"

for all e (where C depends only on n), while

(Sp, * Dau)(x) — D°u(x) (3.4.11)

as i 0, for 0<'al <m and x E R"—E'. Put u, =cp,*u. Each u, EC°° (Rn )nW k ip(R"). Let us denote by (3.4.5), and (3.4.6), the inequalities(3.4.5) and (3.4.6) with u replaced by u,. Since u, is smooth we knowthat (3.4.5), and (3.4.6), are valid. By (3.4.11) and Fatou's lemma, thelower-limit as e 1. 0 of the left-hand side of (3.4.5), and (3.4.6), is greaterthan or equal to the left-hand side of (3.4.5) and (3.4.6). The result of thetheorem will thus follow from Theorem 1.6.1(iî) and Lebesgue's DominatedConvergence theorem when we show for each a with lai = m and r > 0fixed, that the following function of t, 0 < t < 1, is bounded; that is,

t-" f 1Da u,(y)1Pdy < Mr" (3.4.12)B(x,gr)

where M is independent of e.We now proceed to establish (3.4.12). For any measurable subset E of

Rn, we have (when Ian = m)

hence by (3.4.10)

1P

IDa uE (y)I pdy < CPC (" ) I D °ù(z)Idz dy. (3.4.13)E B(y,c)

Thus, when p > 1, we have by Holder's inequality

IDau^(y)Ipdy<CpE-(+^p) IDau(Z)'pdz H P- 1 4E E B(U,^) B (U.e)

3.4. LP-Derivatives for Sobolev Functions 129

so that

1 ID°ue (y)1Pdy < Ce'EIDQu(z)^Pdzdy, (3.4.14)

! f (y,f)

where C depends only on n and p. When p = 1, (3.4.14) follows from(3.4.13).

When tr < 3E, we let E be the ball with center x and radius tr. SinceB(y,e) C B(x , 4E) when y E B(x , tr), (3.4.14) implies that

ID° uE (y)I Pdy < CEf (x,tr)

-n ID°u(z)Ipdzdy.B(x,tr)B(s,4e)

It now follows from (3.4.8) and (3.4.12) holds in the case where tr < 3c.When tr > 3E, we have

I D° uc(1!)Î Pdy = f ID«üc(y)lpdy+ ID° lie (Or dyf (x,tr) B(z,3e) ff<1y-rl <tr

and a double application of (3.4.14) yields

iDa ue(y)IP dy <_ CE -nID"u(z)rdzdy

f(x,tr) B(x,3e)f(x,4F)

+ CE -n ID°u(z)1pdzdyfe<ly-2I<tr B(y,e)

and by (3.4.8)

< C't"rn + Ce -n 1D°ù(w + y)Ipdwdyfe<'V -sl<tr f (O,E)

< C' tn rn + C^t• rn

fB(0 1e) 2e D°u(y)1Pdyy dw

so thatfD°ue(y)IPdy Ci"tnrn.

f(x,tr)

Thus (3.4.12) is established. Q

3.4.2. Theorem. Let 0 < m < k and suppose (k - rn)p < n. Let u E

Wk •P(Rn). Then,

[fB( x . r )

as r l 0, for all x E Rn, except for a set F with

Bk-m,p(F) = O.

r -m1/p

l u(Y ) - PIm ) (y) I p dy ^ 0

1I.


This is the main result of this section. In particular, it states that theintegral average over a ball of radius r of the remainder term involving theformal Taylor polynomial of degree k tends to 0 as r —. 0 at a speed greaterthan r ig at almost every point. If a Taylor polynomial of smaller degree isconsidered, the integral average tends to 0 at perhaps a slower speed, buton a larger set.

Proof of Theorem 3.4.2. When m = 0, the theorem reduces to Theorem3.3.3. Suppose via > O. By Theorem 3.3.3,

ID Q u(y) — DQu(x)Iedy —' OB(s,r)

as r j 0, for all laf = m for all z E R", except for a set F with

Bk_mp(F) = O.

Consider r E R" — F and an a with fal ï m. Define

(3.4.15)

1/p

9(r) = [r_n f ID°'te(y) — D °u(x)f pdy (3.4.16)L()

for r > O. By (3.4.15),î(r) --, Oasr J. hence

(1 — t)m -2 7)(tr)dt —■ 0 (3.4.17)

r j O. The required result now follows from (3.4.16), (3.4.17), and Theorem3.4.1. ❑

3.5 Properties of 11-DerivativesIn this section we consider arbitrary functions that possess formal Taylor se-ries expansions and investigate their relationship with those functions thathave Taylor series expansions in the metric of LP, such as those discussedin the previous section.

3.5.1. Definition. Let E C Rn . A bounded function u defined on Ebelongs to Tk(E), k > 0, if there is a positive number M and for eachz E E there is a polynomial P2 (•) of degree less than k of the form

Px(y) _ ua(x) y( — x)°t, (u o = u) (15.1)a.

In*I>o

whose coefficients uQ satisfy

luck(s)( < M for z E E, 0 < lai < k,

3.5. Properties of Dl-Derivatives 131

andun(y) = D° PS(y) + Ra(x, y)

whenever x, y E E and where Ra (x, y) < Cly — xl k— W, 0 < iai < k.The class tk (E) is defined as all functions u on E such that for eachE E there is a polynomial P= (-) of degree less than or equal to k of the

form (3.5.1) such that for 0 < iaj < k,

Er PIA) = D°`Pt(y) + Ra(z, y),

whenever x, y E E with IR,(x, y)1 < Cly -- xi k-1 'I and

lim - R°' (xky )^ i ` 0y ly i

uniformly on E.As a mnemonic, Tk(E) and t k (E) may be considered as classes of func-

tions that possess formal Taylor series expansions relative to E whose re-mainder terms tend to 0 "big O" or "little O," respectively.

3.5.2. Remark. Clearly, if u E Tk (E) then uo, is locally Lipschitz on E,0 < lai < k. If E is an open set, note that the derivatives Dku exist on E,0 <jal< k, and that

Dau(x) D'PP (x) - ua (x) for x E E.

Since IDaPP (x)I < M for x E E, it follows that u E Wiâ-' '(E) for everyp > 1. The space tk (E) may be considered as the class of functions on Ethat admit formal Taylor series expansions of degree k. Of course, if E wereopen and u E Ck (E), then u would have an expansion as in Definition 3.5.1with

Pr(y) = E 1 D°u(x)(y xr.0<la 1<k

Moreover, if u E Ck (R") and E C R", then the restriction of u to E,ulE, belongs to tk(F) for each compact set F C E. One of the reasons foridentifying the class tk (E) is that it applies directly to the Whitney exten-sion theorem [W11), which we state here without proof. We will provide adifferent version in Section 3.6.

3.5.3. Whitney Extension Theorem. Let E C R" be compact. If u Et k (E), k > 4 an integer, then there exists û E Ck (R") such that for 0 <Pi < k

D4317(x) = DsPP (x) for all r E E.

In view of this result, it follows that u E t c (E) if and only if u is therestriction to E of a function of class Ck(Rn).

as r --• 0. (3-5.3)


We now introduce another class of functions similar to those introducedin Definition 3.5.1 but different in the respect that the remainder termis required to have suitable decay relative to the LP-norm instead of theLQ°-norm. The motivation for this definition is provided by the resultsestablished in Section 4 concerning Taylor expansions for Sobolev functions.

3.5.4. Definition. For 1 <— p < oo, k a non-negative integer, and z E

R", Tk 1P(x) will denote those functions u E LP for which there exists apolynomial Pr (.) of degree less than k and a constant M = Mfr, u) suchthat for 0<r<oo

(L(XF)

When p = oo, the left side of (3.5.2) is interpreted to mean esssup"B(=,r)lu(y) — Pr (y)I. T' (x) is a Banach space if for each u E Tk 'p(x) the normof u, IlulI T k .P(x), is defined as the sum of NG, the absolute value of thecoefficients of Px , and the smallest value of M in (3.5.2).

3.5.5. Definition. A function u E 74,P(x) belongs to t k IP(x) if there is apolynomial of degree less than or equal to k such that

1/p

in(y) — P2(y)1PdY = o(rk)B(z,r)

Note that if u E Tk'P(x) the polynomial Pr is uniquely determined. Tosee this write

u(y) = Ps(y) + R=(y)where

(IB(z,r)

If Ps were not uniquely determined, we would have u(y) = Q x (y) + Rr (y),where R2 satisfies an integral inequality similar to that of R x .

Let SA (y) = P2 (y) —Qx (y). In order to show that S= E 0, first note that

1/p1B (z,r) B(x,r)ISz(y)ldy < FSz(y)Ipdy < Cr', 0 < r < oo.

B(z,r) B(x,r)

Now let Ls be the sum of terms of Sx of lowest order and let Mx = SE — LE .Thus, Ls has the property that for each A E R 1 , Ls (Ay +x) = A° Lz (y + x ),where a is an integer, 0 < a < k — 1. Since Ms is a polynomial of degreeat most k — 1, we have

1MZ(b)Idy <Crk-1 , 0 < r <oo.

1/p

Iu(y) — Px(y)Îpd}f < Mrk _ (3.5.2)

1/p

1Rz(0Ipdy < Mrk .

3.5. Properties of LP-Derivatives 133

It follows from the inequality jLgy), < 1Sz (y)1 + IM=(Y)1 that

r° lLx(y)Idy =8(x,1)

-I^x(y)IdyB (a ,r)

< Crk + Crk-i, 0 < r < oo.

This is impossible for all small r > 0 if a < k - I and L= is non-zero. Ifa = k - 1, then Ms - 0 and the term Crk -1 above can be replaced by 0.

A similar argument holds in case u E tk ie(x).Obviously, t k (E) C tkIP(x) and Tk(E) C Tki'(x) whenever r E E and

p > 1. We now consider the question of the reverse inclusion. For thispurpose, we first need the following lemma.

3.5.6. Lemma. Lei k be a non-negative integer. Then there exists cp ECo (Rn ) with spt cP C { Ix' < 1) such that for every polynomial P on Rnof degree < k and every e > 0,

ci)£ * P=P

where cpe (x) = e ' ço(x/e).

Proof. Let V = C'°(B) where B is the closed unit ball centered at theorigin and let W denote the vector space of all m-tuples (y} whose compo-nents are indexed by multi-indices a = (al, arts ... , a„) with 0 < lai < k.The number m is determined by k and n. Define a linear map T: V -e Wby

thus,

T() = cp(x)xadz ;R"

yet =Sp(x)xadx^n

where 0 < 'al < k and x° = xi' x27 .. -xn^.Note that vector space, range T, has the property that range T = W for

if not, there would exist a vector, a = {a} orthogonal to range T. Thatis,

E aay, = 0 whenever y = { ya } E range T.

This implies,

cp(x)E a.xadx = 0 whenever Sp E V.


Select rl E V such that rl > 0 in (z :1x1 < 1). Now define t' by = E ac,xa11and note that 1i E V. Therefore,

(E a^x°) 2 rJ(x)dz = 0,/R.

which implies E aQx° = 0 whenever 1x1 < 1. But this implies that all mnumbers as = 0, a contradiction. Thus, range T = W. In particular, thisimplies there is Sp E V such that

fRçp(x)dx = 1, (p(x)zadx ï 0, 0 < lai < k.

R"

Since any polynomial Q of degree no greater than k is of the form

Q(z) = E bQza r

o<iaI<k

it follows thatçp(z)Q(z)dz Q(0).

Given a polynomial P = P(x) as in the statement of the lemma, let z =(x — y)/e and set Q(z) = P(x — €z) to obtain the desired result. ❑

The next theorem is the main result of this section. Roughly speaking,it states that if a function possesses a finite Taylor expansion in the 1/-sense at all points of a compact set E, then it has a Taylor expansion inthe classical sense on E. It is rather interesting that we are able to deducea Lw-conclusion from a V'-hypothesis. A critical role is played by theexistence of a smoothing kernel (P that leaves all polynomials of a givendegree invariant under the action of convolution.

3.5.7. Theorem. Let E C Rn be dosed and suppose u E Tk 'P(x), 1 0, with IiurlTk.o(r) < M for all z E E. Then u E Tk (E). Also, if Eis compact and if u E t k .P(x) for all x E E with (3.5.3) holding uniformlyon E, then u E t k (E).

In view of Whitney's Extension theorem (Theorem 3.5.3), note that afunction satisfying the second part of the theorem is necessarily the restric-tion of a function of class Ck (R"). In the next section, we will investigateWhitney's theorem in the context of LP.

Proof of Theorem 3.5.7. Let Sp E Co (R") be the function obtained inLemma 3.5.6 such that

SP, * P(x) = P(z) (3.5.4)

3.5. Properties of IF-Derivatives 135

whenever P is a polynomial of degree less than k, e > 0, and x € R". Notethat (3.5.4) implies

Da çpe * P(x) = cpe * 13' P(x) = D°`P(x). (3.5.5)

Since u E TSCiP(xo) for all xo E E, we have for xo and x E E,

u(y) = P:o(y) + R(xo, y) (3.5.6)

and

where

u(y) = Px (y) + R(x,y) (3.5.7)

(fB(. r)

with x` either x0 or x. Now let e = lx — xol and for 0 < 1131 < k consider

I = DI3 cpe * u(x).

For each fixed z E R", define R; as Rs (x) = R(z, x) whenever x E R.From (3.5.6) and (3.5.5) it follows that

I = D0 cpe * Pzo (x) + Dpcp, * Rzo (x)

= D I3 Pzo (x) + 13''(p * Rso (x)•

Similarly, using (3.5.7) and (3.5.5), we have

I = D'P= (x) + D0 ç0f * Rz (x)

= uo(x) + Ds(p e * Rz(x)•

Therefore,

D I3Ps(x) = DA Pxo(x) + [D13 (pe * (Rzo — Rz)1(x)= DQP=o (x)

+ f (n+IDDK3cP [ (2;

F y)

IR(xo, y) — R(x, y)]dy•

Because cp =— 0 on 1x1 > 1, the last integral is taken over B(x, e). SinceB(x, e) C B(xo, 2e), the integral is dominated by

C 1R(xo, y)Idy + 1R(x, y)Idy e — I01 (3.5.9)IB(tp,2e) $ (x,e)

where C depends on an upper bound for jD13 cpI. Jensen's inequality and(3.5.8) implies that (3.5.9) is bounded by CMe k-1131 , thus proving u ETk(E).

ilpIR(x', y)Ipdy S Mrk , (3.5.8)


A similar proof establishes the second assertion of the theorem. Indeed,as before we obtain

Da Pr(x) = DQPro (x) + (DR cPf * (Rso - Rr)J(x)= D 3 P=0 (x)

+ e-("+I,QI}DB(p (x - y} (R(xo, y) R(x, y)ldyF

< C IR(xo, y)Idy IR(x, y)jcly C —IAI.B(x0,2e) B(x,c)

Since (3.5.3) is assumed to hold uniformly on E, for q > O arbitrary, thelast expression is dominated by i£kÎQ) xulk-I61 provided Ix - xoIis sufficiently small. The compactness of E is used in this case to ensurethat IRQ(x, y)I <- Cox - yak- I 131 whenever x, y E E. ❑

3.6 An LP-Version of the Whitney ExtensionTheorem

We now return to the Whitney Extension Theorem (Theorem 3.5.3) thatwas stated without proof in the previous section. It states that for a corn-pact set E C R", a function u is an element of tk (E) if and only if it is therestriction to E of a function of class C k (R"). The result we establish here,which was first proved in [CZ], is slightly stronger in that the full strengthof the hypothesis u E tk(E) is not required. Instead, our hypothesis requiresthat u E tkup(x) for all x E E with (3.5.3) holding uniformly on E.

We begin by proving a lemma that establishes the existence of a smoothfunction which is comparable to the distance function to an arbitrary closedset.

3.6.1. Lemma. Let A C R" be closed and for x E R" let d(x) = d(x, A)denote the distance from x to A. Let U = {x : d(x) < 1}. Then there is afunction ê E C °° (U — A) and a positive number M = M(n) such that

M - 'd(x) < b(x) < Md(x), r: E U — A,

IDa6(x)I < C(a)d(x)'ìul, x E U - A, > O.

Proof. Let h(x) = Z0d(x), x E U - A, and consider a cover of U - A byclosed balls {B(x, h(x))}, with center x and radins h(x), x EU_- A. FromTheorem 1.3.1 there is a countable set S C U - A such that {B($, h(s)) :s E S} is disjointed and

R" — A D {UB(8,5h(s)) : s ES } ❑ U — A.

3.6. An LP-Version of the Whitney Extension Theorem 137

With a = 13 = 10 and A = 15 , we infer from Lemma 1.3.4 that

3 h(x)/h(s) < 3 for s E Sz . (3.6.1)

Let 0(x) = H°(SM ) < C(n) and let rj : RI -, 10,11 be of class C°° with

rgt) =l for t <I, n(t) = 0 for t > 2.

Now define lb E C°°(R") by i(x) =i(lx1) and y, EC°°(U) by

v,(x) = h(8)0 (5h s8)for s E S, x E U.

Note that apt v, C R(s,10h(s)), v = h(s) on B(s, 5h(s)) and from (3.6.1)that

ilrvs(x)1 < h(s)N(a)[5h(s)j - M"M< N(a)h(x)1 -lìf for E Ss ,

where N(a) is a bound for IDp i1, 1/31 < 101. Now define

b(x) = v,(x) = E v8(x) for x E U.'ES 'ES:

Clearly,d(x) ' h(x)60

< a(x) < 3e(x)h(x) = 20

9(x)d(x)

and

iD°a(i)i < 5- lQ 1 3 1a 1-10(x)N(a)h(x) 1- faM, for x E U - A. ❑

The following is only a prelude to the D'-version of the Whitney exten-sion theorem, although its proof supplies all of the necessary ingredients.Its hypothesis only invokes information pertaining to the spaces Tk.P (x)(bounded difference quotients) and not the spaces t k iP(x) (differentiabil-ity). In particular, the theorem states that if u is Lipschitz on A (the casewhen k = 1) then u can be extended to a Lipschitz function on an openset containing A. This fact is also contained in the statement of Theorem3.5.7.

3.6.2. Theorem. Let A c R° be closed and let U = {x : d(x, A) < 11.If u E Lp(U), 1 < p < oo, and there is a positive constant M such that

< M for all x E A, where k is a non-negative integer, then thereexists û E C - ^ (U) such that DIû(x) = D ' PP (x) for x E A, 0 <101 < k.

Proof. Let b denote the function determined in Lemma 3.6.1. Define Ti = uon A and for x E U- A let

u(x) = cooa(=) su(r) (3.6.2)


where cp is the function determined by Lemma 3.5.6 and where

406(x)(Y) = b(xrnV' b(x) •

Thus, ü is defined at x as the convolution of SPa(=) and u evaluated at z.Because both cP and ô are of class C°° it is easily verified that TIE C°O(U-

A). For x E U, let z' be a point in A such that 4x - x' 1 = d(z) = d(x, A).Because u E Tk 'p(x) we may write

u(x) = Ps . (x) + R. (x) (3.6.3)

where 1 /p

IR1. (z)Wp dz < Mrk .B(x',r)

By substituting this expression into (3.6.2), we obtain

D" (x) = DA [(P6(s) * P=' (x)] + D13 kP6(x) * R.. (z)]

= (DpcP5(2) * P1- (z) + f R 13 (x, y)R'. (y)dy

= cp6(x) * (D0P2•)(z) + f Rp(x , y)R1 . (y)db (3.6.4)

where R13(x, y) D 13 {6(z) -nw[(x - y)5(x) -1 ]}. Applying Lemma 3.5.6 tothe first term on the right side of (3.6.4) we obtain

Dpri(x) = D0Pz . (z) + f Rs(x, y)R7• (y)dy. (3.6.5)

We wish to estimate the remainder term in (3.6.5) which requires an anal-ysis of RS (z, y). It can be shown that

IR13(x, y)1 < C(0)d(x)-n-101

and consequently

1 f Rs(x, y)Rz• (y)dy < C(0)d(x) -n-1 13 1 iRz- (y)Idy. (3.6.6)— B( =,b)

Because 6(x) is comparable to d(x) (Lemma 3.6.1) and Ix - x' I = d(x), itfollows that i(x, ô(x)) C B(x', Kd(x)) for some K > O. Therefore from(3.6.3) and Holder's inequality,

f (s•,Kd(x))IRx• (y)Idy < M[Kd(x)}n+k (3.6.7)

3.6. An Imo-Version of the Whitney Extension Theorem 139

which along with (3.6.5) and (3.6.6) implies,

DAu(x) - D09 P2 . (x) = Sp(x`, x)

(3.6.8)

where1S0(x`,x)( < C(Q, k)MIx - x+lk—IQI .

We emphasize here that for given x E U — A, (3.6.8) is valid only for x' E Asuch that d(x) = ix - x* 1. We now proceed to establish the estimate forarbitrary x' E A.

By assumption IiüfITh.,(x) < M for all x E A. Therefore, we may applyTheorem 3.5.7 to conclude that u E Tk (A). Thus, if xi E A,

Px-(x') u(e)

and

Da Pz . (ac') DaP=; (x') + x'), 0 < lal < k (3.6.9)

whereIR0(4, x')I C(a, k)M1x' - x l lk - lal.

By Taylor's theorem for polynomials, it follows that

k-1-1091 1D^Ps • (x) _ E — D°+aPs. (x+)(x - x')a.

a!

Thus, by (3.6.9) and Taylor's theorem,

k -1 -1091DRP=• (x) _

L--4LrDfi+aP=` (x• ) + ^+S(zi, x+)1(x — x' ) a

lal=0

k - L

r-Ipi

u a !1aj=o

k- 1 - (101 +1091) 1E 7 '

=a

— z' )71

+ Rail? (xT, x•)) (x - x')a.

By Taylor's theorem, it follows that

DpPz (x) _ E t Do+QPx, (xi)(x - xi )a .lal>o

Therefore, since

(3.6.10)

^ x -x'j <ix -x11 and ix' - <- -x1+ix-x,l <21x- x,^,


(3.6.10) becomes (after some algebraic simplification)

(k-))-I$I 1 .DI3 P . (x) - E c7 D0+a Pz' (xi)(x — zir = O(lx — x11 k-Is1 ) .

Ial=a

It follows from (3.6.8), that

(k-1)--101 1

Dpû(x) - E l Dp+aPz; (xi)(x — xi)° = O( I x — xE1kWIs1)1a1= 0

or./Y3û(x) - D"PP ; (x) = 0( lx - x7lk- 1 131 ), (3.6.11)

Thus, (3.6.11) holds whenever act € A and x E U - A and Theorem 3.5.7implies that it also holds with Dsû(x) replaced by us(x) whenever x EA. This implies that DOT is a continuous extension of up and that thisextension has a Taylor series expansion about each point in A. Since û EC°° (U — A) it now follows that ii E Ck-1 (U).

In order to prove that ii E Ck-1,1 (U) it suffices to show that DQû isLipschitz, ff1 = k - 1. We know from (3.6.11) that if a E A, and 101 = k -1

ID 8û(x) - D'û(a)1 < C(k)Mlx - al (3.6.12)

for z E U. Therefore, it is necessary to consider only the case z, y E U - A.First suppose lx - yl > id(y) and let a E A be such that d(y) = la - yl.Then, la - y1 < 21x - yl and

Ix - al <Ix-v1+ly-al S 31x_yl.

Thus, utilizing (3.6.12),

IDsU(x) - D°u(y)I < I D' (x) - D 13ic(a)1 + IDau(v) - Dtt(a)1

< 1DQ17(x) - D' i(a)1 + ID' (v) - D /31 (a)1< C(k)M[Ix - al + Iv - al)< 5C(k)Mlx - vl.

Finally, suppose Ix - yl < Id(y) and d(y) = la - yl. Using (3.6.5) with101= k - 1 and the Mean Value theorem, we have

IDari(x) - D' (y)I = f R(a, z) IRt(x, z) - R1(v, z)J dz

Ix — yl f IDxRs(xo, z}l 'Mat x)ldz (3.6.13)

where xs is a point on the line segment joining x and y. Now apt Rs(xo, y) CB(xa , 6(xo )) and ô(xo ) < Cd(xo). Thus,

ID=Rt(xo, z)I < C(Q)d(xa)-n-k,

3.6. An LP-Version of the Whitney Extension Theorem 141

(Pi = k - 1). Therefore, (3.6.13) implies

!D1(x) - Dike(y)1 5 C(Q)Ix - yld(xo) -n-k IR(a, z)Idz.B(xo,Cd( =o))

(3.6.14)Since Lip(d) = 1, we have

2d(xo ) > d(x) > d(3I) - ix - Ili > d(11) > ix - 1/1.

If z E U, I z - xoI < Cd(xo), then

Iz - aI <lz — xol+Ixo — al< Cd(xo ) +Ixo — al< Cd(x o ) + Ixo - yf + d(y)< Cd(xo) + Ix - 11I + d(1!)< Cd(x0 ) + d(x0) + 2d(x0).

That is,B(x01Cd(xo)) C B(a, (C + 3 )d(xo)).

Therefore, reference to (3.6.7) implies

IR(a, z)I dz < C(d(xo))"+kB(xo,Cd(xo))

and this, along with (3.6.14) completes the proof.

This proof leads directly to the following which is the Whitney extensiontheorem in the context of t k iP(x) spaces.

3.6.3. Theorem. Let A C R" be closed and let U = {x : d(x, A) < 11. Ifu E 17(U), 1 < p < oo, and u E tk'F(x) for all x E A with (3.5.3) holdinguniformly on A, then there exists û E Ck (U) such that DO (x) = DO 13,(x)for z E A, O<WI<k.

Proof. The proof is essentially the same as the one above with only minorchanges necessary. For example, the polynomials in (3.6.8) and (3.6.9) arenow of degree k and the remainders can be estimated, respectively, by

ISO (x' , x )I < o(Ix - x•Ik-1Ø1)

andIRa(xi, x i )I < o(Ix. - x l Ik-Ial ),

thus allowing (3.6.11) to be replaced by

D(x) - DBP=i (x) = o(ix - xi i &-1141 ).

The remainder of the argument proceeds as before.


3.7 An Observation on Di fferentiation

We address the technicality of showing that IuIITk,v(z) is a measurablefunction of x and then establish a result in differentiation theory that willbe needed later in the sequel.

3.7.1. Lemma. Let u E T k .p(x) for all r in a measurable set E. Then,IIt IITc.o(z) is a measurable function of x.

Proof. Recall that the norm IIuIrTR.a(z) is the sum of the numbers llullp,ID°PP (x)I, 0 <'Î < k - 1, and the eh root of

sup r -kp lu(y) — Pz(y)I pdy.

r>Q B(x,r)

Also recall that DaP1 (x) = uQ (x). To show that DaP1 (x) is measurablein z consider the function cp of Lemma 3.5.6 and define

uc(x) = (Pc * u(x).

If we write u(y) - Pz (y) - Rz (y), then

D° u,(x) = D° (çof * Pz)(x) + D a (cPE * R2)(x)

= Da PP(x) + f e -(" +Iak) DaSP [J Rx(x - y)dy.

The above integral is dominated by

Ce- ("+Ian) IR=(x - y)Idy < CE-(n+1aj)ek+n

f(x,c)

= Cek- r a i -0 0 as e -• 0.

This shows that DaPz (x) is the limit of smooth functions Dau( (x) for allx E E, and is therefore measurable. The remainder of the proof is easy toestablish. ❑

3.7.2. Lemma. Let u E LP(Rn ), 1 0 and all r > 0,

1/p

B(z,r)lu(y)Ipdv <- Cra ,

for all x in a measurable set E c R". Then, for almost all z E E,

1/p

lu(y)jpdy = o(r°) as r 1 O.( (r.r)

3.7. An Observation on Differentiation 143

Proof. Without loss of generality we may assume that E is bounded andthat u has compact support. Given e > 0, let A C E be a closed set suchthat I — AI < e. Let U be the open set defined by

U ={x:d(x,A)< 1}.

It will suffice to establish the conclusion for almost all z E A.First, observe that the hypotheses imply that

lien Iu(y)Idy = 0r~0 B(x,r)

for z E A and therefore, u = 0 almost everywhere on A.Let h(x) = ôd(x,A). Recall from Theorem 1.3.1 that there is a count-

able set S C U — A such that {B(s, h(s)) : s E S) is disjointed and

{UB(s, 5h(s)) : s € S} J U - A.

Therefore, since u = 0 almost everywhere on A,

lu(y)1 d dx < I^(y)I y dydxA U ^^ — yln +a A U —A Ix — yl^+a< L E

lu(y)Idy dx sES B(s,5h(s)) I X —

yln +a

f dx A Ix — iJl n+a d y'

= E my)! f(s•5h(s3)

(3.7.1)

Let x, E A be such that Is — x,I = d(s, A) = d(s). Hence, B(s,5h(s)) CB(x,, I s -- x,I + 5h(s)) and Is— x, I = d(s) = 10h(s). By Jensen's inequalityand the hypothesis of the lemma

1 /p l/p

lul < lulp < C Iulp < Ch(s)a.IB(c5h(a)) ( fB..5h(s)) B(x+,15h(s))

(3.7.2)Now for z E A,y E B(s,5h(s)), we have

Ix — vI? Ix — sI —Is — yI>d(s) - 5h(s)= 5h(s).

Hence, for y E B(s, 5h(s)) we estimate by spherical coordinates with originat y,

dx °°< C r'drIx — yln+a i

—^—5^(a)i

< C(a)h(s) — °.1.


This, along with (3.7.2) yields

I11`(v) --tr... < C Q h ^nA irBo,,sh(,)) Ix — yin ()()

Since {B(8, h(9)) : s E S} is disjointed, it follows from (3.7.1) that

lu(y)I+a dydx < C(a) E h(s)" < coIA - sESU I z ^11

and therefore,l u(y)I dv

U Ix — yÎ n+afor almost every x E A. Clearly,

< CO

lu(y)Idv < o0J " -U Ix ^ vl+e+a

for all x E A, and therefore

lu(y)Idy < o0R" Ix — É/1 "+a

for almost all x E A.An analysis of the argument shows that this was established by using

only the fact that

B(x,r)lul < Cra .

If we apply the above argument with u = IuI', our hypothesis becomes

Iv(y)Idy < Cr"

for all x E E and therefore

Iv(y)Idy _ g

l^

u(y)I Pdy R" I üq

^ x''L+ap R" I tl r xi n+ap

for almost all x E E. But, for all such x, and for c > 0,

Iu(y) Ipdv < F for all small r > O.B(s,r) I v — xl"+pa

That is,)1/p

l u(y) Ipdv < eraB(2,0

for all small r. O

00 >

3.8. Rademacher's Theorem in the LP-Context 145

3.8 Rademacher's Theorem in the 17-Context

Recall the fundamental result of Rademacher which states that a Lipschitzfunction defined on Rn has a total differential at almost all points (Theorem2.2.1). We rephrase this result in terms of the present setting by replacingthe hypothesis that u is Lipschitz by u E Tk 'p (x) for all r in some set E.If k = 1 and p = oo, this yields the usual Rademacher hypothesis. Theconclusion we will establish is that u E tkiP(x) for almost all z E E.

3.8.1. Theorem. Let u E Tk 'p(r) for all r E E, where E C R" is measur-able, k a non-negative integer and 1 < p < oc. Then u E tk ,p(x) for almostail x E E.

Proof. By Lemma 3.7.1 and Lusin's theorem we may assume that E iscompact and that IIuIjr.,P( s ) < M for all z E E. Since u E Tk 'D(x) forx E E, we may write u(y) = P1 (y) + Rz (y) where Pz is a polynomial ofdegree less than k and where

lip

IRx (y)1'dy < Mrk, r > 0. (3.8.1)in(r,r)

From Theorem 3.6.2 it follows that there exists an open set U D E andriE Ck- '' 1 (U) such that

Di(z) = Df3Pz (x), 0 < 1131 < k. (3.8.2)

Because Ft is of class Ck- 13 it follows from Theorem 2.1.4 that r E W of (Rn )and therefore we may apply Theorem 3.4.2. Thus, for almost all z E Rn,there is a polynomial QS of degree at most k such that i(y) = Qx (y)+Rx (y)where 1/p

iRx (y)1pdy = o(rk ) as r 10. (3.8.3)(.4.(x,r)

Because ü E Ck-I (Rn ), the argument following Definition 3.5.5 impliesthat

D1317(x) = Df3Q1 (x), O <— IAI < k.Therefore, in view of (3.8.1), (3.8.2), and (3.8.3)

I/pin — i,IP < Cr k

(3.8.4)

(fB(x,r)

for almost all x E E. Appealing to Lemma 3.7.2 we have1 /p

lu —ûI 13 = o(rk) as r1O(fB(r,r)


for almost all x E E. Consequently, for all such z,

1/p l/p

û(y) — = (y)I pd Iu(Y) — U(y)Ip duB(x,r) B(x,r)

/P

+ I u(y) z(âl)I PdY(41(x,r)

< o(rk) as r —. 0,

thus establishing the result. ❑

3.9 The Implications of Pointwise DifferentiabilityWe have seen in Section 4 of this chapter that Sobolev functions possessLP-derivatives almost everywhere. This runs parallel to the classical resultthat an absolutely continuous function f on the real line is differentiablealmost everywhere. Of course, the converse is false. However, if it is assumedthat f' exists everywhere and that 11 , 1 is integrable, them f is absolutelycontinuous (Exercise 3.16). It is natural, therefore, to inquire whether thisresult has a counterpart in the multivariate LP theory. It will be shownthat this question has an affirmative answer. Indeed, we will establish thatif a function has an LP derivative everywhere except for a small exceptionalset, and if the coefficients of the associated Taylor polynomial are in 17,then the function is in a Sobolev space.

We begin the investigation by asking the following question. Supposeu E LP (R") has LP-derivatives at x E R"; that is, suppose u E tk'P(x)where k is a positive integer. Then, is it possible to relate the distributionalderivatives of u (which always exist) to the LP-derivatives of u? The firststep in this direction is given by the following lemma. First, recall thatu E tk , P(x) if there is a polynomial Pi of degree k such that

(JiB (r,r)

1/p

lu(y) — Pz(v)I P dy = o(rk ) as r 0, (3.9.1)

and u E TkiP(x) if there is a polynomial P, of degree less than k and anumber M > 0 such that

/p

(1U(Y) — Px(y)j p dy < Mrk, 0 < r < oc.f (r,r)

3.9.1. Lemma. Suppose u E L7 (R"), p > 1.

3.9. The Implications of Pointwise Differentiability 147

(i) If u E Tk .P(x), then

lim inf Sp i * D°'u(y) > — oo,

with !x -- yi < t, and where Dau denotes the distributional derivativeof u,0<fal <k;

(ii) If u E t k 'P(x), then

lim sup (p i * D'u(y) = Da Ps (x),

with Îx — yi<t, 0 <lai < k.

The function (p i above is a mollifier as described in Section 1.6. Since(Pt E Co (R"), its convolution with a distribution T is again a smoothfunction. Moreover, for small t and lv —xi < t, the quantity (p i *T(y) givesan approximate description of the behavior of T in a neighborhood of x.Indeed, if T is a function, then

lim sup Sp i * T(y) = T(x)tô

1y—x1<t

whenever x is a Lebesgue point for T. This will be established in the proof ofLemma 3.9.3. Very roughly then, the statement in (ii) of the above lemmastates that, on the average, the behavior of the distribution Dau near xis reflected in the value of the coefficient, D°Px (x), of the Taylor seriesexpansion.

Let F(y, t) = cp t s u(y). F is thus a function defined on a subset of R"+' ,namely R" x (0, oo) and is smooth in y. The lower and upper limits statedin (i) and (ii) above can be interpreted as non-tangential approach in R"+ 1of (y, t) to the point (s, 0) whch is located on the hyperplane t = 0.

Proof of Lemma 3.9.1. Proof of (ii). Let

u(y) = P7(y) + R (Y) and Ft (V) = F(y, t)•

ThenD°Ft(y) = ba Nvt * u}(y) = D°'cpt * u(y).

Therefore

DaFi (x + h) = f DacPt(x + h — y)u(y)dy

= f Da Spt (x + h — y)Ps (y)dy + f Da cpt (x + h — 3/)Rx (y)d1!

^ (wt * Ps)(x + h) + D°cp t (x + h — y)Rs (y)dy. (3.9.2)


There is a constant C = C(l DSpl) such that

I D`k 't(x + h - y)I < Cry"

for IaI = k. Consequently, for h E B(0, t), it follows that

If D° Wt(x + h - y)Rx(y)dy < Ct-n-k lRx(Y)Idyf(x+h,t)

< Ct'kIR1(y)Idy --. 0 as t j 0,

a (z,si)

by (3.9.1). Writing Pi in terms of its Taylor series, we have

Px(y) D^Px(x)(y - x)a

1°1=a

and therefore D'P1 (y) = D' P1 (x) for all y E R" if (al = k. Hence,cp e * Da 13,(x+ h) = D°P2 (x), and reference to (3.9.2) yields

lim sup çpt * Dau(x + h) = P1(x), 0 < Ihl < t, (3.9.3)

thus establishing (ii) if Pal = k. However, if 0 < l < k, then u E tt .P(x) andthe associated polynomial is

E D°Px (x)(h -- x)a

faI- oa!

Thus, applying (3.9.3) to this case leads to the proof of (ii).The proof of (i) is similar and perhaps simpler. The only difference is

that because Pz is of degree at most k -1, we have

Da F: (x + h) = 0 + f D°vt(h - EI)Rz(y)dy

if Ial = k. The integral is estimated as before and its absolute value is seento be bounded for all t > 0, thus establishing (i). ❑

The next two lemmas, along with the preceding one, will lead to themain result, Theorem 3.9.4.

3.9.2. Lemma. Let T be a distribution and suppose for all x in an openset S2 C R" that

Ix - yl <_t,lim inf cp e * T(y) > -oo,t -•0tES


where Sc (0, oo) is a countable set having 0 as its only limit point. Let Cbe a closed set such that C n St O. Then there exist N > 0 and an openset I i C Si with C fl S21 o 0 such that SPt * T(1i) > —N > —oo wheneverIv — xl Çt,xECf1Stl,tE S.

Proof. LetF.(x) = inf (SPt * T(y) :ix — < t, t E S}.

Then, F.(x) > —oo for x E Ii since 0 is the only limit point of S and it iseasy to verify that F. is upper semicontinuous. Thus, the sets

Cn ftn ix. F.(x)> —i), i = 1,2, ...,

are dosed relative to C n ft and their union is C n Cl. Since C n fI is of thesecond category in itself, the Baire Category theorem implies that one ofthese sets has a non-empty interior relative to C n ft. ❑

One of the fundamental results in distribution theory is that a non-negative distribution is a measure. The following lemma provides a gener-alization of this fact.

3.9.3. Lemma. Let b > 0, N > 0, and suppose S is as in Lemma 3.9.2. IfT is a distribution in an open set fZ such that

(Pt *T(x)>—N>—oo for x ESl , t ESfl (OA

andlim sup Sp i * T(x) > 0 for almost all xo Eft,

elo^x—xo I<t

then T 18 a non-negative measure in ft.

Proof. Let E O(1l), tsb > 0, and recall from Section 1.7, that the convo-lution Spt * T is a smooth function defined by

SPt * T ( x) = T(T:43 t)

where tPe(i1) = too t (—y) and TxSFe(y) = Ot(y — x). Then,

TOP *Ot) =T *(i!1 *Vt)( 0)_ (T *SPt)*'1;(0 )

= f T * SPe (`g1)til(t!)dl!

= f T * tPt(11)0(âi)dg.


Now V)* CPt —, t/) in g(S1) as t —> 0+. Moreover, since 7/) is non-negativeand coot * T(x) _> —N for x E SZ and t E S n (0, b), it follows with the helpof Fatou's lemma, that

T(0) = elô T(0** (430tES

> lim inf Tto RtES

> n

lim ^rtf TtES

* SPt (l0(11)dEl

* ePt(Y).0(3I)dY

—N(y)dy•

Thus, the distribution T + N has the property that

(T+N)(7i)> 0 for E 2(f1), ip > 0.

That is, T + N is a non-negative measure on IZ, call it A. Let u = v + orwhere v is absolutely continuous with respect to Lebesgue measure and oris singular. Clearly, SZ is the union of a countable number of sets of finitev measure. Thus, by the Radon-Nikodym theorem, there exists f E L'(S2)such that

v(E) = f (x)dxE

for every measurable set E C O. Since T + N = p, it follows that

çot * T (x) + N = çot * (T + N) (x) = Soot * µ(x)

n ipt (x - Of (WY + n ço (x — 11)da(y), (3.9.4)

for x E ft Because or is a singular measure, a result from classical differen-tiation theory states that

a[$(xo, t)] _Urn IB(xo,t)I °

for almost all x o E n, cf. [SA, Lemma 7.1]. Therefore, at all such r0 withIx — xol <t,

f vt(ac — y)da(y) Soe(x — 1/)da(y)f(z,t)

< I ^Pt(x — g1)da(y)?

—< ChPIIoo IB(xo, t)Ia[(B(xo, 2t)]

—• 0 as t —, 0* with Ix — xoI < t.


To treat the other term in (3.9.4), recall that f has a Lebesgue point atalmost all xo € D. That is,

If (I!) — f(xo)1dy —0 as r -+ 0+B(zo.r)

Therefore,

f (Owe (x — y)dil — f (xo) _ if (y) — f (xo)1(Pt(x — y)dvB(x.t) f(x,t)

^ Cq (P1100 11(v) — f (2o)1dy 0 as t -• 0, - ro i Ç t.B(zo ,2t)

Consequently,N < lim sup (p t * T(r) + N = f (xo)

tloIx-xoI<t

for almost all ro E Sl. This implies that

w(E) > NIEI

for all measurable E C Sl. Since p(E) > v(E) it follows that the measurep - N = T is non-negative. ❑

3.9.4. Theorem. Lei T be a distribution in an open set Sl C R" and letf E LL (n). Assume

lim sup SPt * T(y) > f(x), - S t,do

for almost all x E 1, and

lir lô f too t * T(y) > -oo,tES

lx — y j < t ,

for all x E OE Then T - f is a non-negative measure in Sl.

Proof. We first assume that f = 0. Lemma 3.9.2 implies that every opensubset of contains an open subset Sl' such that for some N > 0, cpt *T (x) >-N for z€ T,tE S. Lemma3.9.3impliesthat Tis ameasurein11'.

Let (I I be the union of all open sets Si' c SZ such that T is a non-negativemeasure on I". From Remark 1.7.2 we know that T is a measure in D I .We wish to show that 121 = Si. Suppose not. Applying Lemma 3.9.2 withC = R" — il l , there is an open set 12' C Sl such that w e * T(x) > -Nfor y E C fl S2', Ix - vi < t, and t E S. Let Sl2 = it i U Si' and note that02 — f21 = C fl Sl'. Let

fl3 = SZx n {x : d(x, R" -- n2) > e}


for some e > O. Take e sufficiently small so that 113 n (Rn — 121) A 0.Consider Sp i * T(x) for x E S23 and t < e. Now T is a non-negative measurein Of. Therefore, if d(x, Rn — no > t, SPt *2-(x) > O. On the other hand,if d(x, Rn — 11 1 ) < t, there exists y E R" — [2 2 such that Ix — yl < t.Since B(x, t) C SZ 2 , it follows that y E S22 — SZ1 = C n ST. Consequently,cp t *T (x) > —N. Hence, Spt * T(x) is bounded below for x E Q3 r t E Sn(0, e),and thus T is a measure in SZ 3 by Lemma 3.9.3. But S23 n (Rn — Sl i ) 0 0thus contradicting the definition of O 1 .

For the case f 96 0, for each N > O define

N, f (x) ? Nfw(x)= f(x), — N<f(x)<N

—N, f (x) < —N

and let R be the distribution defined by R = T — fN. Clearly R satisfiesthe same conditions as did T when f was assumed to be identically zero.Therefore, R is a non-negative measure in S2. Thus, for E Cr 1l), lb >15,

R(0) = T (b) — f both dx > O.

Letting N --' oo, we have that

T(?i) — f f idx ^ O.

That is, T — f is a non-negative measure in 0. D

Now that Theorem 3.9.4 is established, we are in a position to considerthe implications of a function u with the property that u E Tk ,p(x) forevery x E 0, where S2 is an open subset of R". From Theorem 3.8.1 wehave that u E tk .P(x) for almost all x E S2. Moreover, in view of Lemma3.9.1 (ii), it follows that whenever u E tk.P(x),

Urn sup * Dau(y) = D°'Px (x), Ix — yl < t,tlo

for 0 < IQl < k. For convenience of notation, let u ,(z) = DaPP (x), andassume uQ E (11). Then Theorem 3.9.4 implies that the distributionDan — u.a is a non-negative measure. Similar reasoning applied to the func-tion —u implies that Da (—u) — (—ua ) is a non-negative measure or equiv-alently, that Dau — u a is a non-positive measure. Thus, we conclude thatDau = ua, almost everywhere in OE That is, the distributional derivativesof u are functions in L"(SZ). In summary, we have the following result.

3.9.5. Theorem. Let 1 < p < oo and let k be a non-negative integer. Ifu E Tk .P(x) for every x E S2 and the LP-derivatives, ua , belong to L"(11),

3.10. A Lusin-Type Approximation for Sobolev Functions 153

0 < f cx} < k, then u E Wk , p(SZ).

Clearly, the hypothesis that the LP-derivatives belong to LP(0) is neces-sary. On the other hand, we will be able to strengthen the result slightlyby not requiring that u E TkIP(x) for all x E Q. The following allows anexceptional set.

3.9.6. Corollary. Let K C R" be compact and let SZ = R" — K. Suppose1-In-1 [7r,(K)) = 0 where the 7r, : Rn , i = 1, 2, ... , n, are n inde-pendent orthogonal projections. Assume u E Tk P(x) for all s E fZ and thatus E IP(ft), 0 < lal < k. Then u E W P(R").

Proof. Assume initially that the projections 7r; are given by

7ri(x) = (x1, • • • , x2, ... , x n )

where (ac t , ... , x„ ... , xn ) denotes the (n— 1)-tuple with the z,-componentdcleted. Theorem 3.9.5 implies that u E Wk1e(11). In view of the assumptionon K, reference to Theorem 2.1.4 shows that u E W L .P(R") since u has arepresentative that is absolutely continuous on almost all lines parallel tothe coordinate axes. Now consider D'u, oaf = 1. Since Daft Ea similar argument shows that Dail E W"P(R") and therefore that u EW2 P(R"). Proceeding inductively, we have that u E Wkie(R1 ).

Recall from Theorem 2.2.2 that u E W k iP(R") remains in the spaceWk•P(R") when subjected to a linear, non-singular change of coordinates.Thus, the initial restriction on the projections in is not necessary and theproof is complete. D

In the special case of k = 1, it is possible to obtain a similar resultthat does not require the exceptional set K to be compact. We state thefollowing [BAZ, Theorem 4.51, without proof.

3.9.7. Theorem. Let K C R" be a Borel set and suppose H" - 1 [7r; (K)] = 0where the r, : R" Rn -1 , i = 1, 2, . _ . , n, arc n independent orthogonalprojections. Let f2 = R" -- K and assume u E L o c (fl) has the property thatits partial derivatives exist at each point of iZ and that they arc in L ôc( 1 ).Then u E Wloc (R") .

3.10 A Lusin-Type Approximation for SobolevFunctions

Lusin's Theorem states that a measurable function on a compact inter-val agrees with a continuous function except perhaps for a closed set ofarbitrarily small measure. By analogy, it seems plausible that a Sobolev

Wk-1,p0-1)


function u E Wk,P(11) should agree with a function of class C k (1/) exceptfor a set of small measure. Moreover, if the requirement concerning thedegree of smoothness is lessened, perhaps it could be expected that thereis a larger set on which there is agreement. That is, one could hope that uagrees with a function of class Cl (n), 0 < l < k, except for a set of smallBk_r,p-capacity. Finally, because Sobolev functions can be approximatedin norm by functions of class C k (11), it is also plausible that the Lusin-typeapproximant could be chosen arbitrarily close to u in norm. The purposeof this and the next section is to show that all of this is possible.

In this section, we begin by showing that if u E W"(R"), then u agreeswith a function, y, of class Ct on the complement of an open set of ar-bitrarily small Bk_,, p-capacity. In the next section, it will also be shownthat Flu -- vH,, p can be made small. The outline of the proof of the exis-tence of y is as follows. If u E W k .p(Rn ) and 0 < t < k, then Theorem3.4.2 implies that u E t e 'p(x) for all x except for a set of Bk_t,p-capacity O.This means that the remainder terms tends to 0 (with appropriate speed)at Bk_t,p-q.e. x E R. We have already established that if a function uhas an 11-derivative of order t at all points of a closed set A (that is, ifu E tt 'p(x) for each x E A) and if the remainder term tends to 0 in L" uni-formly on A, then there exists a function v E Ct (Rn) which agrees with uon A (Theorem 3.6.3). Thus, to establish our result, we need to strengthenTheorem 3.4.2 by showing that the remainder tends uniformly to 0 on thecomplement of sets of arbitrarily small capacity. This will be accomplishedin Theorem 3.10.4 below.

In the following, we will adopt the notation

Mp,Ru(x) = sup iu(y)I dy0<r<R D(z,r)

whenever u E I.p(Rn ), 1 t }] <_ t Ilullk,p

whenever u E Wkip(Rn) and R < 1.

(3.10.1)

Proof. We use Theorem 2.6.1 to represent u as u = gk * f where f E LP(Rnand Iluiik,p N VG- Thus, it is sufficient to establish (3.10.1) with iluf lk,Preplaced by jl f ji p . Since lug < gk * If i, we may assume f > 0. Let

Et = f x : Mp,Ru(x) > t}

and choose r E Et . For notational convenience, we will assume that x = 0


and denote B(0, r) B(r). Thus, there exists 0 < r < R < 1 such that

Iu(y)Ipdy > tFB(r)

orP

gk(y — w)f (w)dw dy > tP.1,30.) R^

Utilizing the simple inequality (a + b)P <_ 2P -1 (aP + b") whenever a, b > 0,it therefore follows that either

P

gk(y — w) f (w)dw dy > 2 1 `P9 (3.10.2)

or

B(r) (iWI>2T

If y E B(r), then from Lemma 2.8.3(i) and the fact that gk < CIk, (2.6.3),we obtain

9k(y — w)f (w)dw <— C I f (w)

-k dwfw^<2r Iy wI

< Crk Mf (y),

where C = C(k, n). Thu.g, in case (3.10.2) holds, we have

tP Ç Cr kP M f (y)Pdy (3.10.4)B(r)

where C = C(k, p, n).We will now establish the estimate

gk (y — w) f (w)dw < C inf1gk(y — w) f (w)dw (3.10.5)

fwI>2r yEB(r) wf>2r

for all y E B(r). Recall that r < 1. Now if y and w are such that I yI < r <2r<IwI<2,we have

3 IwI ? Iw! + IyI ? Iw — yI ? IwI — IyI ? Iw! — IwI

B(r) (j wI< 2r

F

gk(y — w) f (w)dw dy > (3.10.3)

2Consequently, if y 1 and y2 are any two points of B(r), refer to (2.6.3) andthe inequality preceding it to conclude that for some constant C = C(k, n)

C C 9k(w — vi) lw — vt ln^ k ICI"-k

C <Iw -` 1/21"

Cgk (w y2)• (3.10.6)


If }w` > 2 and y E B(r), then IwS > 1w1 + 1 ? 1w — yi > Iwl — EvIIwl — 1 > (w1/2. Therefore, in this case we also have

gk(w - yi) <_ Cgk(w — g2)• (3.10.7)

Our desired estimate (3.10.5) follows from (3.10.6) and (3.10.7). Thus, incase (3.10.3) holds, there is a constant C _ C(k, p, n) such that

Ptp < C inf gk(w —y)f (w)dw

y€ 13 (r) Iwl>2r

< C inf (9k * f(v))p .

To summarize the results of our efforts thus far, for each z E B= thereexists 0 < r < 1 such that either

tp < Crkp1^f f (y)pdy (3.10.8)B (r,r)

ort < C inf gk * f (y). (3.10.9)

yE B(x,r)

Let Gi be the family of all closed balls for which (3.10.8) holds. By Theorem1.3.1, there exists a disjoint subfamily Y such that

Bk,p ({UB : B E g1}] < Bk,p!{UB : B E f]

Ç E Bk,p(B)BEf

5. E (50n-kpB(x,r)EF

< ÊB

Mf (y)p dYBEf

(by Theorem 2.6.13)

< PI li f i^p (by Theorem 2.8.2). (3.10.10)

Let ÿ2 be the family of closed balls for which (3.10.9) holds, then the def-inition of Bessel capacity implies that Bk,p [{UB : B E Gill < (C/tp)II f lif.Thus

Bk ,p lEej < Bk,p [{UB : B E gill + Bk,p [{UB : B E gall < -tF Ilf lip ,

which establishes our result. ❑

We now have the necessary information to prove that integral averagesof Sobolev functions can be made uniformly small on the complement of


sets of small capacity. This result provides an alternate proof of Theorem3.3.3, as promised in the introduction to Section 1 of this chapter.

3.10.2. Theorem. Let 1 0 there exists an openset U C R" with Bk ,p (U) < e such that

lu(v) -u(x)(Pdy -. 0B(x,r)

uniformly on Rf2 - U as r J. 0.

Proof. With the result of Theorem 3.3.3 in mind, we define

Aru(x) = Iu(v) - u(x)IPdy13(z ,r)

for x E R" and r > 0. Select i< such that 0 < i= < 1. Since u E W k.P (Rn ),there exists g E Co (Rn) such that

IIt - gIIi,p < EP+ '/ 2 .

Set h = u = g. Then

Aru(x) 2p-1 [Arg(x) + Arh(x)}+

A r h(x) < 2P-1 Ih(g1)I pdg + ih(x)I pB (z,r)

and therefore,

Aru(x) S C Arg(x) + Ih(y)Ipdy + Ih(x)IPB(x,r)

where C = C(p). Consequently, for each x E R",

sup A r u(x) < C sup Arg(x) + Mp,RIhl(x) + Jh(x)Ip •0<r<R 0<r<R

Since g has compact support, it is uniformly continuous on R" and thereforethere exists 0 < R < 1 such that

sup CArg(x) <T.

0<r<R

whenever x E R". Therefore,

x : sup A,.u(x) > 3T C {x : CMp,RI hI (x) > T} U {x : CIh(x)IP > T}0 <r <R


C {x : CMp ,RI hI (x) > is } U {x : (C7 -1 ) 1IFIh(x)I > 11.

Since h E W k .'(R"), by Theorem 2.6.1 we can write h = yk * f, whereliflip i1hllk.p. Now

{x : (Ce - E)"Ih(x)I > 1) c {x : ( -1)' /p9k *111(x) > 1)

and therefore, by the preceding theorem and the definition of capacity, weobtain a constant C = C(k, p, n) such that

Bk,p({x : sup A r u(x) > 3F-}] < C[E-9hFIt,, + ^^_ Îhllk,p]0<r<R

Fptl< C2E_F

2

< Ce.

For each positive integer i and e as in the statement of the theorem, letF,=C -l et- ' to obtain 0<R; <1 such that

Bk ,n [f. : sup Ar u(x) > 3T< < e2 - `.0<r<R,

Let00

U = ^ x : sup A r z1(2) > 3€;i=1 0<r<Rs

to establish the conclusion of the theorem. ❑

3.10.3. Remark. If we are willing to accept a slightly weaker conclusion inTheorem 3.10.2, the proof becomes less complicated. That is, if we requireonly that

Iu(y) - U(X)14B(x,r)

0

uniformly on Rn - U as r j 0, rather than

lu(y) - u(x)Ipoli -' 0,9(x,r)

then an inspection of the proof reveals that it is only necessary to show

Bk,F f{x ; Mu(x) > t}] < IIuIIk,p'To prove this, let u = gk * f, where II flip ^' 11e4k,p and define

if 1271 < Totherwise.r r (s) = an

3.11. The Main Approximation 159

Then

lu(y)Idy = rr * IuI(x)

< r,* (gk * If 1)(x)< gk * Ml f I (x),

which implies Mu < g k * MI f I. From the definition of capacity,

Bk.p[{x : Mu(x) ? t}] Bk.pRx : gk * Mill(x) ? t})• t - °IIM III IIp• Ct p lI f g, by Theorem 2.8.2,

• Ct -p llulik,p•As an immediate consequence of Theorem 3.10.2 and the proof of The-

orem 3.4.2, we obtain the following theorem which states that Sobolevfunctions are uniformly differentiable on the complement of sets of smallcapacity.

3.10.4. Theorem, Let & k be non-negative integers such that l < k and(k - t)p < n. Let u E Wk .n(Rn). Then, for each e > 0, there exists an openset U with Bk_cp(U) < e such that

r-t [I/p

lu(g) — Prll (b ) Indg -- 0

uniformlyon Rn - U as r10.

Finally, as a direct consequence of Theorems 3.10.4 and 3.6.3, we havethe following.

3.10.5. Theorem. Let t, k be Taon- negative integers such that l < k and(k - e)p < n. Let u E W k Ip(Rn ) and e > 0_ Then there exists an open setU C R" and a C I function y on Rn, such that

Bk_, , (U) < e

andD"v(x) = D"u(x)

for all x E Rn -U and 0< IcYI < B.

3.11 The Main ApproximationWe conclude the approximation procedure by proving that the smoothfunction y obtained in the previous theorem can be modified so as to beclose to u in norm.


In addition to some preliminary lemmas, we will need the following ver-sion of the Poincaré inequality which will be proved in Theorem 4.5.1.

3.11.1. Theorem. Let a E (0,1), t a positive integer, and 1 Q^SZ^,

we have the inequality

I Iu(x)1pdx < Cpt E n

^ku(zedx.I0E=t

3.11.2. Lemma. Let t be a positive integer and let u be a function W t"p(R'n)which vanishes outside a bounded open set U. Let 6,a E (0, 1) and let

E = ôU f1 x : inf IK(xlt) n (Rn — U) >(3.11.1)

a<t<6 in

where K(x, t) denotes the closed cube with center x and side-length t. Letin be a positive integer such that m < t and let e > O. Then there ezrsts afunction v E W m'p (Rn) and an open set V such that

( 1) Ilu — VJ^m,p < ei

(ii) E C V and v(x)= 0 whenxE VU (R"--U).

Proof. For A E (0,1), let Ka denote the set of all closed cubes of the form

((i1 — 1 )A, i1A] x ((i2 — 1)A, i2AJ x ... x Rin — 1)À, in A]

where i1 i i2 , ... , in are arbitrary integers. Let A < 36 and let

K1, K2,..., Kr

be those cubes of 1Ca that intersect E. Let ai be the center of K; and let

pi K(ai, 4A).

Let ( be a C°'O function on Rn, such that 0 < < 1, ((z) = 0, whenx E K(0,1) and ((x) = t when z ft K(0,3/2). Define

VA(X) = n(x) ^ ^ (z _ ai )

Zat =1(3.11.2)

for x E Rn . Clearly va(x) = 0 when d(x,E) < IA, so that, for any A, wecan define y by y = va and find an open set V satisfying (ii).


We keep i fixed for the moment and estimate

flu - UaIlt,p;P, • (3.11.3)

We observe that there exists a constant r, depending only on n and suchthat at most T of the cubes Pi intersect Pt (including Pi ). Denote these by

Pill P12 ,...,Pi.

where s < T . Then, for x E P,

vA(x) = u(x)w(x),

wherea

W (x) = j^ ^[(z - Qik)/24k=1

Now, for r E P, and any multi-index a with 0 <10/1 < t, we have

I D' w(x)I < A1A -1°1 ,

(3.11.4)

(3.1 1.5)

where A l depends only on t and n. Hence for almost all x E P; and anymulti-index 7 with 0 < IryI < t, we have

IYIID7 Va(1)1 A2 E L I D13u(x ) ,,r= o 1A1 =r

(3.11.6)

where A2 depends only on t and n.Let y be a point where K1 intersects E. Clearly, there is a subcube Q; of

P; with center y and edge length 3A. By (3.11.1), u and hence its derivativesare zero on a subset Z of Q. with

IZI > a(3A)"• , (3.11.7)

By applying the Poincaré inequality to the interior of the convex set P, weobtain, when },Q} < t,

< A3ap(1-1131) E 1D(u(x)Iedx (3.11.8)fpi llY3u(x)rdx It1=1 P^

where A3 = A3(t, Q, p, n). But, with a suitable constant A3, (3.11.8) willstill hold when 1/31 = 1. By (3.11.6) and (3.11.8) (since A < 1)

ID7 vx (x)I"dx < A4 IDcu(x)IFdrPf ItI=1 Pi

(3.11.9)


for 0 < < t, where A4 = A4(1, p, a , n). Let

XA= U Pi's=1

Then r

ID'rva(x )Ipd^ < A4 E ^ IDû(^)Ipd;^X,, ItI=t z=1 P;

for 0 <1/I < t. But each point of X), belongs to at most r of the cubes P;,hence

L. 1D 7ua(4 pdz < rA4 E x

ID4u(x)Iedz,k^k=^ a

(3.11.10)

for 0 <171 < L. Now

tlu VAIIt,p 2" E I D7va(r)Ipdi + X ID7 u(:^)Ied.r ,0[1ryl<t Xa a

so that by (3.11.10)

IIu — Va 114 < AS E0 < 171 <1 LAU

ID . u(x) Ipdz (3.1 ] .11)

where A5 - A5 (l, p, a, n). But

X),nUcUr1 {x: d(z,ÔU) < 2/a}.

Hence I(X ), fl U)I —► 0 as A j. O. Therefore by (3.11.11)

IIu — vallt,p -, 0

asA--►0+.The required function y is now obtained by putting y = Va, with suffi-

ciently small A. ❑

3.11.3. Lemma. Let 0 < A < n. Then there exists a constant C = C(A , n)such that

Ix — yl"dx < Cty — zl a ^n (3.11.12)B(z,5)

for all y,z E R' and all 6> 0.

Proof. We first show that there exists a constant C, such that (3.11.12)holds when y _ 0 and z is arbitrary.


When Izl > 36, we have

Izl <_ Ix! + Iz -xI <1x1+ Ix! + 314

so that Iz! < 2141, IxI A-n < Alzla^" , and (3.11.12) holds.When IzI < 36,

ë -n IxIA-n dx 5_ &-n IxlA-ndx = Cba-n

B(:,6) 13(0,46)

Hence, it is clear that (3.1112) holds with y = O.Since we have shown thatÔ-n f IxII-ndx < CIzIa -n

(s,6)

for all z E Rn , the general result follows by a change of variables; that is,replace z by z - y. ❑

Throughout the remainder of this section, it will be more convenientto employ the Riesz capacity, Rk,p , rather than the Bessel capacity, Bk.p.This will have no significant effect on the main result, Theorem 3.11.6. SeeRemark 3.11.7.

3.11.4. Lemma. Let k be a non-negative real number such that kp < n.Let U be a bounded non-empty open subset of Rn and F a subset of ôUwith the property that for each x E F, there is a t E (0,1) for which

IU nB(z,t)I >(3.11.13)

I$(x,t)I -where a E (0, 1). Then there exists a constant C C(n, p, k) such that

Rk,p (U U F) < Ccr -pRk,p (U). (3.11.14)

Proof. Let a, U, and F be as described above. The cases k = 0 and k > 0are treated separately.

(i) We consider first the case where k > 0. Let be a non-negativefunction iii LP(Rn) with the property that

Ix - yl k _n i(y)d1/ > 1 (3.11.15)I(k) f R

for all x E U. Let C1 be the constant of Lemma 3.11.3. It can be assumedthat C 1 > 1. Consider a point b E F and let t be such that (3.11.13) holdsfor z = b. By Lemma 3.11.3,

Ci ly - blk-n >,

Ix - ylk-ndxB(b,i)

3. Pointwise Behavior of Sobolev Functions164

so that

c:Iy - bIk-n ,(y)dyL"

and by (3.11.15),

Hence by (3.11.13),

L (b t) Ix -- ylk-n^(y)dy dz[Le.

> -y(k)t —n IU fl B(b, t)I.

rCa Iy - bl k-n^(^)db ? a.

7(k)JRh1+

Put r? = C2(7 -1 0. Then

17(k) R" Iy — xIk-ni(y)dy ? 1

for all x E F, and therefore,

Rk,(F) < 11711; =(C.2 ) I'

iS IIp.

Thus(C2)P

Rk,p(F) <

Rk ,p (U)•a

The required inequality now follows.(ii) Now let k = 0, so that Rk,p becomes Lebesgue measure.

the collection of all closed balls B with center in F and radiusand 1 such that lung > a.

IIHence, by Theorem 1.3.1, there exists sequence {Br }, Br E B,Br fl B. = 4 when r s and

Let 8 bebetween O

(3.11.16)

such that

oc,FC V Br.

r = t

Thus

ill <E or' = 5n EIBrIr -1 r=1

and by (3.11.16)00

< 5nQf1 E IU fl Br I < 5nalUI.r=1

Since a < 1, the required inequality follows. 0


3.11.5. Lemrna. Let p > 1, k a non-negative real number such that kp < nand I a positive integer. There exists a constant C = C(n, p, k, l) such thatfor each bounded non-empty open subset U of Rn , each u E W 1 'p(R")which vanishes outside U and every e > 0 there exists a C°° function v onRn with the properties

(i) I}u — viit,p < c ,

( 1i ) Rk,p( Spt y) < CRk,p(U) and

(iii) spty C V = Rn n{x: d(x,U) <e}.

Proof. Let U, u, and e be as described above. Since U # 0, it follows thatRk,p (U) > O. Let

E =âU n x: inf IB(x , t) — U}] > 1

-o <t<1/2 in 2(3.11.17)

Then E is closed. By Lemma 3.11.2 there exists a function vo E W t,p(Rn )and an open set Vo such that

IIu - vollt,p < 2^,ECVo and vo (x)=0 when x EVo U (R"—U). Set

F= aU - E.

Then, for each x E F there exists t E (0,1/2] such that

IU nB(x, t)IcrIB(x, t)I —

where a = 1 -- 1/(2a(n)). Let C, be the constant3.11.4. Then

Rk,p(U U F) < 2CRk,p(U),

(3.11.18)

(3.11.19)

appearing in Lemma

(3.11.20)

where C = 2Cla —p. Let

B = R" n {x:vo (x)# 0}.

Then B C U U F and hence Rk,p (B) < i CRk,p (U), so that there exists anopen set W with B C W and

Rk,p (W) < CRk,p (U).

By applying a suitable mollifier to yp we can obtain a C°° function v withsptvCVnWand

(3.11.21)1Ilvo — vllt,p < 2e-

F (3.11.22)Rt-m,p(U) < 1 + C


It follows from (3.11.18) and (3.11.21) that y has the required proper-ties. D

We are now in a position to prove the main theorem.

3.11.6. Theorem. Let t, m be positive integers with m < t, (t - m)p < nand let Si be a non-empty open subset of R". Then, for u E We (n) andeach e > 0, there exists a Cm function v on Si such that if

F = SZ n {x : u(x)o v(x)},

thenRi-,,,p(F) < e and Ilu --- vll,n,p < e.

Proof. It can be assumed that the set A = SZn{x : u(x) # 0} is not empty.Initially, it will be assumed that SZ = Rn and A bounded. We will showthat there exists a Cm function y on R" satisfying the conclusion of thetheorem and that spt v is contained in the set V = Rn n {x : d(x., A) < e}.

Let C be the constant of Lemma 3.11.5. Let u be defined by its valuesat Lebesgue points everywhere on 0 except for a set E with Bt, p (E) =R,,p(E) = O. By Theorem 3.10.5 there exists an open set U of Rn and aCm function h on R", such that U D E,

andh(x) = u(x)

for all x E Rn — U. We may assume that spt h C V and U C V. Bysubstituting t - m for k and u - h for u in Lemma 3.11.5, we obtain a C°°function cP on R" such that

and

Ilu- h -wllm,p< C,

Rt — m ,p(sPt SP) <_ CRt-m,p(U),

(3.11.23)

(3.11.24)

spt cp C V. (3.11.25)

Put y = h +cp. Then the second part of the theorem follows from (3.11.23).Clearly,

F c R" n j {x : h(x) 0 u(x) } u spt (pl c U u spt cp, (3.11.26)

so that by (3.11.24)

Rt-m,p(F) ç (1+ C)Rt-m,p(U).

andsptcP, C Int C,+l•


Thus, the first part of the conclusion follows from (3.11.22). Since spt h andspt cP are both contained in V, it follows that spt v c V.

We now consider the genera] case when SZ is an arbitrary open subset ofR'. Let (Cl r°o be an infinite sequence of non-empty compact sets, suchthatC; C Int C‘ .4. 1(3.11.27)

for i a non-negative integer and

hm C, = SZ. (3.11.28)1—.00

Put C_ 1 = 0. For each i > 0, let (p, be a C°° function on Rn such that0 <(P.*< 1 ,

C1 C int{x : cp,(x) = 11, (3.11.29)

(3.11.30)

Put^Go = (Po and fir, = (pi — gyp,_ 1 (3.11.31)

when i > 1. Then each t' is C°° on R" with compact support and

spt C (IntCs+l) — C1_ 1 . (3.11.32)

Hence, for each x E SZ, i(x) 0 for at most two values of i. Therefore

00

E 0, ( x ) = 1s-o

for all xE U. For each i=0,1,2,... define

(3.11.33)

u=(x) _f{ u(x),M') when x ESZ

when x SZ.(3.11.34)

By the conclusion of our theorem proved under the assumption that SZ =R", there exists for each i > 0 a Cm function v, on Rn with compactsupport such that

andRi-m,p(Fi) < 2:+1

whereF = R" fl {x : ui (x) # vi(x)I•

Moreover,

^

II ZI, -

tl , Il m.P < 21 +1 '

spt v, c (Int C; +l ) — Cc-1 • (3. 1 1 .37)


For each x E SZ, there are at most two values of i for which v i (x) 0 O.Hence we can define 00

v(x) = E.,(x)

for x E SZ. It is easily seen that F C U=°_0 F;, hence

Rt_m,p(F) < e.

AlsoOC

E oui U < E. ai!o

3.11.7. Remark. We have seen from earlier work in Section 2.6, thatRk,p < CBk,p and that Rk,p and Bk,p have the same null sets. However, italso can be shown that Bk,p Ç C[Rk p + (Rk,p )n' (n -kp) t for kp < n, cf. [A5].Therefore, the Riesz capacity in the previous theorem can he replaced byBessel capacity.

Exerc ises

3.1. Prove that the statement

lim u(y)dy = u(x)r^0 B(x,r)

for B 1 ,p-q.e. x E R" and any u E W 1 "F(R") implies the apparentlystronger statement

lirn 1u(y) — u(x)Idy = or^0 8(z,r)

for 1314,-q.e. x E R". See the beginning of Section 3.3.

3.2. It was proved in Theorem 2.1.4 that a function u E W'•P(Rn) hasa representative that is absolutely continuous on almost all line seg-ments parallel to the coordinate axes. If a restriction is placed on p,more information can be obtained. For example, if it is assumed thatp < n — 1, then u is continuous on almost all hyperplanes parallelto the coordinate planes. To prove this, refer to Theorem 3.10.2 toconclude that there is a sequence of integral averages

Ak(x) = u(y)dyB(s,)

which, for each r > 0, converges uniformly to u on the complementof an open set U whose B 1 , p-capacity is less than e. Hence E

Exercises 169

f,>oUU is a set of /31,p-capacity O. It follows from Theorem 2.16.6(or Exercise 2.16) that the projection of E onto a coordinate axishas linear measure O. Note that u is continuous on 7r -1 (t), t ¢ ir(E)where it denotes the projection. Corresponding results for p > n - k ,

k an integer, can be easily stated and proved.

3.3. At the beginning of Section 3.9, an example is given which shows thatu need not be hounded when u E 1,471 ."(B(0, r)}, r < 1. This examplecan be easily modified to make the pathology even more striking. Letu(x) = log log( 1 /IxI) for small ix' and otherwise defined so that u ispositive, smooth and has compact support. Now let

v (z ) =_ E 2 -k u(2 - rk)k=1

where {rk } is dense in Rn. Then y E W 1,n(R") and is unbounded ina neighborhood of each point.

3.4. Use (2.4.18) to show that if u E Wiop (R"), p > n, then u is classicallydifferentiable almost everywhere.

3.5. Verify that IIuIj Tk.p(z), which is discussed in Definition 3.5.4, is in facta norm.

3.6. If u E W 1 .n(R" ), the classical Lebesgue point theorem states that

li rn Iu(x) — u(xo)Idx = 0 (*)r-0 B(xo,r)

for a.e. x o . Of course, u E L 1 (R") is sufficient to establish this result.Since u E W 1 .p(R11 ), this result can be improved to the extent that(*) holds for /31 ,p-q.e. xo E R" (Theorem 3.3.3). Give an examplethat shows this result is optimal. That is, show that in general it isnecessary to omit a B 1 ,p-null set for the validity of (*).

3.7. Prove that (3.3.22) can be improved by replacing p by p' = np/(n -kp).

3.8. A measurable function u is said to have a Lebesgue point at so if

lem B(zo,r)

Iu(y) - u(xo )jdy = O.r --D

A closely related concept is that of approximate continuity. A mea-surable function u is said to be approximately continuous at s o ifthere exists a measurable set E with density 1 at xo such that u iscontinuous at xo relative to E. Show that if u has a Lebesgue point


at x0 , then u is approximately continuous at x 0 . See Remark 4.4.5.Show that the converse is true if u is bounded and that it is falsewithout this assumption.Another definition of approximate continuity is the following. u isapproximately continuous at xo if for every e > 0, the set

A , _ {x : tu(x) — u(xo )I >_ e}

has density 0 at xo. Af is said to have density 0 at xo if

}i IAE n B(xo , r)I = 0.mr ~0 I B(xo , r)I

Prove that the two definitions of approximate continuity are equiva-lent.

3.9. The definition of an approximate total differential is analogous tothat of approximate continuity. If u is a real valued function definedon a subset of R", we say that a linear function L : R" --■ RW is anapproximate differential of u at 10 if for every e > 0 the set

lu(x) — u(xo) — L(x — xo) > €Ix — xol

has density 0 at xo. Prove the analog of Exercise 3.8; show that if uis an element of t 1 "(xo), then u has an approximate total differentialat xo .

3.10. The definition of an approximate total differential given in Exercise3.9 implies that the difference quotient

lu(x) — u(xo) — L(x — xo)Ix — xol

approaches 0 as x --► x0 through a set E whose density at 1 0 is1. In some applications, it is necessary to have more informationconcerning the set E. For example, if u E W I.P(R"), p > n — 1, thenit can be shown that u has a regular approximate total differential atalmost all points x 0 . The definition of this is the same as that for anapproximate total differential, except that the set E is required to bethe union of boundaries of concentric cubes centered at xo . To provethis, consider

ô (t+ z) = u(x0 + tz) — u(so) L(x),tand define

'Yso (t) = sup{Iut(z)i : z E OC}

A , = {x:

exercises 171

where C is a cube centered at xo. Since xo is fixed throughout theargument, let u t (z) = u to (t, z).STEP 1. For each xo and each cube C with xo as center, observe thatu t E W 1 "P(C) for all sufficiently small t > O. With

ckzo(t) =C

(lutI F + IHurI P )dx

prove that czo (t) —* 0 as t ---t 0 for almost all zo .STEP 2. Show that u has a regular approximate differential at all zo

that satisfy the conclusion of Step 1 and for which Du(zo ) exists. Forthis purpose, let L(z) = Du(xo ) • z. Since u t E W 1 'P(C), it followsthat ut E W 1 "P(Kr ) for almost all r > 0 where Kr is the boundary ofa cube of side length 2r. Moreover, from Exercise 3.2, we know thatu t is continuous on all such K. Let

4/t(r) _ utlF + ' Du t I)dHn1 .JKr U

Let Et = [1/2,11P (r : Sp t (r) < azp (t) 1 /2 } and conclude that

1([1/2,11— Et)I < azo(t) 1/2 .

STEP 3. Use the Sobolev inequality to prove that for z E Kr , r E E1 ,

and ,D = (n— 1)/p

1/p

iu t (z)I < Mr -11K Iut[P dH"- 1

KT

1/p+ Mr' IDut1pdHn -1

K r

< Mr-13Spt(r)1 /P - Mr1-13S001(r)1/P

< [M213 + Mlazo (t) 1 /2P ,

where M = M(p, n).STEP 4. Thus, for z E Kr and r E Et ,

yzo (t • r) = r -1 sup{lu t (z)j : z E Kr }< 2[M2s + 11 • axo (t) 1 /2P.

STEP 5. For each positive integer i, let t i = 2- ` and let Et, be theassociated set as in Step 2. Set A = u Et and note that 0 is apoint of right density for A (Step 1) and that 720 (i) --+ 0 as t ---+ 0,t E A.


3.11. Prove that Ti defined in (3.6.2) belongs to C'(U - A).

3.12. Give an example which shows that the uniformity condition in thesecond part of the statement in Theorem 3.5.7 is necessary.

3.13. In this and the next exercise, it will be shown that a function withminimal differentiability hypotheses agrees with a C 1 function on aset of large measure, thus establishing an extension of Theorem 3.11.6.For simplicity, we only consider functions of two variables and beginby outlining a proof of the following classical fact: If u is a measur-able function whose partial derivatives exist almost everywhere on ameasurable set E, then u has an approximate total differential almosteverywhere on E. See Exercise 3.9 for the definition of approximatetotal differential.STEP 1. By Lusin's theorem, we may assume that E is dosed andthat u is its partial derivatives are continuous on E.STEP 2. For each (x, y) E E consider the differences

i(x, y; h, k) = lu(x + h, y + k) - ri(x, y) - hptu(x, y) - kDzu(x, y)l

a l (x, y; h) = lu(x + h, y) - u(x, y) - hD l u(x, y)l

42(x, y; k) = lu(x, y + k) - n(x, y) - kD2u(x, y)}where D 1 = 8/ax and D2 = a fay. Choose positive numbers e, r.Using the information in Step 1, prove that there exists a > 0 suchthat the set A C E consisting of all points (x, y) with the propertythat

l{x+ h : Ai(x,yy;h) < rh, (x+h,y) E E,

a< x< b, lb - al <o-, Ihl < lb -- al}l> (1 ` e)1b- al

satisfies I E - Al < e. Perhaps the following informal description of Awill be helpful. For fixed (x, y), let us agree to call a point (x + h, y)"good" if i 1 (x, y; h) < rh and (x + h, y) E E. The set A consistsof those points (x, y) with the property that if Iz is any intervalparallel to the x-axis containing z whose length is less than o-, thenthe relative measure of the set of good points in Ix is large.STEP 3. Now repeat the analysis of Step 2 with E replaced with A toobtain a positive number a l < o' and a closed set B C A, IA - BI < cwhich consists of all points (x, y) with the property that

I {y + k : 02(x,y; k) <Tk, (x,y +k) E A,

a < y -< b, lb - al <17 1 , lkl < lb- al}' > (1 - E)lb - aI.

Exercises 173

STEP 4. Let o2 < aj be such that

IDIu(x +h2, y+k2) — Dlu(x+ h i , y+kt)1 < T

for any 2 points (x+h2, y+k2), (x+hi, y+k,) in E with 1h2 — hil <112,Ike — k11 <al.STEP 5. Choose (xo, yo) E B and let R = [a l , bul x [a2 , b2J be anyrectangle containing (x o , yo ) whose diameter is less than a2 < a l < a.Let

E2 = {(yo + k) : A2(xo, yo; k) < T1kI,

(xo, yo + k) E A, a2 < yo + k < b211

and for each (yo + k)

E1(yo + k) _ {(xo + h) : O1(xo, yo + k; h) < 7Ih1,

(xo+h,yo+ k) E E}.

Now for any (h, k) such that Vo + k E E2 and xo + h E E1(yo + k),we have (xo + h, yo + k) E E n R and therefore

A(xo, WI; h, k) <Di (xo, Yo; h)+ 42(xo, Yo; k)+ Ihi ID 1144, yo + k) — D1u(xo, yo)I

< T(IhI+ Ik1).

From this conclude that

1B n R n {(xo + h, yo + k) : A (x0, yo; h, k) < 21. (Ihl + IkI)}i

> (1 — E) 2 01 — (2 1)(b2 — ai) = ( 1 — e) 2 IR1-

STEP 6. Take R to be a square with (xo,Vo) as center and appeal toExercise 3.8 to reach the desired conclusion.

3.14. We continue to outline the proof that a function whose partial deriva-tives exist almost everywhere agrees with a C' function on a set oflarge measure. Let u be a real valued function defined on a measurableset E C R", and for each positive number M and x E E let

A(x, M) = En {y : Iu(y) — u(x)I < MIy — xI}.

If A(x, M) has density 1. at z, u is said to be of approximate lineardistortion at x. Our objective is to show that if u is of approximatelinear distortion at each point E, then there exists sets Ek such thatE = Uk 1 Ek and u is Lipechitzian on each of the sets Ek.

174 3. Pointwise Behavior of Soholev Functions

STEP 1. If z 1. and z 2 are any two points of R", then

_ I B(xl, ix2 — z, l) n B(x2, Ix9 - x11)1 a

— IB(x1, Ix2 — x11)1 + IB(x2,1X2 — Z11)1

is a positive number less than 1 which is independent of the choice ofz 1 and z2.STEP 2. For each positive integer k, let Ek be the set of those pointsx E E such that lu(x)l < k and that if r is any number such that0 < r < 1/k, then

IA(x, k) n B(x, r )I > 1 — rx.IB(x, r)I

Prove that E = Uk Ek.

STEP 3. In order to show that u is Lipschitz on Ek, choose any twopoints z 1 , z2 E Ek. If (x2 — x 1 I > 1/k, then

Iu(x2) — u(xl)1 < 2k 2 Ix2 — x11.

Thus, assume that1

0 <Iz2 -z 1 I< k .Let

A1 = A(xl, k) n T3(x1, Ix2 - x1 1),A2 = A(x2, k) n B(zz, Ix2 - xi l).

Prove that IA1 n Al l > O. If ac' E A1 n A2 1 show that

Iu(z") — u(z;)l < klz' — x,l, i = 1,2Ix" — x. l< Ix2 — z l I, i= 1, 2.

Now conclude that

1u(x2) - u(x 1)1 < 2k1z2 - x 1 I.

STEP 4. 1f u has partial derivatives almost everywhere, appeal to theprevious exercise to conclude that u is of approximate linear distor-tion at almost every point. Now refer to Theorem 3.11.6 to find a C'function that agrees with u on a set of arbitrarily large measure.

3.15. Suppose u E 1471 .P (Rn). Prove that for $ 11p -q.e. x E R te , u is abso-lutely continuous on almost every ray A s whose endpoint is z.

3.16. Let f be a measurable function defined on [0, II having the propertythat f' exists everywhere on [0, 1] and that If'l is integrable. Provethat f is an absolutely continuous function.


Historical Notes3.1. The idea that an integrable function has a representative that can beexpressed as the limit of integral averages originates with Lebesgue [LE21.The set of points for which the limit of integral averages does not exist (tlieexceptional set) is of measure zero. Several authors were aware that theexceptional sets associated with Sobolev functions or Riesz potentials weremuch smaller than sets of measure zero, cf. [DL], [ARS1j, [FU], [FL], [C11.However, optimal results for the exceptional sets in terms of capacity wereobtained in [FZ], [BAZ], [ME2], [CFR1. The development in this section istaken from IMIZ1.3.2. The results in this section are merely a few of the many measuretheoretic density theorems of a general nature; see [F, Section 2.10.91 formore.

3.3. Theorem 3.3.3 was first established in [FZ] for the case k = 1, and forgeneral k in [BAZ], [ME2], and [CFR]. The concepts of thinness and finecontinuity are found in classical potential theory although their develop-ment in the context of nonlinear potential theory was advanced significantlyin [AM], [HE2], [HW], [ME31. The proof of the theorem in Remark 3.3.5was communicated to the author by Norman Meyers.

3.4. Derivatives of a function at a point in the LI-sense were first studiedin depth by Calderon and Zygmund [CZ]. They also proved Theorem 3.4.2where the exceptional set was obtained as a set of Lebesgue measure zero.The proof of the theorem with the exceptional set expressed in terms ofcapacity appears in [BAZ], [ME2], and [CFR].3.5. The spaces Tk(E), tk (E), Tk.P(s), and tk•e(x) were first introducedin [CZ] where also Theorem 3.5.7 was proved. These spaces introduce butone of many methods of dealing with the notion of "approximate differen-tiability." For other forms of approximate differentiability, see [F, Section3.1.2], [RR].3.6-3.8. The material in these sections is adopted from [CZ]. It shouldbe noted that Theorem 8 in [CZ] is slightly in error. The error occursin the following part of the statement of their theorem: "If in additionf E tt(xa) for all x0 E Q, then f E bu (Q)." The difficulty is that forthis conclusion to hold, it is necessary that condition (1.2) in [CZ] holdsuniformly. Indeed, the example in [WH] can be easily modified to showthat this uniformity condition is necessary. Theorem 3.6.3 gives the correctversion of their theorem. In order for this result to be applicable within theframework of Sobolev spaces, it is necessary to show that Sobolev functionsare uniformly differentiable on the complement of sets of small capacity.This is established in Theorem 3.10.4.

In comparing Whitney's Extension theorem (Theorem 3.5.3) with theLP-version (Theorem 3.63), observe that the latter is more general in the


sense that the remainder term of the function u in question is required toapproach zero only in LP and not in L°O. On the other hand, the functionis required to be defined only on the set E in Whitney's theorem, while thecondition u E t iP(x) in Theorem 3.5.3 implies that u E L1[B(x, r)] (for allsmall r > 0) thus requiring u to be defined in a neighborhood of r.

3.9. Theorem 3.9.4 and the preceding lemmas are due to Calderon [CA4];the remaining results are from [BAZ].

3.10. The main result of this section is Theorem 3.10.2 which easily impliesthat Sobolev functions are uniformly differentiable in LP on the complementof sets of small capacity. The proof is due to Lars Hedberg. Observe thatthe proof of Theorem 3.10.2 becomes simpler if we are willing to accept asubsequence {r ; , } such that

u(y) — u(x) jpdy —• 0B(x,r, 1 )

uniformly on JP —U where Bk, p (U) < e. This can be proved by the methodsof Lemma 2.6.4. However, this result would not be strong enough to applyTheorem 3.6.3, thus not making it possible to establish the approximationresult in Theorem 3.10.5.

3.11. These results appear in [MIZ]. The main theorem (Theorem 3.11.6)is analogous to an interesting result provcd by J.H. Michael [MI] in thesetting of area theory. He proved that a measurable function f defined ona closed cube Q C R" can be approximated by a Lipschitz function g sucht hat

t{x : f ( 2 ) g(x)} I < e

and IA( f , Q) — A(g, Q)I < e where A(f , Q) denotes the Lebesgue area of fon Q. Theorem 3.11.6 was first proved by Liu [LI) in the case m = t.

4

Poincaré InequalitiesA Unified ApproachIn Chapter 2, basic Sobolev inequalities were established for functions inthe space Wô 'p(St). We recall the following fundamental result which is aparticular case of Theorem 2.4.2.

4.1.1. Theorem. Let 0 Rn be an open set and 1< p<n. There is aconstant C = C(p, n) such that if u E W'F (SI), then

Ilullp•;n < CIIDuIII.p;n (4.1.1)

where p' =np/(n - p).

Clearly, inequality (4.1.1) is false in case u is the function that is identi-cally equal to a non-zero constant, thereby ruling out the possibility thatit may hold for all u E W1'F(1). One of the main objectives of this chapteris to determine the extent to which the hypothesis that u is "zero on theboundary of Stn can be replaced by others. It is well known that there area variety of hypotheses that imply (4.1.1). For example, if we assume thatSZ is a bounded, connected, extension domain (see Remark 2.5.2) and thatu is zero on a set S with ISI = a > 0, then it can be shown that (4.1.1)remains valid where the constant C now depends on a, n, and 1. This in-equality and others similar to it, are known as Poincaré-type inequalities.We will give a proof of this inequality which is based on an argument thatis fundamental to the development of this chapter. A general and abstractversion of this argument is given in Lemma 4.1.3.

There is no loss of generality in proving the inequality with p' replacedby p. The proof proceeds as follows and is by contradiction. If (4.1.1) werefalse for the class of Sobolev functions that vanish on a set whose measureis greater than a, then for each integer i there is such a function u; withthe property that

IIuiIIp;n > iIIDu iIIP;^.Clearly, we may assume that Hu; Ij 1,pp = 1. But then, there exist a sub-sequence (denoted by the full sequence) and u E W l'p(Q) such that u ;

tends weakly to u in W 1 "p(11). By the Rellich-Kondrachov compactnesstheorem (Theorem 2.5.1, see also Remark 2.5.2) u; tends strongly to u inL'(0). Since IIu ; fI1,p;n = Zit follows that IIDu,UI p;n --+ 0 and therefore thatII Dulip;n = O. Corollary 2.1.9 thus implies that u is constant on S2. This

178 4. Poincaré Inequalities—A Unified Approach

constant is not 0 since Ilull p;^ = 1. Now each ui is 0 on a set S, whose mea-sure is no less than a. The strong convergence of ui to u in LP(S2) impliesthat (for a subsequence) u, —' u almost everywhere on S = rlj"_ U°°_ ; Si.Since ISI > a > 0, this contradicts the conclusion that u is equal to anon-zero constant on Il.

A close inspection of the proof reveals that the result also remains valid ifwe assume fI u(x)ds = 0 rather than u 0 on a set of positive measure. Inthis chapter we will show that these two inequalities and many other relatedones follow from a single, comprehensive inequality obtained in Theorem4.2.1.

4.1 Inequalities in a General Setting

We now proceed to establish an abstract version of the argument givenabove which will lead to the general form of the Poincaré inequality, The-orem 4.2.1.

4.1.2. Definition. If X is a Banach space and Y C X a .subspace, then abounded linear map L : X --+ Y onto Y is called a projection if 1! o L = L.

Note that(4.1.2)

for there exists x E X such that L(x) = y and y = L(x) = LIL(x)1 = L(y).

4.1.3. Lemma. Let Xo be a normed linear space with norm It HD and letX C X0 be a Banach space with norm II Il• Suppose It II = II lia + II Ili whereI^ Ill is a semi-norm and assume that bounded sets in X are precompact inXo . Let Y = X n Ix : IIxlII = 01. If L : X -; Y is a projection, there is aconstant C independent of L such that

Ilx — L(z)Ilo < CIILII (4.1.3)for all x E X .

Proof. First, select a particular projection L' : X —, Y. We will prove thatthere is a constant C' = C' (#I L' II) such that

Ilx — L'(x)llo <_ ^'Ilxllr, (4.1.4)

for all x E X. We emphasize that this part of the proof will produce aconstant that depends on L' .

If (4.1.4) were false there would exist x i E X such that

llx■ — L ' (^s)Ilo > =Ilxilti, i = 1,2,... .

4.1. Inequalities in a General Setting 179

Replacing r i by xi/Uri — L i (xi)11 0 it follows that

Ilxi — L'(x.)II o 1 and Ilx^lliLet z,-r,— L'(x t ). Then

Ilzilf 1= IIx — L'(:r )IIi + If L'(x^)IIÎIx=111

since Il L' (xi) II 1 = O. Hence, z, is a bounded sequence in X and therefore,by assumption, there exist a subsequence (which we still denote by { z i } )and z E Xo such that II zi -- z Ila —00. Since I1 zi jf 1 —0. 0 it follows that z, is aCauchy sequence in X and therefore It; — zII —, 0. Note that IIzIIo = 1 andIlzll^ = O. Thus z O, z E Y, and L'(z)=z by(4.1.2). But L'(z ; )—, L'(z)and L'(z,) = 0, a contradiction.

The next step is to prove (4.1.3) for any projection L where C does notdepend on L. Let L : X — Y be a projection and observe that

r— L(r) = x — L'(x) — L(x — L'(x)).

Hence, by (4.1.4),

llx — L(r)Ilo <_ 11x — L'(x)Ilo + IIL(x — L ' (z))Ilo< C

' llxlll + — L`(r))flC'Iirlll + IILII fl(r — L'(x))II

< Cr llrlll + IILII (II(x — L'(z))Ilo + I141}since 1IL'(x)Ii i = 0. Appealing again to (4.1.4) we obtain,

lix — L(x)110 < C' llxlli + IILII [C'llxll l + Il x ll ll(C' + (C' + 1 )IILII)ItxlIi.

Since L is a projection, 1141 > 1 and the result now follows. O

We now will apply this result in the context of Sobolev spaces. In par-ticular it will be convenient to take X = WmiP(Tl).

For notational simplicity, in the following we will let the characteristicfunction of S) be denoted by 1. That is, let xn = 1. Also, let Pk(R") denotethe set of all polynomials in R" of degree k.

4.1.4. Lemma. Let k and m be integers with 0 < k < m and p > 1.Let S? C R" be an open set of finite Lebesgue measure and suppose T E(Wm-k m(SZ))' has the property that T(1) O. Then there is a projectionL : Wm'P(SZ) —. Pk(R") such that for cash u E Wm 'D(SZ) and all < k,

^ 0.

T(Dru ) = T(D°iP) (4.1.5)


where P L(u). Moreover, L has the form

4%0= E T(Pa(Du))x«

«1Sk

where P« E Pk(R"), Du = (D i u, D2 u, . . . ,13.u), and

II nI ? C•GIT(111) k+1

C = C(k,P, PI)•

Proof. If P E Pk(R') then P has the form P(x) = EÉ,71=0 a.:7 andtherefore

D0P(0) = a!aa

for any multi-index a. Consequently, by Taylor's theorem for polynomials,

k-1«1 Da+o p(0 ) ÂDa P(x) = E A'IQ1=o

ork-1n1 (a + Q)!

rAD"P(x) = E 610-1-(3

1o1=o (3!

In particular,D'P(x ) = aaa !

if IcI = k. Thus, in order to satisfy (4.1.5), the coefficients as of the poly-nomial must satisfy

T(Dau)as = a!T(1) , (4.1.6)

if IcI = k. Similarly, if lai - k -- 1 then

D°P(x) = a, + E as+ p (a + 0) 1 xA

al .IQ1=1 p!

Consequently by using (4.1.6), (4.1.5) will hold if

as = T (D"u) -(a + f3 s)! T (x )

a.T(1) as+p a! /3! T(1)

101=1

where lal = k - 1. Proceeding recursively, for any lai < k we have

a T(D°U) - k

- IQ1 a«+A

(a + p)! T (x^) « a!T(1) a!/3! T(1)

1A1=1

(4.1.7)

4.1. Inequalities in a General Setting 181

It is easily verified that L is a projection since L(u) = P implies Da [L(u)] =DaP for any multi-index a. But then,

T(DQu) = T(DQP) = T[Da(Lu)]

and reference to (4.1.7) yields the desired conclusion.In order to estimate the norm of L, let u E Wm.p(1) with Ilulf m ,p;Ç < 1.

Then IILII <_ IIL(u)Ilm,p;n = IIPIIm,p;fl where P(x) = > ki ,rl_o a.y x1'. Now

111316,pin < C(I1I) E la-rI.I-v1= 0

To estimate the series, first consider IaQI, lai = k. Note that for ial = kand any non-negative integer t,

1171 •T(1)r I

+1

a!T(1) a&T(1)1+t C(l, p, iHI) IITI( 1) )

(4.1.8)

because T(1) In particular, this holds for = k. Hence from(4.1.6) it follows that

k+11 '2' 1 <

a&T II < C(k^P, ICI) ^ I

If lal _ k-1, k > 1, then from (4.1.7), (4.1.9) and the fact that IITII/T( 1 ) >

ÎI <_ C(k,p, IÎ)

I0I -1 Ip

IaQI < ÎTII + C(k , 1 11) E aQ+ Q ?,ITII

a.T(1) I0I=1 (1)

< C(k, p, I0I) TII( Tw )

2IIk+ 1

< C'(k,p, IÎ) TII ^.(Z)

l#

In general, if lai = k — i , k > ti, we have

!co <C(k,p,InI) IITII =+1T(1)

<_ C'(k,p, If2I)IITII(1) k+1 -TProceeding in this way, we find that

fill ) k+1

II^

T

(4.1.9)


In the preceding analysis, if we knew that the distribution T was a non-negative measure p, then we would be able to improve the result. Indeed,suppose the measure satisfies the inequality

< Mp(Q) (4.1.I0)fn edp(x)

for every Ic < k. Of course, such an M exists if either SZ or spt a is hounded.Then the estimate of IILII becomes sharper because, with T = p, the termT(70 )/T(1) in (4.1.7) is bounded above by M, thus implying that

IILII < C(k,p,M) I I()

Hence, we have the following corollary.

4.1.5. Corollary. Let k and m be integers with 0 < k < m and p > 1.Let SZ C R" be an open set and suppose p E (Wm-k .p(1l)) . is a non-negative non-trivial measure satisfying (4.1.10). Then there is a projectionL : —> Pk(Rn) such that for each u E Wm.p(1) and all ice' < k,

p(D"u) = p(DaP) (4.1.11)

where P = L(u). Moreover, L has the form

L(u) = E p (Pa( Du ))xaiai<k

where P, E Pk(R11 ), Du = (D i u, Dzu, ... , D"u), and

IILII < C -(Ilpllp(f)) '

C = C(k, p, M).

4.2 Applications to Sobolev SpacesWe now consider some of the consequences of the previous two results whenapplied in the setting of Sobolev spaces. Thus, if 0 < k < m are integers,p > 1 and SZ C R" is a bounded, connected, extension domain (see Remark2.5.2), we employ Lemma 4.1.3 with X = Wm ,p(11) and X0 = W47k.P(0). Itfollows from the Rellich-Konrachov imbedding theorem (see Exercise 2.3)that bounded sets in Wm.P(S)) are precompact in W k,P(11). Set IIuIIo =Il ull k JAI and IlullL = IlDk+i ullm-(k+1),p;f2 where Dk+lu is considered as thevector {Dau) lai = k + 1. Clearly, (lull = Hullo + Hulk is au equivalentnorm on Wm (0) . Moreover, it follows from Exercise 2.7 that llull1 = 0 if

4.2. Applications to Sobolev Spaces 183

and only if u E Pk(Rn ). If T E (Wm -kip(SZ))' with T(I) 96 0, then Lemma4.1.4 asserts that there is a projection L : Wm'P(SZ) Pk (Rn) such that

IILII C (

fli ) k+]

T(1)

Therefore, Lemma 4.1.3 implies

IIu - L(u)Ilk,p;s1 CIILII IID k+ ùllm- (k +1),p;n k + 1^ C 11111 IIDk +lUl l m— (k+l),D;f1T(1)

These observations are summarized in the following theorem

4.2.1. Theorem. Suppose 0 < k < m are integers and p > 1. Let SZ C Rnbe a bounded, connected extension domain. Let T E (Wm -k'P(S2)) 4 be suchthat T(1) O. Then, if L : Wm'P(l) Pk (R1z) is the projection associatedwith T,

k+ 1

IIu - L(u)IIk,D;n <_ C 11Thl II Dk+1Ullm--(k+1),p;r1( (1) )

where C = C(k, p, 12).

(4.2.1)

It will now be shown that the norm on the left side of (4.2.1) can bereplaced by the LP . -norm of u -- L(u), where p' = np/(n - mp). For thiswe need the following lemma.

4.2.2. Lemma. Suppose m > 1 is an integer and p > 1. Let SZ C Rn be abounded extension domain. Then for each integer k, I < k < m - 1, ande > 0 there is a constant C = C(n, m, p, k, r, 1) such that

llD k l4IP;n < CIIuII F;c + ellDmullD;si, u E Wm 'A ( 1l) (4.2.2)

whenever u E Wm .P(11).

Proof. We proceed by contradiction. If the result were not true, then foreach positive integer i there would exist u i E Wm'P(12) such that

IlD k w llp;^ > + (4.2.3)

replacing ui by ui/Iluillm, a we may assume that IIuÎIm,p ;n = 1, i =

1, 2, .... Hence, from Exercise 2.3 there is u E Wm.P(S2) and a subsequence(which we assume without loss of generality is the full sequence) such thatu i — u strongly in Wm -1 "(f2). In particular ui — u in L"(11). Since

II!D^ù^llD;n <_ Ilu: ll m- 1,v;n _< IIuÎI^,D;^, (4.2.4)


it follows from (9.2.3) that u, —, 0 in LP(1) and therefore u = O. But thenui —, 0 in Wm-i , p(S2) and consequently IID k ni lip;n —' 0 by (4.2.4). Thisimplies that IIDmu, ll pp —, 0 or u i —, 0 in Wm'p(SZ), a contradiction to thefact that IIu,II m , p;n =1.

If v E Wm'p(RR) has compact support, then it follows from the funda-mental Sabolev inequalities, namely Theorem 2.4.2, 2.9.1, and 2.4.4 that

I IU IIp• <_ GlI v l lm ,pwhere p' is defined by

and

1 1 m

P . ^ P n

1 < p` <oo

if mp < n,

if rnp = ra,

p' = oo if mp > n.

Since Si C R" is an extension domain, u E Wm'D(1) has an extensionto v E Wm .P(R'1 ) with compact support such that IlVIIm,p < (; II^ IIm,p;i2•Therefore,

Ilullp• ; CI l .!) IIF-< CIIVIIm,p< Cllüllm,p;n< C EIIÎIn + IlDmu llp;nl (4.2.5)

by Lemma 4.2.2. Now apply this to (4.2.1) while observing that D'(L(u))0, lot' = m, and obtain

lin — L(u){ 1p•;0 — L(n)Ilp;n + IlDmullp,nl

< C IITII k+i IIDk+ i Îlm- ( k+^y,P;s^'T(1)

We have thus established the following result.

4.2.3. Corollary. With the hypotheses of Theorem 4.2.1,k+1

Ilu — L(n)llp•:n <_ C (1177/)1 IIDfc+'Ullm-(k+i)p;n.T(1)

4.2.4. Remark. In many applications it is of interest to know when L(u) =O. In this connection we remind the reader the coefficients of the polynomialL(u) are given by (4.1.7) and will be zero if T(D°u) = 0 for 0 < Ial <k. The question of determining conditions under which L(u) = 0 will bepursued in Sections 4.4 and 4.5.

4.3. The Dual of Wm'P(fl) 185

4.3 The Dual of WmtP(1)

In order to obtain more information from inequality (4.2.1) it will be helpfulto have a representation of (Wm'P(1l))', the dual of Wm'P(11). This is easilyaccomplished by regarding Wm.P(Ç) as a closed subspace of the cartesianproduct of LJ'(SZ).

To this end letN=N(n,rn)= E

0<1a1<mbe the number of multi-induces a with 0 < lal < m. Let

N

LN(S/) _ 11L(c).LN(SZ) is endowed with the norm

IIUIIp, N;^2 =

m^ Ilvallp ;n

1 aE—o

max IIUII00;fio <1ai<m

1/p

if 1 <p<oo

if p = oo

where y = {va } E LN(SZ).

4.3.1. Theorem. Let SI C Rn be an open set. Then each linear functionalT E (Wm.a(SZ))', 1 < p < oo, can be represented as

m

T(u) _ E va (x)D°1 u(2)dx for u E Wm 's'(SZ), (4.3.1)

1a1_0,

where y = {va } E LN(SZ).

Proof. Clearly, the right side of (4.3.1) defines an element T E (Wm.p(SZ))'with

11Th see (2.1,5). In order to express T(u) in the form of (4.3.1) first observethat W in I P (n) can be identified as a subspace of LN((). The operatorD: W"(f2)— LN(1) defined by

D(u) = {D°u}, 0 < lai < m

has a closed range since Wm•P(fI) is complete. Define a linear functionalT' on the range of D by

T' f D(u)J = T(u), u E Wm.P(ft).

186 4. Poincaré Inequalities —A Unified Approach

By the Hahn-Banach theorem, there is a norm preserving extension T'of T' to all of LN (I1). By the Riesz Representation theorem, there existsi1 = { va } E L%;(1) such that

m

T' (w) = E f va (x)wa (r)dxrai_o

whenever w = {wa } E LN(SZ). Thus, if u E Wm ''(0), we may regard

Du = {Da u) E LN(1Z) and therefore

T(u) = T' ID(u)] = T'(Du)m_ n ya(x)Day.(x ) dZ. ❑

1 (2 1=0a

In the event that 1Z C R" is a bounded extension doinaiii, the repre-sentation of (Wm°P(SZ))' is slightly simpler, as described in the followingresult.

4.3.2. Theorem. If SZ C R'' is a bounded extension domain and 1 < p <oo, then each element T E (Wm•P(Ç1))' can be represented as

T(u) = + E v0D°u)dx (4.3.2)f ( vu2 l°1-m

where y, va E Lp`(S2), lcxl = m .

Proof. The proof is almost the same as in Theorem 4.3.1 except that nowWm.P(S?) can be identified with a subspace of LN (ft) where N = k(m) + 1,and k(m) = the number of multi-indices a such that lad = M. Thus u EWm.P(SZ) is identified with (u, {Dau}1a1=m). In view of Lemma 4.2.2 thisprovides an isometric embedding of Wm'P(12) into LN(SZ). ❑

It is useful to regard the restriction of the linear functional T in Theorems4.3.1 and 4.3.2 to the space .95(1) as a distribution. Indeed, if Sp E g(ft)is a Schwartz test function (see Section 1.7), then from (4.3.1) we have

T(Sp) _ E j i,a D° Spdx (4.3.3)Ial=0

where v° E Lp' (II). In the language of distributions, this states that T is adistribution in Ç with

m

T = E ( -1 ) 101I7'va1a1= 0

(4.3.4)

4.3. The Dual of Wm'p(SZ) 187

where va E LP G (SI). Similarly, if T is the functional in Theorem 4.3.2, then

T = y + E (-1)I0ID°`ya (4.3.5)Ian=m

where u, va E L (il). However, not every distribution T of the form (4.3.4)or (4.3.5) is necessarily in (Wm.p(S2))'. In case one deals with Wâ 'p (SZ)instead of Wm'p(SZ), distributions of the form (4.3.4) or (4.3.5) completelydescribe the dual space, for if T is a distribution as in (4.3.4), for example,then it possesses a unique extension to Wp "(û). To see this, consideru E Wô 'p (f2) and let {vi} be a sequence in (SZ) such that çp, —, u inWo p (SZ). Then

IT(92i ) — 4.02)1 =-- E f vaDacp i — ya Da vjt101.4)la1= m

E 11Da(cpi — Spj)II r 1IV IIp',nIra1.0

IIl^t — ^PjIlm^pIIvaIIp',o --t O

as i , 3 -a oo. Thus, T (cp;) converges to a limit, denoted by t(u), which iswell-defined. f' is clearly linear and bounded, for if Bp i -. u in W 'P(SZ),then

IT(u)I — llrn IT(Pi)1 lun IITII IIî1lm,p+moo fôo

= IITII II tIIm,p•

The norm IITII in this context is defined relative to the space Wô `"(SZ).These remarks are formalized in the following theorem.

4.3.3. Theorem. Let 1 < p < oo. If SZ c R" is an open set, then the dualspare (Wô '"(SZ))' consists of all distributions T of the form

m

T = E (-1) Ial Da va

Ial=O

where va E L" (fi). If SZ is a bounded extension domain, then (Wo 'p(0))'consists of those T such that

T = v + E (-1)Ia!Dckva

v, va E (S/) •The dual space (Wô 'P(f2))' is denoted by W-m,p'(A).


4.4 Some Measures in (W(cZ))*

We now exploit Theorem 4.2.1 and its Corollary 4.2.3 to derive some of themost basic and often used Poincaré-type inequalities. These inequalitiesare obtained below by considering Lebesgue measure and its variants aselements of (Wm°p(0))'.

In order to demonstrate the method that employs the results of Section4.2, we begin by reproving the inequality

Il Dk ull p <_ CIIDm uilp (4.4.1)

for u E Co (R"), where 0 < k < m are integers and p > 1_ Suppose that thesupport of u is contained in some ball: spt u C B(0, r). Let S2 = B(0, 2r).With this choice of f2, we wish to apply Corollary 4.2.3 by selecting T sothat the associated projection L will have the property that L(u) = O.Then by appealing to (4.2.2), we will have established (4.4.1). Define T E

(Wm + km (f1 )) • byT(w) _vw dx

nfor w E Wm -kmo(0), where v ! XB(0,20- B(0,r)• Since spt u C B(0, r),

T(Dau) = 0 for 0 < lai < k

and therefore L(u) = 0 by Remark 4.2.4. Hence, (4.4.1) is established.In case Il is a bounded open set and u E Co (f2), a similar result can

be established by defining u to be identically zero on the complement of 0and by considering a bail B(0, r) that contains Il. Since Cr (0) is densein WtP(f2) the following is immediate. (Of course, this result also followsfrom the inequalities established in Chapter 2.)

4.4.1. Theorem. Let SI C R" be a bounded set. Let 0 < k < m be integersand p > 1. Then, there is a constant C = C(k, m, p, diam 1-2) such that

II D k ullp,o < CII Dm ullp ;n.for u E Wô 'P(S2).

A slight variation of the preceding argument leads to the following results.

4.4.2. Theorem. Suppose 0 < k < m are integers and p > 1. Let S2 be abounded extension domain. Suppose u E Wm.P(11) has the property that

1 Daudx_0 for 0<lai<k,

where E c Sl is a measurable set of positive Lebesgue measure. Then,

IIt II k,p;O < CIIDk4- Mini-(k+1),pin

4.4. Some Measures in (Wô 'F(f l))'

where C = C(k, m, p, 0, IEI).

Proof. Define T E (Wm`k.p(1))' by

T(w) = f wdz, w E Wm-k,P(0).E

189

Then T(1) 0,T(D°u)=0 for 05IcI< k,

and therefore by Remark 4.2.4 the associated functional L

has the propertythat L(u) = O. The result now follows from Theorem 4.2. 1. ❑

4.4.3. Corollary. If u E W m.P (fû) has the property that Dck u = 0 almosteverywhere on E for 0 5 loll < k, then

Ilullk.p;n < CIIDk+1Ullm-(k+14;f1.

Theorem 4.4.2 provides a Poincaré-type inequality provided the integralaverages of the derivatives of u over a set E of positive measure are zero. Inthe next result, the integral average hypothesis is replaced by one involvingthe generalized notion of median of a function. If the sets A and B beloware of equal measure, then we could think of 0 as being the median of uover A U B.

4.4.4. Theorem. Let fi E Rn be a bounded extension domain and letu E W 1 'e(0), p > 1. Suppose u > 0 on A and u < 0 on B, where A and Bare measurable subsets of D of positive Lebesgue measure. Then

Ilu llp.n <— C IIDüIIP ;c^where C = C(p,n, IAI,IBI)•

Proof. Let

=u dsA

and Q = u dxB

and define T E ( W i .p(Sl))' by

T(w) = vwdx, w E W i ^p(f2)fwhere v = (1/a4A — (1/0)XB. Then T(n) = 0 and the result follows fromTheorem 4.2.1 and Remark 4.2.4. 0

4.4.5. Remark. In the remainder of this section, we will include a smalldevelopment of the notion of trace of a Sobolev function on the boundaryof a Lipschitz domain as well as some related Sobolev-type inequalities

190

4. Poincaré Inequalities—A Unified Approach

(Theorem 4.4.6 and its corollary). This material will be subsumed in thedevelopment of BV functions in Chapter 5, but we include it here for thebenefit of the reader who does not wish to pursue the BV theory.

If Sl C R" is a bounded Lipschitz domain and u E W 1 "F(Sl), 1 < p < oo,it is possible to give a pointwise definition of u on Of) in the following way.Since fl is an extension domain, let ù denote an extension of u to all ofJr where û E W''n(R"). Therefore, it has a Lebesgue point everywhere onR" except possibly for a set of B1,p-capacity zero (Theorem 3.3.3). Sincep > I, we know from Theorem 2.6.16 that sets of B 1 , F-capacity zero are ofH" -1 -measure zero and therefore it is defined H" -1 -almost everywhere onan. We define the trace of u on On by setting u = té on OSi.

We now show that this definition is independent of the extension ü. Forthis purpose, we first show that at each Lebesgue point x o of û, there is ameasurable set A such that the Lebesgue density of A at xo is 1 and thatii is continuous at x° relative to A. Since

I^(x) - Ü(xo)Idx 0 as r -■ 0,

for each positive integer i, there is a number r i such that the set E i =R" fl {x : Iu(x) - u(x o )1 > 1/4 has the property that

IB(xo , r) nE I • < 2 -1 , for r < ri . (4.4.2)

IB(xo, r)I

We may assume that the sequence {ri} is strictly decreasing. Let

00E = UB(xo,rj) - g(xo, ri_ 1 )] n Ei .

i-1

We now will show that the Lebesgue density of E at xo is zero, that is

r---.0 IB(x, r)Ilirn IB(xa, r) n EI = 0. (4.4.3)

Choose a small r > 0 and let k be that unique index such that rk+1 <r < rk . For notational simplicity, let B(r) = B(x o , r). Then from (4.4.2) itfollows that

IB(r) fl El < U [(B(r) n Ei) n ($(ri) - B(ri+1))j+ =k

ao

< 2 -k IB(r)I + E 2- 'IB(ri)Ii=k+1

< 2 -kiB(r)I + 2 -k1B(r)1

4.4. Some Measures in (Wo"(f ))' 191

which establishes (4.4.3). Clearly, if we set A = Rn - E we have that u iscontinuous at xo relative to A and that the Lebesgue density of A at zo isone.

Because 1? is a Lipschitz domain, the boundary of SZ is locally repre-sentable as the graph of a Lipschitz function. Thus, the boundary canbe expressed locally as {(x, f (x)) : x E U), where U is an open ball inRn-1 and f is a Lipschitz function. Recall from Theorem 2.2.1 that a Lip-schitz function is differentiable almost everywhere. Moreover, the function7 : R"' -+ R" defined by

7(x) = (x, f (x))

is Lipschitz and carries sets of Lebesgue measure zero in Rn -1 into sets ofH" 1 -measure zero in R". Consequently, an possesses a tangent plane atall H"" 1 -almost all points of 31l. From this it is not difficult to show that

I B (xo, r) n DI _ 1inn01

2 ,

for H" -1 -almost all xo E an. Since the Lebesgue density of A at xo is equalto one, it follows that

lirn fB(zo '

/

r) n sin A I - 1rÛ I B(xo+ r)I

2.

Also, because u is continuous at z o relative to A, it is clear that

litn u(x) = ü (x0).X ---DZa

rE finA

This shows that the value of ti(r o ) is determined by u in SZ, thus provingthat the trace of u on the boundary of SZ is independent of the extensionü.

In the statement of the next theorem, we will let denote the restrictionof (n - 1)-dimensional Hausdorff measure to Oft That is #(A) = H" -1 (An80) whenever A C R".

4.4.6. Theorem. Let SZ c R" be a bounded Lipschitz domain and supposeu E W'P'(fï), 1 < p Cao. Let

c(u) =udHn -1 = udµ.asp art

Then µ E ( W I 'P(SZ))" and

1/P' (f

f/n([ lu - c(u)lo . dx < C J IDa^lpdx ,(in n


where p' = np/(n - p) and C = C(n,p, fZ).

Proof. Because fZ is a bounded Lipschitz domain, u has an extension üto all of Rn such that IIÙII1 ,p < By multiplying i by a function

E Co (Rn) with too _ 1 on SI, we may assume that ti has compact support.In order to show that E (W 1 .p(cl))*, we will first prove that

f vdp < CIIvII 1,p (4.4.4)

whenever v is a non-negative function in Co (Rn). from Lemma 1.5.1, wehave

where Et = {x: v(x) > t). By the Morse-Sard theorem, for almost all t,Et is bounded by a smooth manifold. We now borrow an essentially self-contained result of Chapter 5. That is, we employ Lemma 5.9.3 and Remark5.4.2 to conclude that for all such t, Et can be covered by balls B(x 1 , r^ )such that

00

E t„n -1 ^ ^:f^n-1Û-1(t)7

i=1

(4.4.6)

where C is a constant depending only on n. Because Of/ is locally the graphof a Lipschitz function, it follows from (1.4.6) that there is a constant Csuch that p(B(x, r)) < Crn' 1. Thus, from (4.4.6) it follows that

00

ti(Et) < EL(B(zj,ri))1 =1

00

C G,C``LL

rnt

-1 < CHn-1 (v-! (t)).

t= 1

Appealing to (4.4.5) and co-area formula (Theorem 2.7.1), we have00

v dN = µ(E: )dtf00

� C Hn - 1 (v -1 (t))dt0

= CIIflvII1• C(sZ)IIDvllp< cllUlll,p,

thus establishing (4.4.4).If y is now assumed to be a bounded, non-negative function in W 1, p(Rn),

we may apply (4.4.4) to the mollified function v E . From Theorem 1.6.I we

00v dµ = µ (4.4.5)f(Ei )dt

4.5. Poincaré Inequalities 193

have that llv€ — vil lip —, 0 and that v,(x) v(x) whenever x is a Lebesguepoint for v. From Theorem 3.3.3 we know that y has a Lebesgue point at allx except possibly for a set of B l ,p capacity zero, therefore of H'-measure0, and therefore of it-measure O. Consequently, by Lebesgue's dominatedconvergence theorem,

f Vc dL —+ fv dµ.

It now follows that (4.4.4) is established whenever y is a non-negative,bounded function in W 1 'P(Rn).

If we drop the assumption that y is bounded, then we may apply (4.4.4)to the functions

vk(x) — k if v(x) > k

v(x) if v(x) < k.

It follows from Corollary 2.1.8 that vk E W 1,P (R") for k = 1, 2, .... Thus,an application of the Monotone Convergence theorem yields (4.4.4) for non-negative functions in W 1,P(R"), in particular, for ü+ and ü`. Hence (4.4.4)is established for u.

From Remark 4.4.5 we have that u = u Hn — l-almost everywhere on 81?,and therefore

J udi — udµ

Ç CiRul11 ,P

Gllulll,p;n.

Thus, we have shown that ti E (W 14)(1l))*, and reference to Corollary4.2.3 completes the proof. ❑

The following is an immediate consequence of Theorem 4.4.6.

4.4.7. Corollary. If SZ is a bounded Lipschitz domain and u E W 1 .P(11),p> 1,then

f u dHn -1 S C ^llüllp • in + 11Dullp,nl

and

Ilullp.;u <_ C [ itLiip ; c dHn-1js1

As mentioned in the beginning of Remark 2.4.5, these inequalities willbe extended to the situation when u E BV, thus including the case p = 1.

4.5 Poincaré InequalitiesHere we further develop the results in Section 4.2 to obtain Poincaré-typeinequalities for which the term L(u) in inequality (4.2.1) is zero. We will


show that this term vanishes provided the set {x : u(s) = 0} is sufficientlylarge when measured by an appropriate capacity.

First, recall from Corollary 2.6.9 that i f' A C R" is a Suslin set, then

B,,p (A) = sup{B,,p (K) : K C A,K compact}.

Moreover, if' K C Rn is compact Theorem 2.6.12 implies that there is anon-negative measure p such that spt P C K,

119c * PIIp' < 1,

andP(R") = [Bcp(K)1 1I P

Now consider u E Wm -km(f2) where fZ C Rn is a bounded extensiondomain. Then u has an extension û defined on Rn such that IIÙII m_k , p <CIlullm_k,p;ç. Without loss of generality, we may assume that u has com-pact support. From Theorem 2.6.1 it follows that ü has the representation

u =9m-k * fwhere f E I1(Rn ) and IJ f II

Now supposesuppose that p is a non-negative measure with the properties thatspt p C 1 and

9m-k*P E LP' (R")

where k is an integer, 0 < k G i. Observe that p can be considered as anelement of (Wm -k(f2))' for if we define T : Wm -k(SI) -' R' by

T(u) = f u dµ,

then,

f pdp = füdµ

— f 9m - k * f dFt

- 9m--k * p • f dx, by Fubini's theorem,

S N9m-k * Pulp' if lip, by Iidlder's inequality,

* ullp+Ilullm -k,p'

Ç CII9rn -k * lltilim-k,p;0- (4.5.1)

Thus, p E (Wm-kip(f2))`.

4.5. Poincaré Inequalities 195

This leads to another application of Theorem 4.2.1 which allows themain constant in the inequality to be estimated by the capacity of the seton which u vanishes.

4.5.1. Theorem. Let SI E R" be a bounded extension domain and letA C R" be a Suslin set with Bm _k,p (A) >0 where0<k<m are integersand p > 1. Then, there exists a projection L: Wm"p(s1) lk(R") such that

1Iu - L(u)Ilk,p,n Ç C (Bm-k,p(A)) -1/F IID 144- 'uIIm-(k+1),p ;nwhere C = C(k, rn,p, 0).

Proof. From the above discussion, there exists a non-negative measure psuch that is supported in A,

119m-k * < 1

andp(fin) ? -

2If we set T = p in Theorem 4.2.1, we have T(1) = p(R") > 0 and from(4.5.1) that

11Th <_ CIIgm-k * pIIp' <_ C.The result now follows from Theorem 4.2.1 and Corollary 4.1.5. ❑

4.5.2. Corollary. Let u E Wm'p(0) and let

N = f1 {x: D °iu(x) = 0 for all 0< Iocl <k} .

If Bm _k,p (N) > 0 then

IIÎIk,P;^ C(Bra_k,p(N)) - LIP IIDk+l ullm-(k+l),p;nand

IIuIIp•,O < C (Bin-k,p(N)) Jl1P Dk+1 ullm-(k+1),pm.

Proof. The coefficients of the polynomial L(u) in Theorem 4.5.1 dependupon

T(Dau) = D°ù dp

for 0 < lkl < k, and thus are all zero, (see Remark 4.2.4). The secondinequality follows from Corollary 4.2.3. ❑

Because of the importance of the case m = 1, k = 0, we state the Poincaréinequality separately in this situation.

4.5.3. Corollary. If u E W l , p(SZ), then

Î^1Ip•;^ ^ C (Bl,p(N)) - ' l° IIDuÎ°,^ (4.5.2)

196 4, Poincaré Inequalities—A Unified Approach

where N — {x : u(x) = 0}.

4.6 Another Version of Poincaré's Inequality

We can improve the inequalities of Corollary 4.5.2 if we allow dependenceon the set N and not merely on its capacity. In particular, if j, k, and m areintegers such that 0 < j < k < m, then the assumption B,n _k,p (N) > 0 willbe replaced by the weaker one, Bm_( k _j ),p (N) > 0, provided an additionalcondition is added which requires dependence on the set N in the resultinginequality.

To make this precise let 0 be a bounded extension domain and let N C S2be a Suslin set with the property that

Bin - (k -j),p(N - Z) > 0

for every set Z of the form

Z n {x: DaP(x)- 0, OO P E Pk( R")}.

These sets comprise a subclass of the class of algebraic varieties. Thus, forany algebraic variety of the form (4.6.2), we require some subset of N ofpositive capacity to lie in the complement of Z.

Let M(N) denote the set of all non-negative Radon measures com-pactly supported in N such that

9m-(k-j) * E LP. (R" ).

Consider all functionals of the form

T(u) = fDau dµ,, u E M(N), (4.6.3)

where Ia l < k— j. We will verify that all such T are elements of (Wm ,P (0) r.Let u E Wm.P(SZ). Since SZ is an extension domain, there is an extension ûof u to R" with IIÜIIm,p < from Theorem 2.6.1, we know that ûhas the representation fa = g," * f, where f E Lp(R") and II flip ^' IIu IIm,pSince is supported in N C 1, it follows that

Daudµ = f D°lidµ.

But D'û E Wm - ial.P(R") and therefore

Da ii = 9m--,a1 * g

4.6. Another Version of Poincaré's Inequality 197

where g E L' (Rn ). By Fubini's theorem we have

f Daü dp = f 9m-1a1 * 9dµ

f 9rn_lol *p.gdx.

It follows from (2.6.2) that

9m-101* th = ge * (gm-(k-1) * P)i

where l = (k - j) - 'al. Since gm_(k_ i ) * a E LP' (R") by assumption andge E L 1 (R'a ), it follows from Young's inequality that gT1-la1 * L E LI? (Rn).Therefore an application of Holder's inequality yields

DaU dp = f 9m-IoI * tz • g dx

Ilgm-iai * ilIIp` IIgilPfl9m- 1oI * PUP' IID"uIlm - Iol.p.

However,iiDaullm-iar,p Ilullm.p - CIIUIIm,p,

thus proving that T E (Wm 'P (OW .Let V C (Wm•P(1))* be the space spanned by all such functionals T as

defined in (4.6.3). Let

Vo={TIPk: TEV},

so that Vo C N. Observe that

dim Vo = dimPk(R")

orV0 = [Pk(R")] * ,

for if this were not true, there would exist OOPE Pk(R") such thatT(P) = 0 for every T E V. This would imply

Da Pdp=O ► Ic:tI <k - j

for all i E M(N). That is, from Theorem 2.6.12, this would imply

Er 13 (x) = 0,

for Bm_( k _j )-q.e. z E N and loci < k - j, a contradiction to (4.6.1) and(4.6.2). Therefore dim Vo = dimPk(R") or Vo = IPk(R")]*. This impliesthe existence of Ta E V such that

Ta (xS) - 6a ,B. (4.6.4)

198 4. Poincaré Inequalities--A Unified Approach

Hence, if we define L ; Wm'p(0) --► Pk(R") by

rn

L(u) = E Tor (u)x°, (4.6.5)1a1=o

then (4.6.4) shows that L is a projection. An appeal to (4.1.3) results inthe following theorem.

4.6.1. Theorem. Let it C R" be a bounded extension domain. SupposeN C St is a Suslin set such that (4.6.1) is satisfied. Then, with L given by(4.6.5), there is a constant C = C(j, k, m, N,1) such that

IIu - L(u)Ilk,p;fi < CII Dk+I tillnt-(k+1),p;n.

The special nature of the projection L is what makes this result inter-esting. For example if we assume that Dau = 0 on N except possibly fora Bm _(k_,),p-null set, then all T of the form (4.6.3) are zero and thereforeL(u) = O. The following is a consequence.

4.6.2. Corollary. Let n C R" be a bounded extension domain. SupposeN c n is a Suslin set such that (4.6.1) is satisfied. If u E Wm •p(SZ) is suchthat

D"u(x) = 0 for Bm _(k _ 3 ),p -q.e. x E N

and all O < la! < k - j, then

IIuIIk,p;n CllDk+1 fllm- (k+1),p;t2

where C = C(j, k, m, N, Il).

4.7 More Measures in (

The general inequality (4.2.1) involves a projection operator L : Wm•p(SZ)Pk(R") which is determined by an element T E (W m-k ip(SZ)) • It is there-fore of importance to have an ample supply of elements in the dual ofWm-k ,P(0) that are useful in applications. In Section 4.4 we have alreadyseen that Lebesgue measure (more precisely, suitably normalized measureswhich are absolutely continuous with respect to Lebesgue measure) andnormalized (n-1)-dimensional Hausdorff measure belong to (Wm -kiP(S1))*.The fact that these measures are elements of (Wm -k •p(Çl))* allowed us todeduce interesting Poincaré-type inequalities. In this section we will per-form a finer analysis to establish that a large class of measures belongto (Wm - k'p(û))•, including those that are obtained as the restriction ofHausdorff measure to sub-manifolds of appropriate dimension in R".

4.7. More Measures in (Wm'p(S2))' 199

We begin with a result that provides a generalization of the Sobolevinequality for Riesz potentials and also gives us a method of exhibiting alarge class of measures that are elements of (Wm -k .D(S1))'. It will dependon the Marcinkiewicz Interpolation Theorem which we state here withoutproof.

Let (po , qo ) and (p i , qi ) be pairs of numbers such that 1 < p; < q; < CO,i = 0,1, po < pi, and qo 0 q t . Let p be a Radon measure defined on R"and suppose T is an additive operator defined on Co (Rn) whose values areµ-measurable functions. The operator T is said to be of weak-type (p i , q,)if there is a constant C i such that for any f E Co (R'), and a > 0,

1.4({x :1(Tf)(x)I > a1) < (ct -i Ctflf IIp. ) 4 ' •

4.7.1. Theorem (Marcinkiewicz Interpolation Theorem). Suppose T issimultaneously of weak-types (po , qo ) and (pi, qi). If 4 < 0 < 1, and

1 -B ^1Ip_ +

Po P i

1 - 1 B BI4 +go Pi ^

then T is of strong type (p, q); that is,

Il^f ll q;,i <_ CCo -BCl Ilflip, f E Co (R"),

where C = C(p„ q i , 9), i = 0, 1.

We are now in a position to prove the basic estimate of this section whichis expressed in ternis of the Riesz kernel, Ik, that was introduced in Section2.6.

4.7.2. Theorem. Let µ be a Radon measure on RR such that for all x E R"and 0 < r < oo, there is a constant M with the property that

01B(x, r)1 < Mr'

where a-q/p(n - kp),k> 0, 1 < p < q < oo, and kp < n. If f EI"(Rri),then

i/4(1 11k * fl°dp < CM'/gI1f11D

where C = C(k, p, q, n).

This inequality is obviously an extension of the Sobolev inequality forRiesz potentials that was established in Theorem 2.8.4. In that situation,the measure is taken as Lebesgue measure. In Remark 4.7.3, we will


discuss further what other measures play an important role in the inequalityof Theorem 4.7.2.

Proof o f' Theorem 4.7.2. For t > 0 let

At= {y: 'k * III(Y)>t}.Our objective is to estimate p(A t ) in terms of III Let p t = µ I A t . Then

tp(At) Ik * If I di. = Ik * IÎI dfpef g

he kt(x)II (x)I dx (4.7.1)L .

where the last equality is a consequence of Fubini's theorem. Referring toLemma 1.5.1 it follows that

Ik * Pe (x) = 7( k)j°° it [B (r,rh/(k_')] dr

°°

7(k)= (n — k ) f t (B(x r))rk_hI_ 1 dr .

For R > 0 which will be specified later, (4.7.1) becomes

ti(A)ar(k) I^ (x) I t[$(x,r)]rk dx dr

JR

Rn_

r f L+ n k c) If(x)Ipt(B(x, r)]r k- a - ldx dr

= I1 + 12. (4.7.2)

Since µ(B(x, r)] < Mr° by hypothesis, the first integral, I t , is estimatedby observing that

pt[B(x,r)] A t(B(x, r)]1!p'(Mra)1/P

and then applying Holder's inequality to obtain

jl < (n — k) I1111 M'1p R

l/p

rk-n-1+(a1p)dr.7(k) o

(fR.ILL[B(x,r)]dx)

(4.7.3)We now will evaluate

Ftt[B(x, r)]dx•

For this purpose, consider the diagonal

D = (Rn x Rn) n {(x,b): x= y}

4.7. More Measures in (Wm'P(Si))'

and define for r > 0 ,

Dr = (Rn x Rn ) (1 {(x> y) Iz -yI < r}.

Finally, let F = xDr. . Then, by Fubini's theorem,

P t f B(x, r) ldx =RdJ^cty)dx

RR " f (s,r)

= F(x, i!)diê (b)dxR^ R"

= F(x, P)dx dpi (^)Lin L.

= I B(Y , r)I dµt(U)R"

cr(n)rn µ(At).

201

Therefore (4.7.3) yields

^ < p(n - k) 7(k)[kp (n a)l

II /Ilpml/pa(n)^/p'µ(At}t/P' Rk - (n - a )p ^

Similarly, by employing the elementary estimate

1jt(B(x, r)] < pt[B(x, r)]1/p'µ(At)1/P }

we have_ ao 1/p'

I^ < (n^±'(

k)k) II /IIpp(At)' /P ^l,t(B(x, r)]dx r^` -n-ldrR R"

< p( kpl7{- k)k

f

r )

IIfIIpti(At)ck(n)l/P'Rk-n/P

Hence

[ M 1 /PP(A) 1 /P'Rk_n_a 1Pt ( ?Il +

< P(n - k) ^(n) 11P ' Ii.f IIp - •y(k) kP - (n - a)

+ A(At)R k -n/P

n - kpIn order for this inequality to achieve its maximum effectiveness, we seekthat value of R for which the right-hand side attains a minimum. An ele-mentary calculation shows that

R = N(At) 1la ►M

202 4. Poincaré Inequalities— A Unified Approach

and the value of the right-hand side for this value of R is

np(n

k )a n ri a(n)11P1^(n) i/p` M 1IvF^(At) i-i l^ll f llp-

7(k)( P)(P + )

Consequently, from (4.7.2)

tp (A t )i/q i tA(At)p(At)119-1

< p(n - k)a a(n)1/p'Mi/v^y(k)(n- kp)(kp-n+a)

((A e )'""(At) III ii )

Ç 7(k)(n - kP)(q - P) a(n)IIp M il4 llt'lp. ( 4.7.4)

Expression (4.7.4) states that the Riesz potential operator Ik is of weaktype (p, q) whenever p and q are numbers such that

1<p< q<oo, kp<n. (4.7.5)

Hence, if (po, qo), (p, q) and (p i , qi) are pairs of numbers such that (po. qo),(pi, qi) satisfy (4.7.5) and for 0 < 0 < 1,

1/p- 1- B + 0Po Pi

1/ _ 1-B + Bq - qo Pi

then the Marcinkiewicz Interpolation Theorem states that Ik is of type(p, q), with

111k* .6l4:µ <_ GM 119 l1 f Ilµ,thus establishing our result. D

4.7.3. Remark. The number a that appears in the statement of Theorem4.7.2 is equal to n when q = np/(n - kp) = p'. In this case the conditionsof the theorem are satisfied by any measure p that is absolutely continuouswith respect to Lebesgue and that has bounded density. In particular, ifwe take p as Lebesgue measure, we can recover Theorem 2.3.6, which isSobolev's inequality for Riesz potentials.

Theorem 4.7.2 also provides an inequality for Riesz potentials restrictedto a lower dimensional submanifold MA of Rn. For example, if Ma is a corn-pact, smooth A-dimensional submanifold of Rn, then it is easy to verify thatA-dimensional Hausdorff measure restricted to M satisfies the condition ofTheorem 4.7.2. That is, if we define p by

p(E) = H(E n MA),

4.7. More Measures in (Wm.p(SZ))` 203

for every Bore] set E C Rn, then there is a constant M such that

p[B(x, r)1 < Mr A (4.7.6)

for every ball B(x, r) C R". Now let f E I"(R") and consider the potential

u = Ik * f.

By Theorem 4.7.2 we havei/A•

uA. dp < Cil fli p

where a` = J1p/(n — kp), n — A < kp < n. In other words,3/ A'

Iu.(x) I A. dHa(x) <— CilfllpU .

(4.7.7)

where C = C(k, p, A, n, M). Note that the constant C depends on M which,in turn, theory, it is sometimes possible to obtain an equality similar to(4.7.7) where the constant is independent of the manifold.

Inequality (4.7.7) is valid for Riesz potentials u = Ik * f and thus doesnot automatically include Sobolev functions. However, it is immediate thatTheorem 4.7.2 and (4.7.7) apply as well to Sobolev functions u E Wk,p(Rn)because Theorem 2.6.2 states that u can be represented as

u= 9k*fwhere gk is the Besse] kernel, f E LP(R") and Ilf ljp , Ilullk,p. Moreover,we know from (2.6.3) that there is a constant C such that gk(x) < CIk(x),x 0.

To reassure ourselves that the integral on the left-side of (4.7.7) is mean-ingful, recall from Theorem 3.3.3 that u is defined pointwise everywhere onR" except possibly for a set A with Rk,p (A) = O. Therefore, by Theorem2.6.16, 1171—kP+e (A) = 0 for every e > 0. By assumption, A > n — kp andconsequently H A (A) = O. Thus, u is defined HA almost everywhere on MAwhich is in accord with inequality (4.7.7).

Also, we observe that if p is a non-negative measure on RR with compactsupport, and otherwise satisfies the conditions of Theorem 4.7.2, then p E(Wk.p(1))` whenever II is a bounded extension domain. To see this, letu E Wk ,p(SZ) and let u be an extension of u to R" such that II/14,p <_Cllullk,pio. Because u E Wk,P(R" ), we have

û- 9k * fwhere Ilf lip ilüiik,p• Hence, by Theorem 4.7.2 and the fact that spt p iscompact,

f lIa'Op 5_ C lu lQdp _<_ C IJf Ilp


= CIIu IJk.p< Cll^ llk,p;n.

This establishes the following result.

4.7.4. Theorem. Let 1 C Rn be a bounded extension domain and supposep is a compactly supported Radon measure on R" with the property that ife > 0, there is a constant M such that

p(B(x, r)) < Mr" - kp+e,

for all z R" and all r > 0, where kp< n,p> 1. Then p E (Wk , p(0))8 .

This result obviously is not sharp and thereby invites the question ofdetermining an optimal condition for p to be an element of (W k .p(Rn))'.By using a different approach, it is possible to find a condition, relatedto the one in the theorem above, that provides a characterization of thoseRadon measures that are elements of (W k ip(R"))'_

For this purpose, we need a few preliminaries. Ifµ is a Radon measure,we will use the fractional maximal operator

Mkp(x) = sup{rk- "µ[B(x, r)] : r > 0}.

There is an obvious relationship between the Riesz potential of and thefractional maximal operator: Mkp(x) < CIk * p(x) for every x E R'1 , whereC = C(k, n). The opposite inequality in integrated form is not so obviousand is implied by a result due to [MW]. It states that for every 1 < p < ocand 0 < k < n, there exists C = C(k, p, n) such that

IIIk *tiIIp <_ CII kp1I . (4.7.8)

The (k, p) -energy of is defined as

,ek.p(µ) _ (9k * p)p dz.

R*

Since the Bessel kernel is dominated by the Riesz kernel, we have

tk.p(p) < Cf * p)p dx"- (fk * p) • ( Ik * 11)1/(p-l)dx

Rn

-- Ik * (Ik * µ) 1 / ( p- 1}dµ, by Fubini's theorem. (4.7.9)f"The expression

Ik * (Ik * /(p -1)

4.7. More Measures in (Wm 'P(f ))' 205

is called the non-linear potential of A.

4.7.5. Theorem. Let p > 1 and kp < n. If is a Radon measure, thenµ E (Wk 'D(ft"))' if and only if

( IB(vir)1 \ 1/(p-1) dr

Rn U r" — pT dµ(y) < oc. (4.7.10)

Proof. In order to avoid technical details involving the behavior at infinity,we will give a proof for measures with compact support.

If it is such a measure with 9k * µ E L", then by Fubini's theorem andwith u = gk * f we can write

f udµ= 9k*fdA

f gk *u f dx

II9k* i4 ^^f IIP< CIfgk * IIIp ' 1Iulik,p ,

which implies that /.4 E (W k 'P (R") )' . Conversely, if E (W k,p (R") )', thenthe reflexivity of LP implies that gk * µ E LP ' . Therefore µ is an elementof (Wk 'p (Rn )) • if and only if 119k * µ4 p' < oo, i.e., if and only if the (k,p)-energy of µ is finite.

We proceed to find a (sharp) condition on µ that will ensure the finitenessof its (k,p)-energy. For each r > 0,

• i/p'A[B(x, r)l ^ C

f

2r 1 PEB(x'o)1i p dtrn —k to — k t

' I /p'< C

(,r µ[B(^,t)1 p dt—

(f: tn —k t

Thus, ,

Jl^t ^< ^ u[B(z, p dt

1Ipkij( ) C_

o t"- kt)^t

Therefore,

tic .p(u) ^ ^ (Ik * µ)p,

dxR"

5.0 (Mkµ)P ^ dx, by (4.7.8),R* ^

< C r t+[B(x, t)^ p dtdx.

— R" p tn —k t


Now to evaluate the last term, we have

fR'p[B(x, t)]prd

R

z = A[B(x, t)] 11(p-1) u[B(x, t)jdx"

< il[B(x, t))1/(p-1) dth(y) dxR" B(z,t)

< p[B(y,2t)l11(p-1)diL(y) dxL . fa(o)

F(x ) y)F4B(11, 2t)] 1/(n-1) dp(y) dxR" R"

< 4B(y, 2t)] 1/(P-1) IB(x, t)Idp(y)R"

and D1 = Rn x R" n ((x, y) : la - yI < t}. Therefore,

£k,p(µ) < C °O (tk ")p' " p[B(y, 2t)] t!(P-1) 1B(x, t)Idj(y) dt

D R

< CR" 0 oo

[B(y, t)] 1/(F-1) dt dA(y)

•to-kp t

Since has compact support and finite total mass, it is evident that theexpression on the right side of the above inequality is finite if and only if

1 ( [B(Ylt)] ) l?(r-1) dt

R" ot"-kp Td(y) < oc.

This establishes the sufficiency of condition (4.7.10).For the proof of necessity, we employ the estimate

gk(x) ? Clxik- "e -21x1 for x E R", z # a(see Section 2.6). As in (4.7.9), we have

t"k,p(N) = (9k * Fl) p dx k * (9k *R JR 'l

(4.7_11)

To estimate the last integral, let f = (gk * p) 1/(-1 ) and use Lemma 1.5.1and (4.7.11) to obtain

009k * .f (x) •? G f (y)dy rk-"e-9r

dr

o B(z,r) r

Clearly, for r > 0,

f(y) ?9k(y - x)dµ(z)B (y,r)

where F(x, y) = XDr

4.8. Other Inequalities Involving Measures in (WkIT

and therefore,

00k-n -2r dr

9k * f (x) ? C f (y)dy r eo B(s,r) r

a, 1/(P-1)dr

(

^^ (rn 9k(y - z)d►l(z) dy Tk-ne- 2r—

0 BfV.r1 T

)1/(p-1)

This implies

I 1 ^^k(l^) ^ ^R" o

( I4B(Y f))) 1/(P_l) e_2Plr dlL(y)Tn-kp r

l^(^1 i/(p-1) ,> c e

- 2p r ^r

dg y)tic, '

T^Bn-kpr)^ r

2 l( 4B(Yr)] 1 /(P -1) dr,^? Ce- P —du(ll)•

R" L I

T

r

4.8 Other Inequalities Involving Measures in(Wk,p)*

We now return to the inequality (4.2.1) for another application. It statesthat

Hu - L(u)Ilk,p;ç C<_ I on ii

where T E (Wm -k,P(11) )' and L : WmiP(SZ) -' Pk(Rn) is the associated

projection, L(u) has the form

L(u) = E T(Pa(D)u)Xa

1 k

where PQ is a polynomial of degree Ici whose argument is D = (D 1 , ... , Dn ).In Corollary 4.5.2 we found that L(u) 0 if Bm _k,p (N) > 0 where

N=ft fl{x: D°u(x)= 0 for all 0< la' < k).

This was proved by the establishing the existence of a measure p > 0supported in N with p E (Wm-k.P(11))• By taking T = p it clearly follows

207

r> C ^,n rk- ne -ardF^(z)o B(y,r)

> Cfow F^rB(^J, T)1 i/(P-I)e-2P , r dr

-

Tn - kp r

dy Tk -n e- 2r dTr

D


that L(u) = E f P° (D)udp = 0. (4.8.1)

Now, if p is taken as any non-negative measure in (Wm -k.P(11))* with theproperty that

D°u dp= 0 for all 0<IaI< k,

then (4.8.1) holds. This observation along with Theorem 4.2.1 and Corol-lary 4.2.3 yield the following result.

4.8.1. Theorem. Let p > 1 and suppose 0 < k < m are integers. LetS1 C R" be a bounded extension domain. If u is a non-negative measure onRn such that p E (Wm - k ,P(fl))*, µ(Rn) # 0 and

f D°u dµ = 0 for ail 0 < k,

then

andIlUIIk,p;i2 < CIIDk+1uIlm—(k+l),p;S2

CIIDlc+l ullm_(k+1)IPP

where C = (k, p, rrb, µ, Sl).

In particular, with k = 0 and m = 1, we have

IIÎIp';n < ÎIDuIIP;nif p E (Wh P(11))* and

f u d = o.

From the preceding section we have found that a non-negative measureµ with compact support belongs to (Wm -km(ft))* if, for some e > 0,

sup n[ B(xk ) )j a :a E R", r > 0 < oo. (4.8.2)

r

Consequently, if a is an integer such that a > n - (m - k)p + c and M A isa smooth compact manifold of dimension a, then H a I Ma is a measure in(Wm- k ,e(11))•. As an immediate consequence of Theorem 4.8.1, we have

4.8.2. Corollary. Let A be an integer such that A > n - (m - k)p + ewhere p > 1 and e > O. Suppose MA is a smooth compact submanifold

4.9. The Case p - 1 209

of dimension JI of R", SZ C R" is an extension domain and suppose u EWmip(1) has the property that

JM AnR D"u dH a = 0 for all 0 < < k

where Ha (MP n SI) # O. Then

Ilulik.p;n < CIIDk+l ullm—(k+1),p;ç

andHullap';ct < CIID k+l ulfm— (k +l),pm

where C = C(k, p, rn, MA , SZ).

4.9 The Case p = 1

The development thus far in this chapter has excluded the case p = 1,a situation which almost always requires special treatment in LP-theory.Our objective here is to extend Theorem 4.7.2 to include the case p = 1.Since the analysis will depend upon estimates involving Wulf ' , it is notsurprising that the co-area formula (Theorem 2.7.1) will play a critical role.We begin with the following lemma that se rves as a first approximation toTheorem 4.7.2 in the case p 1. We will return to this later (in Chapter5) for a complete development in the setting of BV functions.

4.9.1. Lemma. Let > 0 be a Radon measure on R" and q a numbersuch that 1 < q < n/n — 1. If

B x,r 1sup Ilrq( ( _l) ) ! : x E R", r> 0 < M

for some M > 0, then there exists C = C(q, n) such that

whenever u E Co (R").

1/quQdµ CM1/61IIDuII 1

R"(4.9.1)

Proof. First consider q =1 and refer to Lemma 1.5.1 to conclude that

f r<dµ _F ( 4 .9 . 2 )f4Et)dt

whenever u E Co (R") is non-negative. Here Et = {x : u(x) > t}. Becauseu is continuous, OEt C u -1 (t) for each t > 0; moreover the smoothness

210 4. Poincaré Inequalities -A Unified Approach

of u and the Morse-Sard theorem states that u. -1 (t) is a smooth (n - 1)-manifold for almost all t > O. Consequently, Lemma 5.9.3 and Remark 5.4.2imply that for all such t there exists a covering of E t by a sequence of ballsB(x i , r i ) such that

m

Er:1-1<cHn-ipEo <cHn-i(0,-1 ( 0})i =1

where C = C(n). Hence,00

1i(Et) < µ{$(Ti,ri)1,-1

<rnEi=1

< C H 1 [{u -1 (t)}n

where C = C(n). Referring to (4.9.2) and the co-area formula (Theorem2.7.1) we have

fco

udjc = p(Et )dt0 00

< CMHn -1 ({u- '(t)})dt0

= CM I Dul dx .

R"

If u is not non-negative, write lui = u+ - u - , and apply the precedingargument to u+ and u - to establish our result for q = 1.

Now consider q > 1 and let g E Lg ' (p), g > 0. Then, Holder's inequalityimplies

1/q'

9 dµ < ggrdJ1 p[E(i, r)]114B(x,r) (fB(x,r)

<_ ^1/g I1911g' ;^Tn-^

Thus, gp is a Radon measure which satisfies the conditions of the lemmafor q = 1. Consequently, if u E Co (R") we have

J IttIgdµ < CM 1 1 4 Ilsllg';^• Iûld^^n

for all g E Lq! (u), g > O. However, by the Riesz Representation theorem,

Ilullq,^. = supR IuIg dp : II91I q' < 1, g> 0"

4.9. The Case p = 1 211

and our result is established. a

4.9.2. Remark. The restriction q < n/n - 1 in the lemma is not essential.If q > n/n - 1, the lemma would require a Radon measure ti to satisfy

4B(x, r)) < Mr'

for all x E Rn , all r > 0, and some m > n. However, there is no non-trivialRadon measure with this property. In order to see this, let U C R" be abounded open set. Choose e > 0 and for each x E_U, consider the familyGm of closed balls B(x, r) such that 0 < r < e and B(x, r) C U. Defining

ÿ={B: B E ç= ,x E U}

we see that Q covers U and thus, by Theorem 1.3.1 there is a disjointedsubfamily C Ç such that

U CU{B :BE Ç}cU{B: B E.1"}.

Hence, denoting the radius of B, by r ; , we have00

u(U) < E gift) Ç M E ( 5r,)mBE.* i=1

00< M5m

Cm -n Er n.

i=1

< M5m c m-n lUl.

Since U is bounded and e is arbitrary, it follows that ,.(U) = O. D

Our next objective in this section is to extend inequality (4.9.1) by re -

placing on the right side by (IDtuIIi. For this purpose it will benecessary to first establish the following lemma.

4.9.3. Lemma. Lei > 0 be a Radon measure on R", t < n, 1 < q <(n-1 +1)(n - t) and r -1 = 1 - (q - 1)(n -1)/n. Then there ta a constantC such that for all z € R" and r > 0,

rL—(n+l) IIl s *^ II r;B{s,r) <(: SUp{r(i-n)Qµ[B(x, r)j : x E R", r > 0}.

Proof. It will suffice to prove the lemma for x = O. An application ofMinkowski's inequality for integrals yields

dp(y) ri/r

dx(

18(0.?) (

L(O.2?) Ix _ Yi" -1

J (0,r)r1B(y,r) Ix - YI N-1)r

dxfB0 1 i x _ V(n- 1) r

dx dx


Observe thatfB(0.2r)

1/rdx

f (O,r) 4x Î_ (n-1} 7 du(F1)• (4.9.3)

+B(O , r) -B(y ,

r) Ix - YIN 1)r

The first integral can be estimated by

fB(//,r) I x `- 11I (n-1)rI < Crn-r(n-1)dx

and the second integral is dominated by rn-r(n-I) Here we have used thefact that (n - 1)r < n. Thus,

/Boo-) Ix - VI N-11r

dx < Crn-i(n-1)

and therefore from (4.9.3)r 1/r

(far) 9(0,2r) ix - YI N-1)rt-(n+1) dp(3i) d^

< Cr(C-n)gµ[B(O, 2r)j. (4.9.4)If 1x1 < r and M > 2r, then IyI < 21v - xl. Consequently,

r 1/r

(L(D,r)

(Li^ 2 r Ix - yl (nT1)r l-(n+1) dµ(y) dx

< Gr(")-(Q-1)(n-1) f dµ(11) (4.9.5)11ip.2r I11I4-1 •

Appealing to Lemma 1.5.1, we have

OM o0

^-i < (n - 1) µ[B(0, t)^t -ndt.JY I>2r lyl 2r

Now define a measure v on B 1 by y = t (n-l) Q-ndt and write

1413(0, t))i -ndt = µ[B(0, t )] t (t-')gdvfr jr

o00Jr

< G•r (n-t)Q-n+1 sup TO-11)12/4[B(0, 01-

,

r >O

4.9. The Case p = 1. 213

This combined with (4.9.5) and (4.9.4) yield the desired result.

We are now prepared for the main result of this section.

4.9.4. Theorem. Let p > 0 be a Radon measure on R" and let l < n,q > 1. Then if

sup il[B(x1 r)] : x E Rn , r > 0 - M < co,r

there exists C = C(q,n) such that

U.lfg

04 5 GM 1/g if D 1 uI11

whenever u E Cô (Rn ).

Proof. If l = n, then for u E Cô (Rn) and x E Rn ,u(x) < f IDn uldy,

Rn

from which the result follows.If 1 = f < n, the result follows from Lemma 4.9.1.Next, consider the case l < n, t > 1 and q > n/(n - 1). Since u E

Co (R"), it follows that u E Kt— l' n/(n_1)

(Rn ) and therefore u = 91-1 * f,f E L"/("-1)(Rn) with Ilfllnf(n-1) - IIuIIl-1,n/(n-1) ^^ IIDr-lulln/(n_1).Thus, Theorem 4.7.2 implies

Ilullq:µ <— CM 1/a IID1-lulln,(n-1 ).

Since We-i lk/n-1 < CIIDrulll by Theorem 2.4.1, our result is establishedin this case.

Finally, consider l < n, t > 1, and q < n/(n - 1). We proceed byinduction on t, assuming that the result holds for derivatives of orders upto and including t - 1. As in (2.4.5),

L. lu(,),,,, ) < C Rn it ID

Iz(_iu

yI(in)1 1)dy dFe(x)

5 CgIDuI Iul g-l fi * µ dy (by ^bini's theorem)Rn

< CgllulIn/(1n_ O II IDulfl * µll T (by Holder's inequality)

where r -1 - 1 - (q - 1) (n - l)n - 1 . By Sobolev's inequality,

11u1r, f(n -i) < CIiD luqi- . (4.9.6)


To estimate * µ HT let m be a measure on R'" defined by m =(I1 * µ)T dx and apply the induction hypothesis to obtain

II IDuII^ * IIIr = II IDuI II=;m< Csup{r (1-1)- nni1D(x, r)rT : r > 0,x E Rn } IID 1 uII1= C sup{r (r--1)~ "IIIj * µjlT ; ^(z ,T) : r > 0, x E R"} IJD 'uII I •

This combined with (4.9.6) and Lemma 4.9.3 establishes the proof. D

Exercises

4.1. Give a proof of Corollary 4.5.3 based on the argument that imme-diately precedes Section 4.1. You will need the material in Section2.6.

4.2. The following provides another method that can be used to definethe trace of a Sobolev function u E 14/ 1 'P(S1) on the boundary of aLipschitz domain S1,STEP 1. Assume first that u E C 1 (Si), u > O. For each xo E (3S1 andwith the (n + 1)-cube centered at zo with side length 2r denoted byC(xo , r), we may assume (after a suitable rotation and relabeling ofcoordinate axes) that there exists r > 0 such that C(xo, r) fl an canbe represented as the graph of Lipschitz function f where the unitexterior normal can be expressed as

(Di f,...,Dn, f,1)^fl + ID fl 2

H"-a.e. on C(xo, r) n 011. With en+1 = (O,... ,1) and under the as-sumption that apt u C C(xo, r), appeal to the Gauss-Green theorem(see Theorem 5.8.7) to conclude that there exists a non-negative con-stant C, depending only on the Lipschitz constant of f , such that

u dH" < C (uen+1) • v dH"fn an

= C div(ue n+ t )dr.n

< C IDuldz.n

If u assumes both positive and negative values, write lui _ u+ + u -

to obtain

IuI dHn < C n IDul dz.

f812

Exercises 215

STEP 2. With no restriction on spt u, use a partition of unity toobtain

lul dHn <_ C n (lul + IDuj)dx.an

STEP 3. Prove that

lulPdHn Ç C5 (lulp + IDulp)d^

8S2

by replacing lui by lulp in the preceding step.STEP 4. Now under the assumption that u E WLP(11), refer to Ex-ercise 2.17 to obtain a sequence of smooth functions u,E such thatl^ uk — ulli,p;1t —' 0 and

fa I iuk--uelpdH" —+ 0 as k,l oo .

The limiting function u' E I'(011) is called the trace of u.

4.3. Prove that u' obtained in the preceding exercise is equivalent to thetrace obtained in Remark 4.4.5.

4.4. Prove the following Poincaré-type inequality which provides an esti-mate of the measure of nul > k} in terms of WWDull1. Let u E W"" 1 (B)where B is an open ball of radius r and suppose IA is a measure oftotal mass 1 supported in B n {x : u(x) = 0}. Then, if k > 0,

kl {x : lu(x)l > k }l < Cr IDuI + Cr" (Il * ^) •• 'Dui,B s

where R 1 is the Riesz kernel (see Section 2-6). Hint: Choose x, y E Bwith u(y) = 0 and obtain

iu(x)i <_ iu(x) — u(z)i+ u(z)

whenever z E B. An application of polar coordinates yields

u(s) < Cll r * (xs IDul)(x) + (x$ . 'DOM].

4.5. The technique in the preceding exercise yields yet another proof ofCorollary 4.5.3 which is outlined as follows. Let u E W 1 •p(B(r)) whereB(r) is a ball of radius r and let N = {x : u(x) = 0}. Let coo be a non-negative smooth function with apt cp contained in the ball of radius2r and such that cp is identically one on B(r). Select x E B(r) inaccordance with the result of Exercise 3.15 and define h = cp[u(x)—u].Then, for each y E R" ,

V(v)u(x) = (P(v)ü(sr) + h(Y)-


Recall from Theorem 2.6.1 that the operator J : r1(R") —■ W'.p(R")defined in terms of the Bessel kernel gi by J(f) = g1 * f is an isom-etry. Therefore, with ti as any non-negative Radon measure µ, anapplication of Fubini's theorem yields

f h dA= f91 *(J - 'h)dµ (J -rh)• gt * Adz.

Thus, if is concentrated on B(r) n N and satisfies Ilgl * µll pr < 1, itfollows that

µ[$(r) n N]Iu(z)I <_ it.r'[v(u - u(x)))IIp < CIEç(u -- u(x)II,.

Taking the supremum over all such 1.4 leads to

Bi,p[B(r) n N)lu(x)I" < C r -p Iu(x) — u(y)Ipdyf (2r)

+ IDulpdg .B(2r)

Use Exercise 3.15 to estimate

Iu(x) - u(g)IpdyB(2r)

in terms of the norm of Du.

4.6. Poincaré's inequality states that if u E W h .p(SZ) and u vanishes ona set N of positive Bi,p -capacity, then IIuIIp•;n < CIIDuIIP;n, whereC depends on Si and the capacity of N. In the event that more isknown about u, this result can be improved. Using the indirect proofof Section 4.1, prove that if u E W 1"p(0) is a harmonic function thatvanishes at some point zo E Si, then there exists C = C(zo , St) suchthat

IIuIip•;a < CIIDuIIp;^4.7. Lemma 4.2.2 is one of many interpolation results involving different

orders of derivatives of a given function. In this and the next exercise,we will establish another one that has many useful applications. Provethe following: Let g be a measurable function on Rn , and let 0 < ar <n,O<e<1.Then

Ira£(g)(x)I <— C(M9(x)) 1-E . (!a(191)(z))`,where Mg is the maximal function of g. Refer to the proof of Theorem2.8.4 and choose 6 in(2.8.4) as

act + 1.(191)(x)

Mg(x)


4.8. Let f = !.(g), g > O. Prove the interpolation inequality

IIDkf II* < CIlf IlâlcrIlf Ilia— / k i },,

where k is any multi-index with 0 < Iki < ar, 1/r = lkI/ap+(1-Ikl)/s,and p < a < oo. Use the previous exercise to prove

Iliac(9)11r <— CIIgIIP-ÌIIQ(19l)I1:

where 1 < p < oo, 1/r = (1 - e)/p + E/s, p < a < oo. Then letf = IQ (g) and observe that

IDkf (W)I <— Icr—k(g)(x).

4.9. Prove the following as a consequence of Theorem 4.2.1. Let Sl C R"be a bounded, connected, extension domain. Suppose rr E LP` (Il),n 0, p > 1. Prove that there exists C = C(p, v , 0) such that

iluflp^^ CIIDullp;n

whenever u E W hip(1l), fn uv dx = O.

4.10. When I C R"+' is a Lipschitz domain, Exercise 4.2 shows one wayof defining the trace, u' E I"(00), when u E W'"p(11), p > 1. Notethat

l u•1pdN" < c(Iulp + IDulp)dz.n

Let v E I^ (Oft), u O. Prove that there exists C = C(p, v, fl) suchthat

IIuIIP:n < CILDuIIp;^whenever u E W"p(S?), fafl u'v'dH" = O.

4.11. At the beginning of this chapter an indirect proof of the followingPoincaré inequality is given: if u E W'.p(1l) and u = 0 on a set ofpositive measure S, then llullp:n < CllbullP;n. Show that essentiallythe same argument will establish the same conclusion if it is onlyassumed that u = 0 on a set of positive B1, p -capacity.

Historical Notes4.1. Lemmas 4.1.3 and 4.1.4 provide the main idea that serves as the key-stone for the developments in this chapter. They are due to Norman Meyers(ME4} and many other results in this chapter, such as those in Sections 2,5, and 6 are taken from this paper. It should be emphasized that Lemma4.1.3 is an abstract version of the usual indirect proof of the basic Poincaréinequalities.

fa.


4.4. In Remark 4.4.5 an approach to the subject of trace theory on theboundary is indicated which is based on the material in Chapter 3 con-cerning the property of Sobolev functions being defined everywhere in thecomplement of small exceptional sets. Another approach to this subject ispresented in [LM].

In the proof of Theorem 4.4.6 it is not necessary to use the Morse-Sardtheorem if we are willing to use the full strength of the "Boxing Inequality"[GU] and not the version reflected in Lemma 5.9.3. The inequality in [GU]states that there is a constant C = C(n) such that any compact set K c R"can be covered by a sequence of balls {B(r ; )} such that

E rt.1-i < CH" --l (ax).i=1

This inequality could be used to establish (4.4.6) if K is taken as E t andby observing that aE t C v -1 (t) since tr is continuous.

4.5. The proof of the Poincaré inequality here is, of course, based on thematerial in the previous sections, particularly Theorem 4.2.1. This proof iscontained in [ME4]. There are several other proofs of the Poincaré inequal-ity including the one in [P] which is especially interesting.

4.7. All of the Sobolev-type inequalities discussed thus far are in termsof inequalities defined on R". There also are similar inequalities that holdfor functions defined on submanifolds of R". For example, in minimal sur-face theory, Sobolev inequalities are known to hold for functions defined onsubmanifolds where the inequality includes a term involving the mean cur-vature of the submanifold, cf. [MS]. In case of a minimal surface, the meancurvature is O. Theorem 4.7.2 is a result of the same ilk in that the leftside involves integration with respect to a measure which can be takenas a suitable Hausdorff measure restricted to some submanifold. However,it is different in the respect that the right side of the inequality involvesthe li-norm of the gradient relative to Lebesgue measure on R" and notthe norm relative to Hausdorff measure restricted to the submanifold. Thisinteresting result was proved by David Adams [AD2]. Theorem 4.7.4 statesthat measures with suitable growth over all balls are elements of the dualof W k ip(R"). Thus, Theorem 4.7.2 is closely related to (4.2.1).

Theorem 4.7.5 which yields a characterization of those measures in thedual of Wk.P(R") is due to Hedberg and Wolff [HW] although the proof wegive is adapted from [AD7].

4.9. Inequality (4.9.1) is due to Meyers-Ziemer NZ] in case q = 1. Theproof for the case 1 < q < n/(n — 1) is taken from [MA3]. This inequalityis also established in Chapter 5 in the setting of BV functions, cf. Theorem5.12.5. Corollary 4.1.5 is an observation that was communicated to theauthor by David Adams. This result when applied to Theorem 4.5.1 yields


more information if Lemma 4.1.4 were used. This is an interesting exampleof the critical role played by the sharpness of a constant, in this instance, theexponent of Bm _k ,p (A) in Theorem 4.5.1. Indeed, in the work of Hedberg[HE2I, it was essential that the best exponent appear. He gave a differentproof of Theorem 4.5.1.

5

Functions of BoundedVariationA function of bounded variation of one variable can be characterized asan integrable function whose derivative in the sense of distributions is asigned measure with finite total variation. This chapter is directed to themultivariate analog of these functions, namely the class of L' functionswhose partial derivatives are measures in the sense of distributions. Justas absolutely continuous functions form a subclass of BV functions, so itis that Sobolev functions are contained within the class of BV functions ofseveral variables. While functions of bounded variation of one variable havea relatively simple structure that is easy to expose, the multivariate theoryproduces a rich and beautiful structure that draws heavily from geometricmeasure theory. An interesting and important aspect of the theory is theanalysis of sets whose characteristic functions are BV (called sets of finiteperimeter). These sets have applications in a variety of settings because oftheir generality and utility. For example, they include the class of Lipschitzdomains and the fact that the Gauss—Green theorem is valid for themunderscores their usefulness. One of our main objectives is to establishPoincaré-type inequalities for functions of bounded variation in a contextsimilar to that developed in Chapter 4 for Sobolev functions. This willrequire an analysis of the structure of BV functions including the notion oftrace on the boundary of an open set.

5.1 Definitions

5.1.1. Definition. A function u E L'(0) whose partial derivatives in thesense of distributions are measures with finite total variation in SZ is calleda function of bounded variation. The class of all such functions will bedenoted by BV (Sl). Thus u E BV (i2) if and only if there are Radon (signedmeasures) measures p ] , P2 , ... , An defined in 12 such that for i 1, 2, ... , n,I Dpi+( 11 ) < oc and

f uD;cod_fcod i(5.1.1)

for all coo ECo (S)).The gradient of u will therefore be a vector valued measure with finite

5.1. Definitions 221

total variation:

- sup{u divvdx : v=(v 1 ,...,v„) E Cp (fZ; Rn ),n

lv(x)l < 1 for x E f2} < oo. (5.1.2)

The divergence of a vector field y is denoted by div y and is defined bydiv v = En 1 D i v; . Observe that in (5.1.1) and (5.1.2), the space Cô (11)may be replaced by CVO). The space BV (0) is endowed with the norm

IIU IIBV = IIuIII,n + (5.1.3)

If u E BV(0) the total variation IlDulI may be regarded as a measure,for if f is a non-negative real-valued continuous function with compactsupport 12, define

IlDu!!( f ) = sup{udiv v dx : i, = (v i , . . . , v,^) E Co (S^; Rn ),c^

Iv(x)1 < f (x) for x E sl }. (5.1.4)

5.1.2. Remark. In order to see that 'Dull as defined by (5.1.4) is in facta measure, an appeal to the Riesz Representation Theorem shows it issufficient to prove that 'Pull is a positive linear functional on C0(l) whichis continuous under monotone convergence. That is, if { f; } is a sequenceof non-negative functions in C0 (0) such that fi j g for g E C0(12), thenrrDull(f;) I^DuÎ(g), cf. (F4, Theorem 2.5.51. In order to prove that IlDulihas these properties, let = Du and refer to (5.1.1) to see that p satisfies

u div cpdx = —f ço • dg

where 5o E Co (S2; R"). Therefore, we may write (5.1.4) as

IIDuII( f) = sup{f y • dp : U = (v 1 , ... ,v„) E Co(Si; Rn )

lv(x)I < f(x) for x E S2}. (5.1.5)

To show that Pull is additive, let f, g E C0(0) be non-negative functionsand suppose y E Co(f2, R") is such that lvl < f + g. Let h = inf{ f, Ivl} anddefine

{

h(x) ^(^) !v(T)I 0 0w(x) = i ( )I

o Iv(x)1= a.

222 5. Functions of Bounded Variation

It is easy to verify that w E C0(f1) and Iv — WI = M — h < g. Therefore,since Iwl = h < f

v•dµ=w•dµ+ f (v—w).durz 0

IlDull(f) + IlDull(9)-

This implies that IlDull(f + g) < IlDuII(f) + IIDuII(g) . The opposite in-equality is obvious and consequently it follows that IlDull is additive. Itis clearly positively homogeneous. It remains to show that it is contin-uous under monotone convergence. For this purpose, let f, I g and lety E Co(SZ, R") be such that M < g. Also, define h; = inf{ f,, M} and

hi(x) 1)

Iv(x)I 0w i (s) = IU(x)Io Iv(x)I =0.

Note that wi E Co(f2), Iivil = h; < f,, and that Iv — w11 = IVI — h, 1 Q asi —4 oo. Since Iv — w; I = M — h i < 214 Lebesgue's Dominated ConvergenceTheorem implies

f v di.4 = Du - y = lim Du - w, < lim II Dull (hi).s—.00 I--60o

By taking the supremum of the left side over all such y it follows thatIIDuII(9) _< lim,—.0.0 IIDuII(h , ). Since h i < g for all i = 1, 2, ..., we haveIIDuII(g) = lim,_00 II Dull (h,). This establishes that IIDuII is a non-negativeRadon measure on ft.

We know that the space of absolutely continuous u with u' E L' (R 1 )is contained within BV(R 1 ). Analogously, in R" we have that a Sobolevfunction is also BV . That is, W 1 " 1 (St) C BV (11), for if u E W 1,1 (11), then

I "

udivv dx i — EDiuvdxn i`1

and the gradient of u has finite total variation with

II Du II(s^) = f Dudx.

5.2 Elementary Properties of BV Functions

In this section we establish a few results concerning convergence propertiesof BV functions. We begin with the following which is almost immediatefrom definitions, but yet extremely useful.

1

5.2.1. Theorem. Let 11 C R" be an open set and u E BV (11) a sequence

5.2. Elementary Properties of BV Functions 223

of functions that converge to a function u in 140 ,01). Then

lirn inf ['Dui II (U) ?.IIDuII (U)3-.00

for every open set U C S2.

Proof. Let v be a vector field such that v E Cp (U; R") and Iv(x)I < I forT E U. Then

I udivv dx = limu ; divv dx < ZirninfIlDuiIl(U)•J-9013 U i-60o

The result follows by taking the supremum over all such v. ❑

5.2.2. Remark. Note that the above result does not assert that the limitfunction u is an element of BV(SZ). However, if u E L 1 (SZ) and we assumethat

sup{ I,DuiOM) : t = 1,2,. _.} < 00

then u E BV (S2). Indeed, if pp E Co (S2), and Du; is any partial derivativeof u i , then

lim cP Duidx _ — limu i Dcp dxno ^

and therefore

u DSp dxft

u DSp da < sup Iol lim inf Wu; IRO) < oo.{^00

Since Cr(l) is dense in the space of continuous functions with compactsupport, we have that

Du((p) _ — ( uD,dxn

is a bounded functional on COI). That is, Du is a measure on O.Theorem 5.2.1 established the lower semicontinuity of the total variation

of the gradient measure relative to convergence in LL We now will provean elementary result that provides upper semicontinuity.

5.2.3. Theorem. Let {u 1 ) E BV(S1) be a sequence such that u i —. u inLL(S1) and

fini 1IDu, I I( 1l) IlDull(^)1 00

Then,

U,.

lim sup IIDuII(U n n) < IIDuII(U n si)


whenever U is an open subset of 11.

Proof. Since V = f1 — U is open, it follows from Theorem 5.2.1 that

IlDuII(U) < limnf IlDuill(U)

IIDuII(V) <— liminf IlDuill(V)•i-0oc

But,

IlDull(v n s2) + IlDull(V) = = lim IlDuill (n)..^^

> lim sup II n i2) + liminf II Dud! (V)iôo i-00

> limsup [1 Dud! (U n f2) + 1IDu11(V)• ❑i-+00In view of the last result and Theorem 5.2.1, the following is immediate.

5.2.4. Corollary. If {u,} E BV(0) is a sequence such that u, -i u inLL(fl), limi-. 00 IIDu1II(f1 ) = IlDul1(1), and 11Dull(tU) = co, where U is anopen subset of f1, then

lim Il Duill(U ) = I lDull(U)•iôo

5.3 Regularization of BY Functions

Here we collect some results that employ the technique of regularizationintroduced in Section 1.6. Thus, for each e > 0, cp, is the regularizing kerneland of = u * cp c . From the proof of Theorem 1.6.1, it follows that if U Cc U,and u E Li c (U), then lluell1;u < IIuII i;n for all sufficiently small e > 0. Inthis sense, regularization does not increase the norm. We begin by showingthat a similar statement is valid when the BV norm is considered.

5.3.1. Theorem. Suppose U is an open set with U C S2 and let u E BV (f1).Then, for all sufficiently small e > 0,

IIuCII$v(u) IIUIIBV(n)-

Proof. In view of Theorem 5.2.1, it suffices to show that IIDuclI(U) <IIDull(i2 ) for all sufficiently small e > O. Select v E C0 (U, Rtm) with Id < 1.Choose rq > 0 such that {x : d(x,U) < rt} C f2. Note that Iv,' < 1 andspt of C {x : d(x, U) < q} for all small e > 0. For all such e > 0, Fubini'sTheorem yields

u(div v dx = u(div v dx ï u(div v) E dxfu R L

fnudi vEdx < IlDull(u)•v

5.3. Regularization of BV Functions 225

The result follows by taking the supremum over all such al. a

5.3.2. Proposition. Let u E BV(n) and f E Co (n). Then fu E BV (1l)and D(f u) = D f u + f Du in the sense of distributions.

Proof. Let U be an open set such that apt f C U C U C SZ. Then,ue f E C(U) with D(f ue ) = (D f )u, + f Du e at all points in U. However,NE — uIl i ; u —► 0 as e —4 0. (Of course, we consider only those e > 0 for whichu1 (x) is defined for r E U.) In particular, when considered as distributions,u£ u. That is, uE —+ u in 0'(U) and therefore Due —4 Du in . (U), (seeSection 1.7). Since f E C(U), it follows that f Du, —► f Du in 2'(U).Clearly, (D f )u( (D f )u in 2'(U). Finally, with the observation thatfu, — fu in i'(U) and therefore that D(f ue ) —> D(fu) in 0'(U), theconclusion readily follows. D

We now proceed to use the technique of regularization to show thatBV functions can be approximated by smooth functions and thus obtain aresult somewhat analogous to Theorem 2.3.2 which states that C°°(11)fl{u :

Iluljk,p;c^ < oo} is dense in Wk'D(1). Of course, it is not possible to obtaina strict analog of this result for BV functions because a sequence {ui} EC°°(1) that is fundamental in the BV norm will converge to a function inW"(SZ). However, we obtain the following approximation result.

5.3.3. Theorem. Let u E BV(f2). Then there exists a sequence ful l EC°° (Q) such that

limi

lu ; — uldx = 0i—.00° and

lirn II Du= II(o) ^ IlDull(^)^ -•oo

Proof. In view of Theorem 5.2.1, it suffices to show that for every e > 0,there exists a function vE E C°°(1) such that

1 lu, — vf jdr < F and IIDuFII(n) < + e. (5.3.1)

Proceeding as in Theorem 2.3.2, let 11i be subdomains of S/ such thatSZ ; CC S2i+1 and U,_ oili = O. Since Mull is a measure we may assume,by renumbering if necessary, that 1lDulI(S1 — no ) < E. Let Uo = 11, andU; = for i = 1, 2, .... By Lemma 2.3.1, there is a partition ofunity subordinate to the covering Ui = 52 ;+ , — S2^_ a , i = 0, 1, .... Thus,there exist functions L such that f, E Co (Ui), 0 < fi < 1, and E°°_ o fi i 1on h. Let yo, be a regularizer as discussed at the beginning of this section.Then, for each i there exists ei > 0 such that

spt(( fiu) f , ) C Ui, (5.3.2)


I(liu)E, — fiuldx < E2 —(i + 1)n

I(uDf,)E, — uD f,Idx < c2 -" 1) .n

Define00

vc = E(ufi )^ :

i=oClearly, yr E C° (1) and u = EMI ufi . Therefore, from (5.3.3)

(5. 3.3)

(5.3.4)

I EIve — uldx < I(ufi)e, ufildx < E.$-o 11

(5.3.5)

Reference to Proposition 5.3.2 leads to

Dv, = E(fiDu)E, +E(uDfi)E,1 —o i=o00 00

E(f1Du)E, + ERuD fi)^ : — uD fi).i=0 +=o

Here we have used the fact that E D f, = 0 on ft. Therefore,00 00

I DuE Idx < f I(f,Du)E,Idx + ^ I(uD f,) E , — uD f,ldx.i=i i = o

The last term is less than E by (5.3.4). In order to estimate the first term,let V E Cp (ft; R") with sup I'iI < 1. Then, with we *

IfJ ÇOc * (fi Du) dx = f f tIEfid(Du) by Fubini's theorem,

= udiv(t f fi)dx

<— IIDuII(Ui)since spt rj' fi C U; and 11/),L1 < 1. Taking the supremum over all suchy ields

I(fiDu),, Idx < IIDuII(Ui), _ = 0, 1 , . . . .Therefore, since each x E SZ belongs to at most two of the sets fl„

00

IDvEIdx < E IiDuII(U;} + ei-o

CO

<_ IIDuII(^}) + E IIDuII(Ui) + Es =1

< Ilb^lll(^r) + 2 IIDuII(ll — fa ) + e-< IlDu ll(SZ) + 3e.

I

11)

JI

5.3. Regularization of BV Functions 227

Since e > 0 is arbitrary, this along with (5.3.5) establishes (5.3.1). ❑

5.3.4. Corollary. Let 1 c R" be a bounded extension domain for W 13 (1).Then BV(f) n {u : By < 1} is compact in L'(1).

Proof. Let u ; E BV(11) be a sequence of functions with the property that< 1. By Theorem 5.3.3 there exist functions v, E C°°(I) such that

l v, - u,Idx < i` 1 and n

j lDviIdx< 2 .

Thus, the sequence {IIvill,,l;n} is bounded. Then, by the Rellich-Kondra-chov compactness theorem (Theorem 2.5.1), there is a subsequence of {v,}that converges to a function u in L' (1). Referring to Remark 5.2.2, weobtain that v E BV(f ). ❑

In Theorem 2.1.4 we found that u E W'.P if and only if u E LP and u hasa representative that is absolutely continuous on almost all line segmentsparallel to the coordinate axes and whose partial derivatives belong to L".We will show that a similar result holds for BV functions.

Since we are concerned with functions for which changes on sets of mea-sure zero have no effect, it will be necessary to replace the usual notion ofvariation of a function by essential variation. If u is defined on the interval[a, the essential variation of u on la, bJ is defined as

ess Vâ (u) = sup 1E lu(ti) — u(ti- i )fi= 1

where the supremum is taken over all finite partitions a < to < t i ... tr < bsuch that each t i is a point of approximate continuity of u. (See Remarks3.3.5 and 4.4.5 for discussions relating to approximate continuity.) From Ex-ercise 5.1, we see that u E BV (a, b) if and only if ess Vâ (u) < oo. Moreover,ess Vâ (u) = IIDull [(a, b)). We will use this fact in the following theorem. Asin Theorem 2.1.4, if 1 < i < n, we write x = (±,x,) where i E R"' andwe define u;(x,) = u(l , x i ). Note that u= depends on the choice di. but forsimplicity, this dependence will not be exhibited in the notation. Also, weconsider rectangular cells R of the form R = (a l , b 1 ) x (a2, b2 ) x • • . x (a,,, b").

5.3.5. Theorem. Let u E LL(R"). Then u E BVoc (R") if and only if

ess V^ ^ (u, )dz < o0JRfor each rectangular cell R C R" - ', each i = 1, 2, ... , n, and a, < b,.

Proof. Assume first that u E BV1Qc (R"). For 1 < i < n it will be shown

5. Functions of Bounded Variation

ess Vb ' (ui ) dx < 00

for each rectangular cell R C Rn- 1 and at < 14. For notational simplicity,we will drop the dependence on i and take R of the form R = R x [a, b].Now consider the mollified function u E * u and note that

1 luE — uldx —+ a as

andlimsup 1Du r ds < oo (Theorem 5.3.1).E^0 R

Consequently, with u,,, (x,) uE (x, x,), it follows that u,-, iu, in L' (a, b)for H" -1-a.e. E R. Theorem 5.2.1 implies that l m infe. o ^^Du^^^[(a, b)] >Ì Du; II [(a, b)] and therefore, from Exercise 5.1,

ess Vb (u ; ) < lim inf eas Vb(uE ,;)E-0

for E R. Fatou's lemma yields

f ess Vb (ui)dH" -1 (2) < lirnE inf

R ess Vâ (u^.,,)dHn -1 (x)

= lirn infE-6 o R

< hm sup DuIdx < oo.E-o R

For the other half of the theorem, let u £ Lloc (R") and assume

ess V6 (u, )dH" -1 (i) < 00R

for each 1 < i < n, a < b, and each rectangular cell R C R" -1 . Choosecp E G (R), kol < 1, where R R x (a, b) and employ Exercise 5.1 toobtain

u Dog dx ess ab (u;)dHn -1 (i) < oo.RR

f

This shows that the partial derivatives of u are totally finite measures overR and therefore that u E B Voc (Rn ). O

228

that

1

5.4 Sets of Finite PerimeterThe Gauss-Green theorem is one of the fundamental results of analysisand although its proof is well understood for smoothly bounded domains

5.4. Sets of Finite Perimeter 229

or even domains with piece-wise smooth boundary, the formulation of theresult in its ultimate generality requires the notion of an exterior normal toa set with no smoothness properties in the classical sense. In this section,we introduce a large class of subsets of R" for which the Gauss-Greentheorem holds. These sets are called sets of finite perimeter and it will beshown that they possess an exterior normal which is defined in the samespirit as Lebesgue points of LP-derivatives. The Gauss-Green theorem inthe setting of sets of finite perimeter will be proved in Section 5.8.

5.4.1. Definition. A Borel set E C R" is said to have finite perimeterin an open set f provided that the characteristic function of E, XE, is afunction of bounded variation in 0. Thus, the partial derivatives of XE areRadon measures in f and the perimeter of E in 0 is defined as

P(E , n) = IIDXEII(1)•A set E is said to be of locally finite perimeter if P(E, SZ) < oo for everybounded open set O. If E is of finite perimeter in R", it is simply called aset of finite perimeter. From (5.1.4), it follows that

P(E, S?) = supdiv v dx : y = (v1i . . . , ^n ^ E ^D (^s R^)+ Iv(z)1 Ç 1 •'E

(5.4.1)5.4.2. Remark. We will see later that sets with minimally smooth bound-aries, say Lipschitz domains, are of finite perimeter. In case E is a boundedopen set with C 2 boundary, by a simple application of the Gauss Greentheorem it is easy to see that E is of finite perimeter. For if v E Co (1l; R")with I1vII < 1, then

divv dx ^ U • v dH" -1 < H" - ` (II n 0E) < ooJE 8E

where v(x) is the unit exterior normal to E at x. Therefore, by (5.4.1),P(E, Il) < oo whenever 0 is an open set.

Moreover, it is clear that P(E, S2) = n 8E). Indeed, since E is aC2 -domain, there is an open set, U, containing OE such that d(x) d(x, E)is Cl on U - OE and Dd(x) = (x - (x)) jd(x) where (x) is the uniquepoint in OE that is nearest to x. Therefore, the unit exterior normal v toE has an extension t E Co(Rn) such that PI < 1. Hence, if v = rjv withq E Co (S2), we have,

div v dx - div rjv dx _ LEThis implies

P( E, SZ) > sup rtdHn - 18E

H" -1 (1-2 f? OE).

n E Co (f), IqI < 1


Intuitively, the measure DXE is nothing more than surface measure(H"' 1 -measure) restricted to the boundary of E, at least if E is a smoothlybounded set. One of the main results of this chapter is to show that thisidea still remains valid if E is a set of finite perimeter. Of course, since weare in the setting of measure theory, the topological boundary of E is nolonger the appropriate object of study. Rather, it will be seen that a subsetof the topological boundary, defined in terms of metric density, will carrythe measure DXE.

In Theorem 2.7.4 we observed that the isoperimetric inequality lead tothe Sobolev inequality via the co-area formula. Conversely, in Remark2.7.5 we indicated that the Sobolev inequality can be used to establishthe isoperimetric inequality. We now return to this idea and place it inthe appropriate context of sets of finite perimeter. We will establish theclassical isoperimetric inequality for sets of finite perimeter and also a localversion, called the relative isoperimetric inequality.

5.4.3. Theorem. Let E C R" be a bounded set of finite perimeter. Thenthere is a constant C = C(n) such that

EI(n-1)/n < CIIDXE II(R") = CP(E). (5.4.2)

Moreover, for each ball B(r) C R",

min {fB(r) n El, IB(r) - E)I} (" -1)/n < CIIDXEII(B(r)) = CP(E, B(r)).

(5.4.3)Proof. The inequality (5.4.2) is a special case of the Sobolev inequality forBV functions since XE is BV. We will give a general treatment of Sobolev-type inequalities in Section 11. If u E BV (R" ), refer to Theorem 5.3.3 tofind functions u ; E Co (B") such that

lim f lu ; - uldx = 0 ,

1.-yoo

lim Du, 11(R" ) = IlDu ll(R" ).t-y00

By passing to a subsequence, we may assume that ui -' u a.e. Then, byFatou's lemma and Sobolev's inequality (Theorem 2.4.1),

llull "/(n-^ ) < liminf Iluilln/(n-1)i --• 00

< lim CIIDu1II(Rn)

< CÌIDulI(R").To prove the relative isoperimetric inequality (5.4.3), a similar argument

along with Poincaré's inequality for smooth functions (Theorem 4.4.2),yields

Ilu - tl(r)1Ift/("-1):B(r) Ç C' llDu ll($(r))

5.4. Sets of Finite Perimeter 231

where u(r) = fÈ(r) u(x)dx and B(r) is any ball in Rn. Now [et u = XE andobtain

n-^)

( IB(r)Ef

n1(n-1)

B(r) lu(x) —

^(T)In/(

dz — IB(r)l

IB(r) n El

IB(r) n El ) fl/(n1)

+ IB(r)I

IB(r) — El.

If IB(r ) — El > IB(r) n El, then (IB(r) — EIV(IB(r)I) and

CIIDX&II(B(r)) = CIiDuII(B(r)) ? Ilü -- ri(r)Iin/(n-1);$(r)

^ ( lB(r) —El \IB(r) n EI(n.-1)/n

IB(r)I

> 1 min f B(r) n El ^ IB(r) — El (n -1)/n

2 IB(r)I IB(01A similar argument treats the case IBM n Et > IB(r) — El. ❑

We now return to the topic of the co-area formula which was provedin Theorem 2.7.1 for smooth functions. Simple examples show that (2.7.1)cannot hold for BV functions (consider a step function). However, a versionis valid if the perimeters of level sets are employed. In the following, we let

Et =S2 n {x : u(x) > t}.

5.4.4. Theorem. Let S2 C Rn be open and u E BV(fI). Then

MuII( 11 ) _R

^f DXEg !I (s^)dt.'

Moreover, if u E L' (S2) and Et has finite perimeter zn S2 for almost all twith

II DX Es II ( 0 )dt < oo,JR i

then u E BV (1Z).

Proof. We will first proof the second assertion of the theorem. For eacht E R', define a function ft : Rn --► R' by

}e =if t 70—X Rn _E t if t < 0.

Thus,u(x) = Ît(x)dt, a E R^.

R^


Now consider a test function cp E Co (S2), such that sup kol < 1. Then

u(x)Sp(x)dx = ft(x)cp(x)dtdxf „ f^ RI

=1...ft(x)Sp(x)dxdt. (5.4.4)

Now (5.4.4) remains valid if Sp is replaced by any one of its first partialderivatives. Also, it is not difficult to see that the mapping t --> I DXE, 11( 2 )is measurable. Therefore, if coo is taken as cp E Co (SZ; Rn) with snp Icpl < 1,we have

Du(cp) _ --u • div cp dx = — R

ft(x)div cp(x)dxdtf n ' J^

< D ft (cp)dt <R

IIDXE,11(12)dt < co. (5.4.5)Rz 1

However, the sup of (5.4.5) over all such cp equals IIDuII(SZ), which estab-lishes the second assertion.

In order to prove the opposite inequality under the assumption thatu E BV(1), let {Pk} be a sequence of polyhedral regions invading St andLk : Pk --+ R 1 piecewise linear maps such that

lim 1Lk — uldx = 0k-oao Pk

(5.4.6)

andlim

k--ooI D LkIdx = 11Dull(n), (5.4.7)

(see Exercise 5.2). Let

E^ = Pk n (x . Lk(x) > t),

xt = —Et •From (5.4.6) it follows that there is a countable set S C R' such that foreach j = 1, 2, .. .

limF IX t (x) — Xt (x)Idx = 0 (5.4.8)k~cio iwhenever t it S. Thus, for t S, and E > 0, refer to (5.4.1) to findW E Co (SZ; R”) such that Icpl < 1 and

IIDXE, II(Sa) — div cp dx < 2 . (5.4.9)fr

Let M = fR„ Idiv cpldx and choose j such that spt cp C P) . Choose ko > jsuch that for k > ko ,

EIXt -- Xt l dx<2M

.Pt

5.5. The Generalized Exterior Normal 233

For k > !co ,

div cp di - div cp dxf , E^ <MP IXt —Xtldx< e

i.

^

(5.4.10)

Therefore, from (5.4.9) and (5.4.10)

II DXE, 1I (i) < div Sp dx + €Et

<_ IIDXE: II(fl)+ e.

Thus, for t S,IIDXE, II(n) < lku inf IIDXE; II (n)•

Therefore, Fatou's lemma implies

Ro IIDxE, II(n)dt < lkrn inf

Ri DX E .:, I((SI)dt

<licn inf H " -1 [Lk 1 (t ) n fZ)dt (by Remark 5.4.2)k —• oo R'

< lim inf I DLkldx (by (2.7.1))_ k Pk

= (by (5.4.7)). ❑

5.5 The Generalized Exterior Normal

In Remark 5.4.2 we observed that a smoothly bounded set has finite perime-ter. We now begin the investigation of the converse by determining the reg-ularity properties possessed by the boundary of a set of finite perimeter.

5.5.1. Definition. Let E be of locally finite perimeter. The reduced bound-ary of E, 0- E, consists of all points x E R' for which the following hold:

(i) IIDXEIU[B(x, r)1 > 0 for all r > 0,

(ii) If v,.(x) = -DXE[B(x, r))/IIDXEIj[B(x, r)], then the limit v(x) _v,, (x) exists with Iv(x)I = 1.

v(x) is called the generalized exterior normal to E at x. We will employthe notation v(x) = w(x, E) in case there is a possibility of ambiguity. Thenotation ô` is used in 0-E to indicate that the normal to E is pointing inthe direction opposite to the gradient.

Observe that v(x, E) is essentially the Radon-Nikodym derivative ofDXE with respect to IIDXEII. To see this, let p(x) be the vector-valued


function defined by

P(x) - - lim DXE[ t (x, r)]

r-POIIDXE IIIB(x ,r )^ .

From the theory of differentiatiou of measures in Chapter 1 (see Remark1.3.9) this implies that p is the Radon- Nikodym derivative of DXE withrespect to II DXE II and that

DXE(B) = —JB P(1H

for all Borel sets B C Fr. Moreover,

I dive dx = - f v(x) p(x)dIIDXEII

whenever y E Cp(Rn; Rn). Consequently, by (5.1.2), Ip(x)I = 1 for IIDXEII-

a.e. x E R" and therefore, p(x) = v(x, E) for IIDXEII-a.e. x E R". Thus,we have

DXE(B) = - Y(x, E)4IDXEII(x),Bna -E

IIDXEII(R" - ô E) = O.

The next lemma is a preliminary versiou of the Gauss-Green theorem.

5.5.2. Lemma. Suppose E is of locally finite perimeter and let f E Co (R" ).Then, for almost all r > 0,

Al dx = - f d(D1XE) + f (11)vi (V, B( r))din- I (y)

EnB(r) B (r) En8(B(r))

where B(r) = B(x,r) and v,(y, B(r)) is the i th component of the unitexterior normal.

Proof. To simplify notation, we will take x = O. From Proposition 5.3.2,we have that fXE E BV(1 ). Let S be the countable set of r such thatII DI(fxE)II [a(B(r))] O. Select r S and let of be a piecewise linearfunction on (0, oo) such that rt e z- 1 on (0,r] and rk =_ 0 on (r + e, oo).Since Di [ f X E] is a measure, we have

— lie(IxI)d(Ds[fXE])(x)R"

- Di(fxE)[B(r)]

— ne(I xI )d(Di[f xE])(f(r+i) -B(r)

f (x)XE(x)Difrje(IxI)1dx =

x).

5.5. The Generalized Exterior Normal 235

Therefore

1 ff (x)XE(z) dxJD(r+)_B(r) 1x1

= -Di(fXE)[B(r)] - vk(IxI)d(D [fXEJ)(x)•f(r+c)-B(r)

Since r S and IrJI < 1, the integral on the right converges to 0 as e J. O.By the co-area formula (Theorem 2.7.3), the integral on the left can beexpressed as

1^ f (z)XE(x) x dxJB(r+c)—B(r) I 1

Éf(x)+ dHn1(x)dt.

f?+C

I n8(B(r)) I ITherefore

f (x)XE(x) 2t dZ ^ - f (x) ^ dHn-1(x)fBfr+c)—B(r) I i L8(B(r)) ^

which implies

f (x) Î I dH n-1 (x) = Di (f XE)[B(r)1fn8(B(r))

for almost all r > O. Moreover, from Proposition 5.3.2,

D If XEJ(B(r)) = (Dif)XE(B(r)) + f DiXE(B(r))Di f (z)dx + f d(DtXE). ❑

f f1B(r) B (r)

5.5.3. Corollary. If E has finite perimeter in ii, then for almost all r > 0with B(r) C SZ,

PE fl 8(0,f2) < P(E, B(r)) + H' 1 [E n a(B(r))].

Proof. Choose E Co (S1, R") with lvi < 1 and let r > 0 be a number forwhich the preceding lemma holds. Then

div v dx - - v • d(DXE)f f1B(r) B(r)

+ v(x) , 1.(xf B(r))dHn- (x)J fZâ(B(r))

Ç IIDXEII(B(r)) + H"- '[E n ô(B(r))j.


Taking the supremum over all such u establishes the result. ❑

Remark. Equality actually holds in the above corollary, but this is notneeded in the immediate sequel.

The next lemma will be needed later when we begin to investigate bound-ary regularity of sets of finite perimeter.

5.5.4. Lemma. Let E be a set with locally finite perimeter. Then, for eachz € 0-E, there is a positive constant C = C(n) such that for all sufficientlysmall r > 0,

r' IB(x, r) n E > C, (5.5.1)

r -n IB(s, r) - El ? C, (5.5.2)C S r i

-n IIDXEII(B(x, r)) 5 C l .- (5.5.3)

Proof. To simplify the notation, we may assume that ï = O. Since 0 E8- E, there is a positive constant C = C(n) such that

li/T(0)I = IDXE($(r))IIIIDxEII[B(r)J ? C (5.5.4)

for all small r > O. For almost all r > 0, it follows from Lemma 5.5.2 that

DXE(B(r)) -fEn5(.(R), I x

sldHn`1 (s)

and thereforeDXE(B(r))I < H" -1 [E n O(B(r))J.

Consequently, (5.5.4) implies

IIDXE 1(B(r)) < C -1 1P -L [E n ô(B(r))) < C-1rn-1. (5.5.5)

Note that (5.5.5) holds for all small values of r since the left side is aleft-continuous function of r. This establishes the upper bound in (5.5.3).

To establish (5.5.1), recall from Corollary 5.5.3 and (5.5.5) that for almostall r > 0,

P(E n B(r)) < P(E, B(r)) + Nn-1[E n 8($(r)))and

P(E,B ( r )) < C-1Hn-1[E n 0(B(r))]•

Thus, an application of the isoperimetric inequality (Theorem 5.4.3) andthe previous two inequalities lead to

IE n B(r)I (n -1)in < CP(E n B(r)) < CH" -1 (E n 8(B(r))1,

5.6. Tangential Properties of the Reduced Boundary 237

for some constant C = C(n). Let h(r) = 1E n B(r)I and observe that theco-area formula (Theorem 2.7.3) yields

T

h(r) =EID(Ix[)Idx = Hn -1 [E n a(B(t)))dt.rZB(r) o

Hence, h i (r) > Ch(r) ("-1 )/" and therefore that h(r) (1 /" )-1 h'(r)n(h l/"(r))' > C. This implies h(r) 1 /n > Cr, thus establishing (5.5.1).

Note that (5.5.1) implies (5.5.2) since P(E) = P(Rn - E) and vE

VRP EThe lower bound in (5.5.3) follows immediately from (5.5.1), (5.5.2), and

the relative isoperimetric inequality (Theorem 5.4.3)

IIDXEII(B(r)) > Crain ( IB(r) n EI 'BO-) - EI ("-1)/"

rn - I rn rn

5.6 Tangential Properties of the ReducedBoundary and the Measure-Theoretic Normal

Now that we have introduced the definition of the unit exterior normal to aset of finite perimeter, we ask whether the existence of the exterior normalimplies some type of regularity of the boundary. In order for the theoryto run parallel to the classical development, the hyperplane orthogonal tothe generalized normal in some sense should be tangent to the reducedboundary (see Definition 5.5.1). Although it cannot be expected that thisplane is tangent in the usual sense, it will be shown that it is so in themeasure-theoretic sense.

For this purpose, we will employ a "blow-up" technique which views thelocal behavior of a set at a point by examining a sequence of dilations of theset at the point. Specifically, let E be a set of locally finite perimeter andsuppose for notational simplicity that 0 E rÎ-E. For each c > 0, considerthe dilation T(x) = x/E and let EE = Te (E). Note that XE, = XE 0 T'and that the scaling of DXEI is of order n - 1. That is,

DXEc [B(r/e)} = 6 1-"DXE[B(r)] for r > 0(5.6.1)

IIDXE^ ii[B(r/e)1 - e 1-n IIDXEII[B(r)) for r > O.

The proof of the second equation, for example, can be obtained by choos-ing a sequence (u 1 ) E CO° [B(r)) such that u; XE in L 1 [B(r)] and

fB(r) ID% Idx -. IIDXEII [B(r)] (Theorem 5.3.3). However,

IDu1,eIdx = E 1-" 'Dui PSB(r/e) f (r)


where u;,e = u ; o TE 1 , Then, u; E -► XE o TT 1 = XE, in L 1 IB(E/r)I and byTheorem 5.2.1,

lim inf Wus e ldx > IIDXE, II[B(r/e)].'~°° J (*/e)

Hence,IIDXEE II[B(r/e)] < e l-n II DXEI [B( r )] .

The reverse inequality is obtained by a similar argument involving a se-quence of smooth function approximating XE a T' .

5.6.1. Definition. For x E 8- E, let r(x) denote the (n -1)-plane orthog-onal to v(x, E), the generalized exterior normal to E at x. Also, define thehalf-spaces

N+(x) = {y : v(x) • (y - x) > 01

H- (x) = {y : v(x) . (y - x) < 0).

5.6.2. Theorem. If E is of locally finite perimeter and 0 E 8- E, then

XE, —► XH- in LÎoc(R') as e . 0

andIIDXOI(U) -, IIDXH- II(U)

whenever U is a bounded open set with 11 11-1 [(8U) fl 7r(0)1 = O.

Proof. Without loss of generality, we may assume that the exterior normalto E at 0 is directed along the x n -axis so that vn (0) = 1 and v i (0) = ... =Vn -1(0) = O. It is sufficient to show that for each sequence {e ; } -+ 0, thereis a subsequence (which we denote by the full sequence) such that

IXE^ , - XH- Idx --; 0 and IIDXE., 11(U) IIDXH- 11(U) (5.6.2)

as e; O.From (5.6.1) and (5.5.3) we obtain for each r > 0,

IIDXE. II[B(r)] =(5.6.3)

and

IIDXE, II[B(r)] - e t-nIIDXEIIIB(er)) ^ Ce l-n (er') n-1 = Crn-1 (5.6.4)

for all sufficiently small e > O. Thus, for each B(r), Crn -1 <_ IIXEc II ev(s(r ^}ffi < C-1 rn -1 for all sufficiently small e > 0. Therefore we may invoke thecompactness of BV functions (Corollary 5.3.4) and a diagonalization pro-cess to conclude that XE1i —+ XA in LL(Rn). For each bounded open

e l-nIID^ E II[B(er)) < C-l e l-n(er )n - 1 = C-t rn-1 1


set S-2, XEc. --0 XA in 2'01) (in the sense of distributions) and thereforeDXEL DXA in 9'(S3). Note that DXA 0 from (5.6.4). MoreoverDXEC, -i DXA weakly in the sense of Radon measures and therefore, forall but countably many r > 0,

DXE1, [B(r)) --; DXA[B(r)1. (5.6.5)

From (5.6.1), and the definition of the generalized exterior normal,

lim DiXEC[B(r)#/IIDXEr II[B(r)1E^0

= Jim DIXE[B(Er)1/I1 DXEII[B(ir)1 = o , i = 1,2, ... , n - 1, (5.6.6)e-s0whereas

li ô DnX E, [B(r)1/IIDXE, 1I [B(r)j = -1. (5.6.7)

Thus, from (5.6.7) and (5.6.3),

lira IIDXE, I1[B(r)1 = - lim Dn XE. , [B(r)1 = -DnXA[B(r)1. (5.6.8)iôo ■-•oo

From the lower semicontinuity of the total variation measure (Theorem5.2.1) we obtain

l=m ôô f II DXE^, II [$(r)] >- II DXA II [B(r)1

and therefore IIDXAII[B(r)1 < -D„X A [B(r)) from (5.6.8). Since the oppo-site inequality is always true, we conclude that

IIDXAIIIB(r)1 = - D„XA[B(r)] (5.6.9)

for all r > 0_ Therefore, by Theorem 1.3.8 and Remark 1.3.9,

IIDXAII(B(r)) = - DnXA(B(r)) = v,(x,A)d1IDXAII(x)-B (r)

This implies that v. (x, A) = 1 for IIDXAlI-a.e. x and thus that v v (x, A) = 0for IIDXA II

-a.e. x, i = 1, 2,... ,n - 1. Consequently, we conclude that the

measures D i XA are identically zero, i = 1, 2, ... ,n - 1. Hence, XA dependsonly on x,^ and is a non-increasing function of that variable. Let

a = sup{x n : XA (z) = 1}.

Since DXA # 0, we know that A oo. The proof will be completed byshowing that A = 0. If A < 0, we would have B(r) C Rn - A for r < AlIand since XE, —■ XA in LL(R"),

0 = 18(r) n Al — slim IEe , n B(r)I = i11m a nI B(re ; ) n El

= lim r"(rei) - 'IB(re;) n El


which contradicts (5.5.1). A similar contradiction is reached if A > O. There-fore, A = H - and by (5.6.8) and (5.6.9),

lim I1DxE„ II [B(r)1 = IIDrIxN - 1I[B(r)}t-400

for all but countably many r > 0. If U is an open set with U C B(r) forsuch an r > 0 and

II DXH - II(aU) = Hn-1 [ir(0) n au] o,

then Corollary 5.2.4 implies IIDXE.,11(U) IIDXH- 11(U).We now will explore the sense in which the hyperplane 7r(x) introduced in

Definition 5.6.1 is tangent to a- E at r. For this, we introduce the following.

5.6.3. Definition. Let y E Rn with Ivl = 1. For r E Rn and e > 0, let

C(x, E , v) = R" n {y = I(y - x) - vi > Ely - xI}.In what follows, it will be clear from the context that both r and v arefixed and therefore, we will simply write C(e) = C(x, e, w).

C(e) is a cone with vertex at x whose major axis is parallel to the vectorv. If M were a smooth hypersurface with v normal to M at x, then foreach e> 0

C(e) n M n B(x, r) = 0 (5.6.10)

for all r > 0 sufficiently small. When M is replaced by 0 - E, Theorem 5.6.5below yields an approximation to (5.6.10).

Before we begin the proof of Theorem 5.6.5, we introduce another con-cept for the exterior normal to a set. This one states, roughly, that a unitvector n is normal to a set E at a point r if E lies completely on one (theappropriate) side of the hyperplane orthogonal to n, in the sense of metricdensity. The precise definition is as follows.

5.6.4. Definition. Let E C Bn be a Lebesgue measurable set. A unitvector n is called the measure-theoretic normal to E at x if

limr -"IB(x,r)n {y:(y—ac)-n<O,y0E}I = 0r-•0

andlirnr'IB(x,r) n {y: (y —x)-n> 0, y E E}I =

The treasure-theoretic normal to E at x will be denoted by n(x, E) and wedefine

a'E = {x : n(x, E) exists).


The following result proves that the measure theoretic normal existswhenever the generalized exterior normal does. Thus, 0 - E C â' E.

5.6.5. Theorem. Let E be a set with locally finite perimeter. Suppose0 E 8- E. Let y be the generalized exterior normal to E at 0 and 7r(0) thehyperplane orthogonal to v. Then,

Ern r"-IIIDXEII[G(e) n B(r)} = 0,r

(5.fî.11)ô

lim r -nIE n H+ n B(r)I = 0, and (5.6.12)r—.0

lim r -n I(B(r) -- E) n H -I - 0. (5.6.13)r —.0

Proof. Again we use the "blow-up" technique that was employed to obtain(5.6.1). Thus, let Tr (x) = x/r and recall that

IIDXErII[ 3 ( 1 )] = r1- "IIDXEII[D(r)}.

Note that Tr [C(E) n B(r)) = C(e) n B(1). Therefore

1I 1XErII1'(e) n B(1)) = r l- "I1DxEII[G(e) n B(r)),

and by Theorem 5.6.2,

T1—"IIDXEII[C(E) n B(r)J -. Hn 1 [C(e) n B(1) n 7r(0)) = O.

This proves (5.6.11).Similarly,

r-nIE n B(r) n H + I = 'ET n B(1) n H+ Iand since XE,. —. XH- in LL(Rn) (Theorem 5.6.2),

li mIEr nB(1)nH+I=1H - n B(1) nH+1=0.

This establishes (5.6.12) and (5.6.13) is treated similarly. D

The following is an easy consequence of the relative isoperimetric in-equality and complements (5.5.3).

5.6.6. Lemma. There exists a constant C = C(n) such that

D Bxrliminf Ix EII ( )1 > C

r—.o rn-1

whenever x E a'E.

Proof. Recall from Definition 5.6.4 that if x E 9'E, then

11mo r -n1B(x, r) n E n H+(z)1 = 0


andtinâ r-NB(x,r) -- El n H - (x)1 = ü

where H+ (x) and H- (z) are the half-spaces determined by the exteriornormal, n(x, E). Since B(x, r)nH - (x) = ([B(x, r)-EjnH - (x))u(B(x, r)nE n H - (x)), the last equality implies that

IB(r) n Ej 1B(x, r) n E n H- (x)1 1lim inf I > lim = - .

r-00 I B (r r~0 IB(Z, r)I 2

Similarly,

lim inf IB(r) - El > 1

r I B(r)I - 2and consequently,

lirn IB(r) n EI _ hill

- El _ 1r^0 IB(r)I r-P0 IB(r)I 2 -

The result now follows from the relative isoperimetric inequality (5.4.3). ❑

This result allows us to make our first comparison of the measures IIDXEIIand Hn- ' restricted to û'E.

5.6.7. Theorem. There is a positive constant C such that if E is a setwith locally finite perimeter, and B C as is a Bord set, then

H" - '(B) < CIIDXEII(B)•

Proof. For each x E B we obtain from Lemma 5.6.6 that

liminf II DXEI ^[B(x, r)j > Cr-4) r

Our conclusion thus follows from Lemma 3.2.1.

5.6.8. Corollary. If E is a set with locally finite perimeter, then

H" (0 *(ô ' E - ô- E) - O. (5.6.14)

Moreover, IIDX E II and the restriction of Hn- ' to i'E have the same nullsets.

Proof. From the discussion in Definition 5.5.1, we have that IIDXEII(R" -

8- E) = 0 and therefore IIDXEII (a' E - 0- E) = O. Thus, (5.6.14) followsfrom the previous theorem. Moreover, if B C S- E with H" -1 (B) = 0, thenIIDXEII (B) = 0 because of the second inequality in (5.5.3). This establishesthe second assertion. ❑

5.7. Rectifiability of the Reduced Boundary 243

5.7 Rectifiability of the Reduced Boundary

Thus far, we have shown that the measure-theoretic normal to a set Eof locally finite perimeter exists whenever the generalized exterior normalexists (Theorem 5.6.5). Moreover, (5.6.11) states that the measure IIDXEIIhas no mass inside the cone C(e), at least in the sense of measure density.This indicates that the reduced boundary may have some appealing tan-gential properties. Indeed, it will be shown that H" -1 -almost all of a-Ecan be decomposed into countably many sets each of which is containedwithin some Cl manifold of dimension (n - 1).

5.7.1. Definition. A set A C R'n is called countably (n - 1)-rectiflcable ifA C AD U [U°_0 fi(Rn- ' )] where Fr' (A 0 ) = 0, and each fi : R" -1 - R" isLipschitz, i = 1, 2, .... Because a Lipschitz map defined on an arbitrary setin Rn -1 can be extended to a[] of R"'' (Theorem 3.6.2), countable (n-1)-rectifiability is equivalent to the statement that there exist sets Ei C Rn- ànd Lipschitz maps fi : Ei --► Rn such that A C Ao U (U_° (E)].i1fé s

The next result is an easy consequence of Rademacher's theorem andTheorem 3.6.2, concerning the approximation of Lipschitz functions.

5.7.2. Lemma. A set A C Rn -1 is countably (n - 1)-rectifiable if andonly if A C 14 120 A, where Hn -1 (A0 ) = 0, and each A,, i _> 1, is an(n - 1)-dimensional embedded C L submanifold of Rn .

Proof. Obviously, only one direction requires proof. For this purpose, foreach Lipschitz function f, in the Definition 5.7.1, we may use Theorem3.10.5 to find C' functions g i,, j = 1, 2, ..., such that

f,(R" -1 ) C [Li gi,.i(Rn-`)j=1

where 11" 1 (NO = O. Let Co denote the critical set of g, ^

Cla = R" -1 n {y : Jgi,j(y) = 0),

where Jg;,; (y) denotes the Jacobian of gij at y. By an elementary areaformula, see [F4, Theorem 3.2.31, Hn -1 [gi,,(C, ,j)I = 0 and therefore theset(e. 00

Ao -u Ni U U g' ,/ (C',1) = 01=1 1,3 =1

has zero H" -1 measure.For each y E R" -r - Ci,1 an application of the implicit function the-

orem ensures the existence of an open set Ui.j (y) containing y such that

00


g,d I U1,i(y) is univalent and that g,,i(U,,,(y)) is an (n - 1)-dimensional C 1

submanifold of R". Clearly, there exists a sequence of points y 1 , y2, ... inRn —1 — C,,i such that U l U=d (yk) D Rn —1 — C;2 and

001J 9j(U(Yk)) ^ g. ,^(Rn-1 - C=,,i).

k-1

Therefore, for each i,00

f,(Rn ' `) - AD C U gi,; (Uo (yk));, k= 1

from which the result follows. ❑

5.7.3. Theorem. If E C Rn is of locally finite perimeter, then S - E iscountably (n - 1)-rectifiable.

Proof. Clearly, in view of Corollary 5.5.3, we can reduce the argument tothe case of E with finite perimeter. Now recall from the proof of Lemma5.6.6, that if x E ô- E, then

lim r-n IB(x, r) E n H+(x)I = 0r—•0

andlim r -nl[B(x,r) - n H (x)I = O.r—*0

Since B(x, r) n H- (x) = ([B(x, r) - E] n (x)) U (B(x, r) n E n H- (x)),the last equality implies that

lirnIB(x, r) n E n H - (x)1 1

r—*o l B(x, r) I _ 2 .

Therefore, with the aid of Egoroff's theorem, for each 0 < e < 1 andeach positive integer i, there is a measurable set F, C 8- E and a positivenumber r, > 0 such that IIDXEII[(a E) - F,) < 1/(2i) and

IE n H+(x) n B(x, r)I < ()" IB(x , r)I (5.7.1)

IE n H- (x) n B(x, > 4IB(x,r)I (5.7.2)

whenever x E F, and r < r,. Furthermore, by Lusin's theorem, there is acompact set M, C F, such that IIDXEII[F, -M,] < 1/(2i) and the restrictionof v(., E) to M, n 0- E is uniformly continuous. Since H n— ' restricted to0 - E is absolutely continuous with respect to IIDXEII (Theorem 5.6.7), ourconclusion will follow if we can show that each M, is countable (n - 1)-rectifiable.

5.7. Rectifiability of the Reduced Boundary 245

We will first prove that for each x E M;,

C(z,e,v(x, E)) n M, n B(x,r0 = (5.7.3)

where C(x, e, v(x, E)) is the cone introduced in Definition 5.6.3. Thus, wewill show that I v(x) . (x — y)I < elx — y1 whenever x, y E M, and Ix — yl <(1/2)r;. If this were not true, first consider the consequences of v(x) • (y —x) > efx — yI. Since the projection of the vector y — x onto v(x) satisfiesIproj,, ( ) (y — x)j > Elx — y1, it would follow that B(y,elx — y1) C H+ (x).Also, since E < 1,

B(y, eix — yl) c B(x, 2 Ix — yl )and therefore

B(y, Ejx — y1) c 11+ (x) n B(x, 2 1x — yl)• (5.7.4)

However, since 2Iz — yl < r,, it follows from (5.7.1) and (5.7.2) that

r }nI n H + (x) n B(x , 2 Ix — vI)I < 4 l2/ IB(x, 2Ix — yl)Î

< 4 enIB(0,1x — 11I)I (5.7.5)

and

1E n B(y, eIx — yl)I > IE n B(y, Eli — vl) n H (y)I

? 41B(y, E1x — 3/I)I

= 4 E n 1$(0 , Ix — y1)1•Thus, from (5.7.4), a contradiction is reached because

4 E n I$(0, lx — yI)I < IE n B(u,Elx - yI)I

(5.7.6)

< IE n H + (r) n B(x, 21z — yl)I < 4 e n 1$(0 , IT — yl)I•

A similar contradiction is reached if v(x) • (y — x) < —Elx — vI and thus,(5.7.3) is established.

We will now proceed to show that each M, is countably (n-1)-rectifiable.In fact, we will show that M. is finitely (n — 1)-rectifiable. First, recall thatM; is compact and that v(•, E) is uniformly continuous on M. It will beshown that for each x o E M, there exists a t > 0 such that M; n B(xo, t)is the image of a set A C Rn-1 under a Lipschitz map. For this purpose,assume for notational simplicity that "(xo , E) = v(ro) is the nth basisvector (0,0,—. , 1). Let ir(xo) be the hyperplane orthogonal to v(x0) and


let p : M1 -o r(x0 ) denote the orthogonal projection of M, into 1r(xo). Theconclusion will be established by showing that p is univalent on B(xo , t)nM1and that pr' Ip[B (xo, t) n M= ] is Lipschitz.

To see that p is univalent, assume the contrary and suppose that y, z EMi are points near xo with Iz-yI < IN and p(y) = p(z). Let u = z-y/Iz-yIand note that Iv(x o ) uI = 1. Since y is continuous, it would follow thatIv(y) • uI > e if y were sufficiently close to xo . However, (5.7.3) implies thatIv(y) • uI < e, a contradiction. Thus, there exists 0 < t < 1r, such that p isunivalent on B(xo , t) n Mi .

Let L be the inverse of p restricted to p[B(x o , t) n M,1 and let y, zp[B(xo, t) n Md. Then

IL(z) - L(y)I _ IL(z)- L(y)I

I z - yI

(I L(z) --- L(y)I 2 - Iproj,(z0)[L(z) - L(y))I2)1/21

(1 IPro:6(. 0 (L(x)-L(1ry)I 2 \

Using again the continuity of y, the last expression is close to

11 IProiy( v) IL(r)-L(y)J12 }

1/2

IL(x) —L(01 3

provided that y is close to xo. by (5.7.3), (5.7.7) is bounded above by1/(1 - e 2 ) 1 / 2 , which proves that L is Lipschitz in some neighborhood of r o .Since M, is compact, this proves that M, is finitely (n - 1)-rectifiable. ❑

The following is an immediate consequence of Lemma 5.7.2 and the pre-vious result.

5.7.4. Corollary. If E C R" is of locally finite perimeter, thenOD

c7 - Ec UM;UNt=1

where H" — ' (N) = 0 and each Mi is an (n - 1)-dimensional embedded C'submanifoid of R".

5.8 The Gauss-Green Theorem

In this section it will be shown that the Gauss- Green formula is valid onsets of locally finite perimeter. The two main ingredients in the formulationof this result are the boundary of a set and the exterior normal. Since weare in the setting of sets of finite perimeter, it should not be surprising

(5.7.7)

5.8. The Gauss-Green Theorem 247

that the boundary of a set will be taken as the reduced boundary and theexterior normal as the measure-theoretic exterior normal.

In Definition 5.6.4, we introduced the notion of the measure-theoreticexterior normal and demonstrated (Theorem 5.6.5) that

a-E C E. (5.8.1)

Moreover, from (5.6.14),

H'(a'E - 0 - E)= 0. (5.8.2)

One of the main objectives of this section is to strengthen this result byshowing that if B C a' E, then

Hn-1 (B) = IIDXEII(B). (5.8.3)

This is a crucial result needed for the proof of the Gauss-Green theorem.

5.8.1. Theorem. If E C Bn has locally finite perimeter, then

Hn-:(B) = IIDXEII(B)

whenever B C a' E is a Borel set.

Proof. If r E a- E, it follows from Theorem 5.6.3 that

r l-n IIDXEII [B ( x ► r)] = IIDXET 1)] -0 IIDXH - lI[$(xt 1)]= Hn-1 [B(x, 1) fl ir(x)]= a(n - 1)

where ir(x) is the hyperplane orthogonal to v(x, E). Therefore,

lim IIDXEII[B(n, r)] =1, x E a-E. (5.8.4)

r^0 a(n - 1)r

Since Hn (a'E - E) = IIDXEII[aÈ —19-4 = 0 (Corollary 5.6.9) wemay assume that B C a- E and B C U1__, Mi , where each Mi is an (n- 1)-manifold of class C l (Corollary 5.7.4). Fix i and let p = Hn -1 I Mi. SinceA is smooth,

zB , rhm }^ 1 = 1, x E B fl Mi,r^0 (Y(f9 - 1)T

and therefore, by (5.8.4),

,li. DX

Bxr( B ) r = 1, x E B f1 M;.

II EIE( (, )I


By the Besicovitch Differentiation Theorem (Theorem 1.3.8 and Remark1.3.9),

Hn-1(B n = p(B) = IIDXEII(B n Mi ).

The result easily follows from this. ❑

We now are able to establish the Gauss-Green theorem in the context ofsets of finite perimeter.

5.8.2. Theorem. Let E be a set with locally finite perimeter. Then,

1 div V dx = n(t, E) • V (x)dHnr 1 (x)8• E

whenever V E Co (Rn; Rn).

Proof. Choose a ball B(r) containing spt V. Then

div V dx = — V . d(DXE) (from Lemma 5.5.2)JE= V(s) • v(x, E)dIIDXE II (from Definition 5.5.1)fa_ E=V (X) •• n(x, E)dHn-! (x) (by the preceding theorem).

a• E

5.8.3. Remark. The Gauss-Green theorem is one of the basic results inanalysis and therefore, the above result alone emphasizes the importanceof sets of finite perimeter. Therefore, a question of critical importance ishow large is the class of sets of finite perimeter. The definition alone doesnot allow easy identification of such sets. However, it is not difficult to seethat a Lipschitz domain, SZ, is a set with locally finite perimeter. An outlineof the proof will be given here while details are left as an exercise, for thereader. We may assume that ft is locally of the form

fl={(w,y). 0 <y< g(w)}

where g is a non-negative Lipschitz function defined on an open cubeQ C Rn -1 . Since g admits a Lipschitz extension (Theorem (3.6.2) we mayassume that g is defined on R" -1 . Let gE be a mollifier of g (Section 1.6)and recall that

for all c > 0. Each set

^Dgejd^ < f IDg ldz4 Q

(5.8.5)

52E ={ (w,3/):0 y<9E(w), wEQ}

5.9. Pointwise Behavior of BV Functions 249

is obviously of finite perimeter because the classical Gauss Green theo-rem applies to it (see Remark 5.4.2). Let X E denote Xç and observe thatllDXE Il(Rn) = Hn-1 (ô1l ). Since

I + iDgE lZdx = Hn-11{(w, t/) = 9E(w)+ w E Q}],

it follows from (5.8.5) that IIDXf 11(Rn) < C where C is some constantindependent of E. We may apply the compactness property of BV functions(Corollary 5.3.4) to conclude that X is BV in R n , thus showing that it islocally of finite perimeter. Moreover, Rademacher's theorem on the almosteverywhere total differentiability of Lipschitz functions (Theorem 2.2.1)implies that the measure-theoretic normal is Hn- t -almost everywhere givenby

n(z, f2) r (Dg(w),1)

1% 1 + IDg(w)1 2(5.8.6)

where x = (w, g(w)).We conclude this section by stating without proof a useful characteriza-

tion of sets of finite perimeter. This will be stated in terms of the measure-theoretic boundary.

5.8.4. Definition. If E C Rn is a Lebesgue measurable set, the measure-theoretic boundary of E is defined by

8,E = {z. D(E,x) >0} n fx.D(Rn—E,z)> 0}.

If we agree to call the measure-theoretic interior (exterior) of E all pointsx for which D(E, x) =1 (D(E, x) = 0), then 8,E consists of those pointsthat are in neither the measure-theoretic interior nor exterior of E. SeeExercise 5.3 for more on this subject. In Lemma 5.9.5, we shall see that0'E and ô, E differ by at most a set of Hn - l -measure O.

5.8.5. Theorem. Let E C Rn be Lebesgue measurable. Then E has locallyfinite perimeter if and only if

Hn-1(K n3M E) < o0

for every compact set K C R".

The reader is referred to [F4, Theorem 4.5.111 for the proof.

5.9 Pointwise Behavior of BV FunctionsWe now begin a treatment for BV functions analogous to that developed forSobolev functions in the first three sections of Chapter 3. It will be shown


that a 13V function can be defined by means of its Lebesgue points every-where except for a set of Hn - '-measure zero and a set that is analogousto the set of jump discontinuities in R 1 .

In the definition below, the following notation will be used:

A t = {x : u(x) > t},

Bt = {x:u(x)<t},

-15(E, x) = lim sup IR n B(x' )Ir-.o IB(x,r)I

andD(E, x) = urn i nf I E n B(x, r)

r~O IB(x , r)I •In case the upper and lower limits are equal, we denote their common valueby D(E, z). Note that the sets A t and Be are defined up to sets of Lebesguemeasure zero.

5.9.1. Definition. If u is a Lebesgue measurable function defined on R",the upper (lower) approximate limit of u at a point x is defined by

ap lirn sup u(y) = inf {t : D(A e , x) = 0}y-•x

(ap Lim inf u(y) = sup{t : D(BE , x) = 0}).3+-•x

We speak of the approximate limit of u at r in case

aplimsupu(y) = apliminfu(y)•v-x v-x

u is said to be approximately continuous at x if

ap lim u(y) = u(x).

5.9.2. Remark. If u is defined on an open set S1, reference to the definitionsimply that u is approximately continuous at r if for every open set Ucontaining u(x),

D[71-' (U) n 1,xj = 1.

An equivalent and rather appealing formulation is the one used in Remark3.3.5. It is as follows: u is approximately continuous at x if there exists aLebesgue measurable set E containing z such that D(E, x) = 1 and u1 Eis continuous at x. It is clear that this formulation implies the previousone. To see the validity of the opposite direction, let r 1 > r2 > r 3 > ... bepositive numbers tending to zero such that

1 11B(x, r) n y : Iu(y) - u(x)1 > k -k =1

Clearly, u I E is continuous at x. In order to complete the assertion, we willshow that 1)(É, x) = O. For this purpose, choose e > 0 and let J be suchthat Er j z < E. Let r be such that 0 < r < rj and let K > J be theinteger such that rK+1 < r < rK. Then,

l( R"00

- E) f1 B(x, r)1 < E 103(x, rk) - B(x, rx+l )}k=K

n fy : 1u(v) -10)1 > k< 18(s, r)1

+ °° I B(x, rk )12k

k=K +lIB(x, r)I + -°°` I$(x, r)1

- 2K 2kk=K+1

00

< IB(x ) r)I Z., 2 k^ K

I B(x, r)IE ^

which yields the desired result since e is arbitrary.One of the main results of this section is that a BV function can be

defined in terms of its approximate limits If" -1 -almost everywhere. Forthis, the following is needed.

5.9.3. Lemma. Let n > 1 and 0 < T < 1/2. Suppose E is a Lebesguemeasurable set such that D(E,x) > r whenever x E E. Then there exists aconstant C = C(r, n) and a sequence of closed balls B(xi, ri) with x; E Esuch that

00

E C 1J B(zer): =1

and

E(r,)"-1 < CIIDxEII[ R"].

i =1

Proof. For each x E Rn , the continuous function

IB(r, r) n__ E1 f(r) IB(x,r)I


assumes the value r for some r s > 0 because it exceeds this value for somepossibly different r and approaches zero as r -. oo. Since r < 1/2, therelative isoperimetric inequality, (5.4.3), implies

[ro(n)rx1(n-1)/n < CIIDXEII[B(x , rs)].

Now apply Theorem 1.3.1 to the family of all such balls B(r, r= ) to obtain asequence of disjoint balls B(x„ ri) such that U 1 B(r , 5r ; ) D E. Therefore,

CC 00

ITa(n)](11 -1)/n

L^`(5r,)n-1 < 5(n -1)G*G^` IIDXEII{B(x, ri}]i= 1 i= 1

< 501-1)CIIDxEIIERn].

In addition to (5.5.3) concerning the (n - 1)-density of the measureIIDXEII, we will need the following.

5.9.4. Lemma. Let E C Rn be a set with locally finite perimeter. Then,for Hn-1-almost every x € Rn — as E,

lirnsup ÎIDXEII{B(x,r)1 _ O.r—)

a ( rt - 1) rn-1

Proof. For each positive number a let

1IDXEIIIB(x ^ r)l A = (Rn — c3'E) fl x:lim sup >a .r-40 cY ( n - 1 ) rn-i

It follows from Lemma 3.2.1 that

IIDXEH{(A) > C)Hn -1 (A).

Therefore Hn -1 (A) = 0 since IIDXEII(A) = 0, thus establishing the con-clusion of the Lemma. ❑

This leads directly to the next result which is needed to discuss thepoints of approximate continuity of BV functions. Recall the definition ofthe measure-theoretic boundary, 0.E, Definition 5.8.4. The next result,along with (5.8.2) shows that all of the boundaries associated with a setof finite perimeter, 0-E, 0' E, and 0,E, are the same except for a set ofHn -1 -measure zero.

5.9.5. Lemma. Let E C R'1 be a set with locally finite perimeter.. Thena•E c a E and lin -1 (0M E - a* E) = O.

Proof. It follows immediately from Definition 5.6.4 that a' E C âM E. Inorder to prove the second assertion, consider a point z E °M E such that

5.9. Pointwise Behavior of 13V Functions 253

D(E, z) > b and D(Rn - E, z) > b where 0 br --•p

andliminf f(r) =1 -D(E,z)<1 -b,

r ---•0

with b < 1 - b. Hence, there are arbitrarily small r > 0 such that b <f (r) < 1- b and for all such r, the relative isoperimetric inequality, (5.4.3),implies

rbct(n)rn}(n-1) /n < Ci IIDxEillB(z+ r)l'Thus,

limsup IIDXEII [B(z, r)i > 0,

r-• cx(n - )rn-1

and reference to Lemma 5.9.4 now establishes the conclusion.

In the next theorem, it is shown that a BV function is approximatelycontinuous at all points except for a set of Hn_

i -measure zero and a count-ably (n - 1)-rectifiable set E which, roughly speaking, includes the pointsat which u has a jump discontinuity (in the sense of approximate limits).It is also shown that at H"- '-almost all points of E, u has one-sided ap-proximate limits. Later, these results will be refined and stated in terms ofintegral averages.

Recall from Definition 5.9.1 that At _ {x : u(x) > t}.

5.9.6. Theorem. Let u E BV (Rn). If

4x) = ap lim sup u(y),I,—s

A(s) apliminfu(y),y^ a

andE = R" n {z : A(x) < µ(x)},

then

(i) E is countably (n - 1)-rectifiable,

(ii) -co < A(x) S p(x) < oo for 11' 1 -almost all z E Rn ,

(iii) for 11n -1 -almost all z E E, there is a unit vector y such that n(z, A,)v whenever .\(z) < s < µ(z).


(iv) For all z as in (iii), w ith -oo < A(x) < p(x) < oo, let H - (z) - {y :

(y - z) • v < 0} and H+ (z) = {y : (y - z) • y > 0}. Then, there a reLebesgue measurable sets E - and E+ such that

Elim

IE_ n H - (z) n B(z,r)I = 1- lim I+ n H+ (x) n B(z, r }I

r^0 IH—(z) n B(,Z, T)I r—PD 111+(z) n B(z, Ty

and

urn u(x) = p(z), Jim u(x) = a(z).zEE'rzH - (a) zEE+f11i+(a)

Proof. Applying Theorems 5.4.4 and 5.7.3, there exists a countable densesubset Q of R 1 such that H(A t ) < oo and 8* A i is countably (n - 1)-rectifiable whenever t E Q. From Remark 5.9.2 we see that

Hn -1 [{U(BM A t - a. At) . t E Q}] = 0.

It follows immediately from definitions that

{x : A(x) < t < p(x)} C A i for t E R I , (5.9.1)

and therefore E C {UaM A t : tEQ},Hn` 1 [E- {Uô'A t : tEQ]]= O. Thisproves that E is (n - 1)-countably rectifiable.

Let I = {x : A(x) = -oo} U {x : p(x) = oo}. We will show thatH" - `(I) = O. For this purpose it will be sufficient to assume that u hascompact support. First, we will prove that H" -1 [{x : A(x) = oo}] = O. LetL t = {z : A(x) > t} and note that D(L t , x) = 1 whenever x E Lt . Nowapply Lemma 5.9.3 to conclude that there is a sequence of halls {B(r)}whose union contains L t such that

00

E Ti -1 <- GIIDXL, II•1=1

Since u has compact support, we may assume that diam B(r,) < a, forsome positive number a. Therefore, Theorem 5.4.4 implies

Hâ -1 1{x:A(x)=oo})=Hâ -l [{nL 1 :tE R'}l<Cl^rni fIIDXLIII(R") =o.

From this it easily follows that Hn -1 [{x : A(x) = oo f] = O. A similar proofyields H" - ' [{x : p(x) _ -oo}] = O. Thus, the set {x : p(x) - A(x) } is well-defined for H" -1 -a.e. x and the proof of (ii) will be concluded by showingthat Hn -1 [{x : p(x) - A(x) +oo}] = O. Since E is countably (n - 1)-rectifiable, it is a-finite with respect to Hn -1 restricted to E. Therefore,

5.10. The Trace of a BV Function 255

we may apply Lemma 1.5.1 to obtain

00

(1,4 - A)dHn- i _ Hn `' [{x : A(s) < t < µ(x) }idt0

00< Hn-1(0A., A t )dt (by 5.9.1)

000< lin-1 (a'A t )dt (by Lemma 5.9.5)

o00

_< CIIDXA, Ildt (by Theorem 5.6.7)0

< CIIDuII(Rn) (by Theorem 5.4.4)

< oo, since spt u is compact.

We will prove that (iii) holds at each point

zE E--{U(a.4 At -0* A t ) :t E Q}.

If t E Q with A(z) < t < µ(z), then z E am A t and therefore z € O A t . Con-sequently, n(z, A t ) exists. But is must be shown that n(z, A t ) = n(z, A 8 )whenever A(z) < 9 < i(z). It follows from the definition of the measuretheoretic exterior normal (Definition 5.6.4) that

D(A t , z) = 1/2 = D(A., z). (5.9.2)

If s < t, then A. D A t and therefore D(A, - A t , z) = 0, which implies thatn(z, A t ) = n(z, A,).

For the proof of the first assertion of (iv), let z E E-- I and choose e > 0such that A(z) < µ(z) - e < A(z). Observe that D(A, (z ) €, z) = 0 while

lim IAy (,) -E fl H - (z) n B(z, r)I = 1

r—.o IH—(z) fl 11(z, r)I '

from (5.9.2). Therefore

hillu-ri^(z) -- e, µ(z) + e] n H - (z) n B(z,r)I - 1

r-0

By an argument similar to that in Remark 5.9.2, this implies that thereis a set E - with the desired properties. The second assertion is provedsimilarly. 0

1

IH-(z) n B (z,r)I

5.10 The Trace of a BV Function

For a given set SZ C R" with suitably regular boundary and u E BV(n),we will show that it is possible to assign values to u at Hn -1 -almost all


points of OA even though u, when considered as a member of L 1 (it), isdefined only as an element of an equivalence class of functions. Recall thattwo measurable functions are called equivalent if they differ at most ona set of Lebesgue measure zero. The difficulty with defining the trace ofa function on the boundary is that an may have zero Lebesgue measure,precisely where the function may be undefined. The theory requires furtherdevelopment in order for this difficulty to be circumvented. The approachwe use for this is as follows. For a certain class of domains S2 C Rn (calledadmissible domains below), if u E BV(A) is extended to all of R" bydefining u E 0 on R" - A, then an easy application of th e co-area formulashows that u E BV(R"). By means of Theorem 5.9.6 we then are ableto define u H" -1 -almost everywhere including E, the set of approximatejump discontinuities.

5.10.1. Definition. A bounded domain SZ of finite perimeter is said to beadmissible if the following two conditions are satisfied:

(i) Hn-1 (ÔA - ôMA) = 0,

(ii) There is a constant M = M(11) and for each x E au there is a ballB(x, r) with

H"-1 [(aM E) n (aMmj < MH"- ! pM E) n q (5.10.1)

whenever E C iZ fl B(x, r) is a measurable set.

5.10.2. Remark. It is not difficult to see that a Lipschitz domain is ad-missible. For this purpose, we may assume that SZ is of the form

A ={(w, y ):OÇ y Ç 9(w)}

where g is a non-negative Lipschitz function defined on an open ball B CRn-1 . From Remark 5.8.3 we know that A is a set of finite perimeter. LetE C A be a measurable set and we may as well assume that H" -1 (S2 flam E) < oo for otherwise (5.10.1) is trivially satisfied. Since 0„E = (aM Enail) U (A n OM E) and H" -1 (011) < oo, we conclude from Theorem 5.8.5that E has finite perimeter. Hence, we may apply the Gauss Green theorem(Theorem 5.8.2) with the constant vector field V = (0, 0, ... ,1) and (5.8.6)to obtain

V • n(x, SZ)r1Hn -1 (x) + V • n(x, E)dH"-1 (3• ) = U.(b• E)n(Bn) (8 • E)nS2

Therefore, if a is the Lipschitz constant of g, we have

1+

2 Hn-1i(a.E) n (a^)j < H"-1 ((a ' r) nl al

onI

5.10. The Trace of a BV F unction 257

and reference to Lemma 5.9.5 establishes the desired conclusion.

5.10.3. Definition. Whenever u is a real valued Lebesgue measurablefunction defined on an open Set S2, we denote by uü the extension of u toRn:

uo ( x ) = { u(r)Q

x E S2x E .Rn - S2_

Observe that uo is merely a measurable function and is therefore de-fined only almost everywhere. Later in the development, we will consideru E BV(S2) where S2 is an admissible domain, and then we will be ableto define uo everywhere except for an H" -1 -null set. If S2 is a smoothlybounded domain and u E BV (0), it is intuitively clear that uo E BV (R" )because the variation of uo is greater than that of u by only the amountcontributed by H" - ' (Oft). The next result makes this precise in the contextof admissible domains.

5.10.4. Lemma. If 11 is an admissible domain and u E BV(1l), thenuo E BV(Rn) and Iluollsv(R.) < CIlullsv(a) where C = C(f2).

Proof. It suffices to show that u0 is BV lu a neighborhood of each point ofan because an is compact. If we write u in terms of its positive and negativeparts, u = u+ - u - , it follows from Theorem 5.3.5 that u E BV (S2) if andonly if u+(0), It- 01) E BV(0). Therefore, we may as well assume that u isnon-negative. For each x E Oft, let B(x, r) be the ball provided by condition(ii) of Definition 5.10.1. Let cP be a smooth function supported by B(x, r)such that 0 < < 1 and cp 1 on B(x, r/2). Clearly, youo E By (S2) andTheorem 5.4.4 and Lemma 5.9.5 implies

lin -1 1S2f1 aM A r ldt = IIH(cPuo)ll(SZ) < co (5.10.2)

where A t = {x : Spuo (x) > t). Since !A t -fflB(x, r)I = 0 for t > 0, (5.10.1)and (5.10.2) imply

Hn - 1 (am A t )dt < GIID(0uo)II(S2 ) < oo.

Hence, by Theorem 5.4.4, Spun E BV (R") with

IlDuoil[$(r, */ 2 )l11 <_ IID(puo)II(Rn) < cIID(taüo ) II(n)-However, by (5.1.2),

IInOvuo)II(SZ) = sup J u

o(pdiv V dr : V E Co (SZ; R"), IVI < 1n

1 00


upcp div V dx =n

u div(^pV)dx — n u

DDv • V dx.

Therefore

IIDuoIIf B(x, r/2)]II <_ CIID(vuo)II(n) < CIIDuIl(n) + c(r)IIuII I ; S2s r + c(r)]IIuIIev(n) •

This is sufficient to establish the result because OS/ is compact. 0

We now are able to define the trace of u on the boundary of an admissibledomain.

5.10.5. Definition. If 0 is an admissible domain and u E BV (11), thetrace, u', of u on an is defined by

IL' (x) = puo (x) + Auo (x)

where µ,yo (x) and Auo (x) are the upper and lower approximate limits of u oas discussed in Definition 5.9.1 and Theorem 5.9.6.

5.10.6. Remark. We will analyze some basic properties of the trace in lightof Theorem 5.9.6. Let E = {x : Auo (x) < Puo (z) ), A t = {x : uo (x) > t },and select a point x o € E fl WI where (iii) of Theorem 5.9.6 applies. Thus,there is a unit vector u such that

n(xa , A t ) = 4r whenever Au° (xo ) < t < puo (xo).

We would like to conclude that

y = ±n(x 0 , SZ). (5.10.3)

For this purpose, note that 0 E [Auo (zo), A t" (z0 )] and A t C 0 for t > O.If t > 0 and u $ ±n(zo , SZ), then simple geometric considerations yield

limsup I(At _n B(xa, 01I > O.f'o IB (x o, ) I

This is impossible since A t C O. On the other hand, if t < 0 and y $±n(xo, SZ), then

lim sup I(Bt — n) n B(xo, r)I > 0,f--00 I B(xo, r)I

an impossibility since Bt = {x : uo (x) < t } C D. Hence, (5.10.3) is estab-lished.

and

I

5.10. The Trace of a BV Function 259

Also, observe that

if y - n(x o , Si), then A 0 (x0) = 0. (5.10.4)

For, if a uo (r0 ) < 0 there would exist t < 0 such that Au° (To) < t < ilup (xa).Because y = n(xo , A i ), it follows that

D(A t n{x: (x—x o )•v >0} , zo) = O.

But t < 0 implies (R" - 12) - A i l - O. This, along with the fact thatu = n(xo , SZ) yields

D(A 1 n {x : (z - x0 ) • v > 0}, z0 ) > 1/2,

a contradiction. Therefore, ALO (xo ) < O.On the other hand, if A 0 (xo) > 0, there would exist t > 0 such that

D(A i , xo ) = 1. This would imply that

D(A t C1 {z:(x - zo )•v>0},xo )'1/2

which is impossible since

A t -S2 = 0 and

D(SZ n {z: (z-z0)•v>0},xo)= 0.

Thus, (5.10.4) follows and a similar argument shows that

if y = -n(xo, f2), then pun (x 0 ) = O. (5. 10. 5)

Later, in Section 5.12, after certain Poincaré-type inequalities have beenestablished for BV functions we will he able to show that if f2 is admissibleand u E BV (f2), then

Iint iu(Y) - u` (i)i"/("-1)dy = 0 (5.10.6)r-60

for 11" - '-almost all x E BSZ.We conclude this section with a result that ensures the integrability of

u` over an.

5.10.7. Theorem. If f2 is an admissible domain, there is a constant M =M(I1) such that

"-

1

8^ n

whenever u E BV(f2).

Proof. Since by definition, if A„. + pue , it suffices to establish theinequality for the non-negative function p = A u., the case involving Abeing treated in a similar manner. As in the proof of Lemma 5.10.4, we

fB ( r)flO


need only consider the case when it is replaced by (pp, where coo is a smoothfunction with 0 < cp < 1, cp = 1 on B(x, r/2), and spt u C B(x, r), whereB(x, r) is a ball satisfying the condition (5.10.1).

First, with A t = {x : (pµ(x) > t), observe that A t n 0,,,11 c (âM A t ) n(0,,4 0), for t > 0. Indeed, let x E A t n OM fl and suppose x Om A t . Theneither D(A t , x) = I or D(A t , x) = 0. In the first case D(f , x) = 1 since(A t - SZ = 0 for t > 0. Hence, x axi s-2, a contradiction. In the secondcase, a contradiction again is reached since the definition of cpi..t(x) impliesz ¢ A t . Therefore, we have

H"-' (At n âMn1 < Hn -1 [(aMAt) n (akin)i,

Thus, we have

µ dHn-1 < dHn-if (a,r%2)naM L2 8(x,r)f18M ^3

t > 0. (5.10.7)

00< Hn-t (At n 0,,4 3-1) (by Lemma 1.5.1)

000

o H"-1[(aMAt) n (8M f)jdt (by 5.10.7)

< M Hn-`[(a,,,At) n f2)dt (by 5.10.1)

< MIID(cpp)11(SZ) (by Theorem 5.4.4)

< Mlllullsv(n)•

Since A u almost everywhere, the last inequality follows as in the proofof Lemma 5.10.4. O

5.11 Poincaré-Type Inequalities for BV FunctionsIn this section we prove the main inequality (Theorem 5.11.1) from whichessentially all Poincaré-type inequalities for 13V functions will follow. Thisresult is analogous to Theorem 4.2.1 which was established in the contextof Sobolev spaces. In accordance with the previous section, throughout wewill adopt the following conventions concerning the point-wise definition ofBV functions. If u E BV (R") set

u(x) = 2 [au (x) + µu (z)] (5.11.1)

at any point where the right side is defined. From Theorem 5.9.6(iî), weknow that this occurs at H" - t-almost all s E R" . If u E BV(SZ), St admis-sible, then we know by Lemma 5.10.4 that n o E BV (R") and therefore upis defined H" -1 -a.e. on R". Thus, we may define u on II in ternis of up as

5.11. Poincaré-Type Inequalities for BV Functions 261

follows:(5.11.2)

At first glance, it may appear strange to define u(x) = 2u.0(x) on OS),but reference to (5.10.4) and (5.10.5) shows that this definition implies theintuitively satisfying fact that at H' -a.e. x E an, either u(x) = puo (x)or u(x) = A uo (x). Consequently, u is a Borel function defined H" - -a.e. onSi Note that we have for H" -1 -a.e. x,

au(x) + bu(x) = (au + bu)(x)

whenever a, b E R 1 .

5.11.1. Theorem. Let Si be a connected, admissible domain and supposeu E BV(SZ). If T E [BV (S2)]' and T(Xn) = 1, then

11u - T(u)IIn/(n-1);52 <_ CIITII IIDuII(i 1 ), (5.11.3)

where 11TII denotes the norm of T as an element of [BV(S2)J', and CC(Sl, n).

Proof. It suffices to show that

Ilu — T (u )I I 1;0 <_ cI I TII IIDu11(n ) , (5.11.4)

for if we set f = u - T(u), then by Sobolev's inequality and (5.11.4),

IIIIIn/(n--1) ;s1 = IIfOIin/(n-1);R" < C,IIfollav(Rn) <_ CII f IIBv(Z).The last inequality follows from Lemma 5.10.9. Also, note that the Sobolevinequality holds for fo because it holds for the regularizers of fo, whose 13Vnorms converge to IIfolIBy(R^) by Corollary 5.2.4. Thus, in view of the factthat 11TII >_ ICI - ` ^ (5. 11 .3 ) follows from (5.11.4).

To prove (5.11.4), it is sufficient to assume fSZ u(x)dx = 0 since theinequality is unchanged by adding a constant to u. With this assumption,(5.11.4) will follow if we can show

Ilulll;^ < CIIDuII(IP) (5.11.5)

because

Ilu - 7'(u)Il1sz IIuIl1;i1 + ISiI 1I 7'II IIuhIBV(s1)< (1 + ICI IITIII(IIull1;n + IIDuII(i1))<_ 21n1 IITII(IlulIi , ^1 + IlDull(S))•

If (5.11.5) were not true for some constant C, there would exist a sequenceuk E BV (S2) such that

uo(x) x E_ S2u(x ) r 1 2u 0 (x) x E 80.

I uk(x)dx = 0, Iluk 1, and IlDuk1l(11) --> 0. (5.11.6)


For each uk, form the extension uk, o by setting u = 0 on Rn — S2. FromLemma 5.10.4, it follows that the sequence {Iluk,ollBv(R^)} is boundedand therefore an application of Corollary 5.3.4 implies that there existu E BV(Rn) and a subsequence of (uk,o} (which will still be denoted bythe full sequence) such that uk, o u in L i (0). Therefore lûlli

;st = 1from (5.11.6). But (5.11.6) also show that llDull(11) = 0 and therefore uconstant on 11 since SZ is connected. Consequently, u - 0 which contradictsIlulli;a = 1. Thus, (5.11.5) and therefore (5.11.4) is established. Q

5.11.2. Corollary. Let 12 be a connected, admissible domain. Let c andM be numbers such that

lao + aicl < MIlao + a[ull av(n)

for all aa , a i E R 2 . Then there exists C = C(I1) such that

Hu — 1);n Ç CM llDull(Rn ). (5.11.7)

Proof. Define a linear map To on the subspace of BV (S2) generated byX s-1 and u by T0(x0) = 1 and To (u) = c. From the hypotheses, the normof To is bounded by M and therefore an application of the Hahn-Banachtheorem provides an extension, T, to BV (S2) with the same norm. Nowapply Theorem 5.11.1 to obtain the desired result.

5.12 Inequalities Involving Capacity

We now will investigate the role that capacity plays in Sobolev-type in-equalities by considering the implications of Theorem 5.11.1. Recall thatthe BV (S2) is endowed with the norm

IIullBv(n) = liuill;fl + IuDull(Il).However, for notational convenience, we will henceforth treat BV(Rn) sep-arately and its norm will be given by

HullBv(R°) = llDull(Rn )-In Section 2.6, Bessel capacity was introduced, developed, and subse-

quently applied to the theory of Sobolev spaces. Because of the irreflexivityof L i , it was necessary to restrict our attention top > 1. The case p = 1is naturally associated with BV functions and the capacity in this case isdefined as follows:

7(E) = : v E Y(Rn), E C int {v > 1 }},

wherey(Rn) = Ln/(n-l)(Rn) n BV(Rn).

lin -1 (am {hj > t })dt+ co

1_m

5.12. Inequalities Involving Capacity 263

Note that W1,1(Rn) c Y(R") and by regularization (cf. Theorem 5.3.1),that

RILL' (n-1) <_ CI1DuII(Rn ) (5.12.1)for u E Y(R"). A simple regularization argument also yields that in caseE is compact,

7(E) = inf{IIDvIj I : v E Co (R"), E C int{v > 1 }}. (5.12.2)

5.12.1. Lemma. If E C R" is a Suslin set, then

y(E) = sup{y(K) : K C E, K compact}.

Proof. Referring to Theorem 2.6.8, we see that it is only necessary toshow that y is left continuous on arbitrary sets since right continuity oncompact sets follows directly from (5.12.2). Thus, it suffices to prove thatif E 1 C E2 C ... are subsets of Rn, then

^

y (U E; — lirni-.00

i= 1

For this purpose, suppose

a = limy(Ei)< oo, and e>O.•-I00

Choose non-negative v i E (R") so that

Ei C int{x : v i (3) > 1) and II Dv ; II(R" ) < 7(E; ) +e2',

and let h i = sup{v1, v 2 , ... , v;}. Note that h, E Y(R") and

hi sup{h3 _ 1 , vj }, Ei_1 C int{x ?: 1).

Therefore, letting Ij = inf {hj _ i i v3 ), it follows from Theorem 5.4.4 (whichremains valid for functions in Y(R")) and Lemma 5.9.5 that

II Dh1II (R" ) +'r( E;-1 ) <_ IIDh1II(R" ) + IIDI,Î(R^)

f +oo

+ H n 1(a%,{1j > t })dt.^

It is an easy matter to verify that

ax, {hi > t} U 9M {Ij > t} C aM {hj _ 1 > t} U a {vj > t}

8,K {hj > t} n 8M {I1 > t} c r3M {hj _1 > t} n aM{vj > t}.


Consequently,

Hn - 1 (Om {hj > t}) + Hn - 1 (â,,, {Ij > t})

< H" -1 (0,M {hj_1 > t}) + H" -1 (0, 4 {vi > t})

and therefore,+oa

Ilbhjll(Rn ) +7(Ej-1) <_ Hn- I (aM{hj-, > t })dtf00

+m

+ Hn- 1 (ô,^{v) > t })dtf ^- IIDhj - 1II(R" ) + IIDU,II(R" )< IIDhj-1II(Rn ) +7(E,) + E2 -? .

It follows by induction that

IlDhall(Rn ) < 7(Ej) + E e2 - s.i =1

Therefore, letting w = timj._.œ hi, (5.12.1) implies

IIwII("+1)/n = ?^ ^ Ph/ II (n+1)/n tim ^pCIID►t,II (R") _< ^(a + E },

whereas the proof of Theorem 5.2.1 implies

ODwII(R") limiuf IlDh,II(R") < oo.1 ^ o0

Thus, w E Y(R") and

00

7 u E, < IIDwII(R") < 1 m e llbh, ll(R") < a+ e. El1,-i

In addition to the properties above, we will also need the following.

5.12.2. Lemma. If A C R" is compact, then

7(A) = inf{P(U) : A C U,U open and lUI < oo}. (5.12.3)

Proof. Let 71(A) denote the right side of (5.12.3).7(A) < 7 1 (A): Choose rt > 0 and let A C U where U is bounded, open

andP(U) < y 1 (A) + rj.


Let X = Xu and X E Xu * ço,, where cp, is a regularizer. Then X E > 1 onA for all sufficiently small e > 0 and

7(A) <R IDX,ldx (by (5.12.2))"

< IJ DXu II( R") (by Theorem 5.3.1)< + rt.

This establishes the desired inequality since n is arbitrary.71(A) < 7(A): If q > 0, (5.12.2) yields u E C(R) such that u > 1 on

A andIDuldx < -y(A) + rt.

By the co-area formula,co

IDuldx = fin-1 l

u -1 (t)ldtRn 0

l

H" -1 [u -1 (t)ldto

> Hn-1(u—' (to)]

for some 0 < to < 1. Since a{u > to } C u -1 (to), with the help of Lemma5.9.5, it follows that

71(A) < P({u > to}) < H" -1 [i9{u > toil < Hn- 1 [11- I(to )]< y(A) + q. fl

We now are able to characterize the null sets of 7 in terms of Hn-1 .

5.12.3. Lemma. If E C R" is a Suslin set, then

ry(E) = 0 if and only if H"- '(E) = O.

Proof. The sufficiency is immediate from the definition of H" -1 and thefact that 7[B(r)J = Cr''.- In fact, by a scaling argument involving x -+ rx,it follows that 7[B(r)] = y[l3(1)]r" -1 .

To establish necessity, Lemma 5.12.1 along with the inner regularity ofH" - 1 1F4, Corollary 2.10.481 shows that it is sufficient to prove that ifA C R" is compact with 7(A) = 0, then lift -1 (A) = O. For e > 0, theprevious lemma implies the existence of an open set U ❑ A such thatP(U) < e. Lemma 5.9.3 provides a sequence of closed balls {B (r ; ) } suchthat U i'° 1 B(r,) DU ❑ A and

CCO

>2r1 ^ CP(U) < C. ❑


We proceed with the following result which provides some informationconcerning the composition of [BV(R`t))' .

5.12.4. Theorem. Let II be a positive Radon measure on Rn . The followingfour statements are equivalent.

(I) H n— I (A) = 0 implies that µ(A) 0 for all Borel seta A C Rn andthat there is a constant M such that I f udpl 5 MII uIlev(Rn) for allu E BV(R").

(ii) There is a constant M such that p(A) < MP(A) for all Borel setaACR" with IAI<oo.

(iii) There is a constant M1 such that t(A) < M j 7(A) for all Borel setsACR".

(iv) There is a constant Mi such that p[B(x,r)) _< Mar" -1 wheneverx ERn and rER' .

The ratios of the smallest constants M, M 1 , and Mf r have upper boundsdepending only on n.

Proof. By taking u = XA, (ii) clearly follows from (i) since

IIuIIBV(R*) = IIDuUI(Rn ) = P(A).

For the implication (ii) = (iii), consider a compact set K and observethat from the regularity of p and (ii),

p(K) = inf {µ(U) : K C U, U open and IUI < oo}< Minf{P(U) : K C U, U open and IUI < co).

Lemma 5.12.2 yields p(K) < M7(K). The inner regularity of p and Lemma(5.12.1) give (iii).

Since 7[B(r)) Crn -1 , (iii) implies (iv).

Clearly, (iv) implies that p vanishes on sets of Hn - l-measure zero. Con-sequently, if u E BV(Rn), our convention (5.11.1) implies that u is definedp-a.e. If u is also non-negative, we obtain from the co-area formula

00

IIDuII(Rn ) = IIull$v(R.) = P(A t )dt (5.12.4)f

where A t = {x : u(x) > t). In particular, this implies that for a.e. t, A t hasfinite perimeter. For all such t, define

Ft = A t n {x : D(A t ,x) > 1/21.

For x E A t , the upper approximate limit of u at x is greater than t (see5.11.1), and therefore

A t - Ft C {x : 0 < D(A t , x) < 1/21. (5.12.5)


Therefore, A t -- Ft C aM A t . In fact, A t - Ft C a„,r A t - ()silt becausex E ô' A t implies that D(A t , x) = 1/2. Therefore, H" -1 (At - Ft) 0by Lemma 5.9.5. Thus, we may apply Lemma 5.9.3 to F t and obtain asequence of balls (B(r; )} such that Ft C Us_ 1 B(r i ) and

00

E 7- 7,1- 1 < CP(Ft )-

i =1

Therefore (iv) yieldsµ(Ft) < CMZ P(Ft ). (5.12.6)

Now, P(A t ) = P(Ft ) and since p vanishes on sets of N" - '-measure zero,we have µ(A t ) = p,(FF ). Thus, Lemma 1.5.1, (5.12.6), and (5.12.4) imply

f 00

udp =p(A t )dt < CM2IIuIIBV(R")-0

If u is not non-negative, apply the above arguments to IuI to obtain (i). ❑

5.12.5. Remark. A positive Radon measure p satisfying one of the con-ditions of Theorem 5.12.4 can be identified with an element of [BV(R")j•and M can be chosen as its norm.

Suppose SZ is an admissible domain and p a positive measure such thatspt p C SZ. In addition, if p E [BV (1 ))', then there exists a constantC C(1, p) such that

udp < CIIuIIBV(p) < CIIuIIBv(R„)

whenever u E BV(Rn). Thus, p E [BV(R")]' and Theorem 5.12.4 applies.On the other hand, if spt p C R and one of the conditions of Theorem 5.12.4holds, then p E (BV(St))' because of Lemma 5.10.4. Thus, for measures psupported by SZ, p E [BV (0)]* if and only if one of the conditions ofTheorem 5.12.4 holds and in this case there is a constant C = C(11) suchthat

C`'IIIPII[BV(R")]• < 1114 11[13V (On' <_ CI' pII[BV (Rn )]'. (5.12.7)

For the applications that follow, it will be necessary to have yet anotherformulation for the capacity 7.

5.12.6. Lemma. If A C R" is a Suslin set, then

i(A) = sup{p(A)} (5.12.8)

where the supremum is taken over the set of positive Radon measures p E[BV(R0)]* with IIPIIiBV(R")l• < 1.

Proof. Because of the inner regularity of -y (Lemma 5.12.1) it suffices toconsider the case when A is compact. Referring to the Minimax theorem


stated in Section 2.6, let X denote the set of all positive Radon measuresp with spt p C A and p(Rn) = 1. Let Y be the set of all non-negativefunctions f E Co (Rn) such that IIDf II► < 1. From the Minimax Theorem,we have

One of the most frequently used Poincaré-type inequalities is

IIullp;n < C(U)IIDüllp;n

where p > 1, u E W 1 P(11), and An u(x)dx = O. Inequalities of this type weretreated from a general perspective in Section 4.2. In the next theorem, wewill obtain a Poincaré-type inequality for BV functions normalized so thattheir integral with respect to a measure in [BV(Q)j• is zero. That is, themeasures under consideration are those with the property that p[B(x, r)] <Mr" -1 for all balls B(x, r). For example, this includes Lebesgue measurerestricted to a bounded domain or (n — 1)-Hausdorff measure restricted toa compact smooth hypersurface in R" .

5.12.7. Theorem. Let SZ be a connected admissible domain in R" and letp be a non-trivial positive Radon measure such that apt µ C U and for someconstant M > Q that

14B(x, r)] < Mr"'for all balls B(x,r) in R". Then, there exists a constant C = C(1) suchthat for each u E BV(U),

II^ - u(0IIn/(n --1);11 G IIDuII(n)µ(

^ )

where û(µ) _ 71s

f u(z) dp,(x).

Proof. Theorem 5.12.4 states that u E [MTV)] * and because S2 is admis-sible, (5.12.7) shows that may be regarded as an element of IBV(11)j`.Therefore, Theorem 5.11.1 is applicable and the result follows immedi-ately.

This leads directly to the Poincaré inequality for BV functions.

5.12.8. Corollary. Let SZ be a connected, admissible domain and let A C Sibe a Suslin set with H" -1 (A) > O. Then for u E BV(S2) with the property

sup inf f f dµ = inf sup f f dt.t.f EY PEX pEX fEy

It is easily seen that the left side is equal to the reciprocal of 7(A) whereasthe right side is the reciprocal of the right side of (5.12.8).


that u(x) = 0 for H" -1 -almost all x E A, there exists a constant C = C(f2)such that

Ilulln/(n-1);n 7( A) ÎDuII (^)•

Proof. From Lemma 5.12.6 we find that A supports a positive Radon mea-sure µ E [BV(1)1' such that µ(A) > 2 -1 7(A) and IIIiIIiBv(R.)i• < 1. Thus,for any u E BV (S2) with the property in the statement of the corollary,f u dp = O. Our conclusion now follows from the preceding theorem. ❑

We now consider inequalities involving the median of a function ratherthan the mean. The definition of the median is given below.

5.12.9. Definition. If u E BV (S2) and a positive Radon measure in[BV (S2)]', we define med(u, p) as the set of real numbers t such that

µ[S2 n {x : u(x) > t}] < 2µ(SZ)

µin {x : u(x) < t}] <

It is easily seen that med(u, A) is a non-empty compact interval and thatif as and al are constants, then

med(a o + al u, p) = ao + a 1 med(u, it) . (5.12.9)

If c E med(u, p), then µ(Ac ) > Îµ(S2) where A, = SZ n {x : u(x) > c}].Consequently,

;p(n) _< cµ(k) 5jlu(x)Id(x).

Similarly, if c Ç 0, then µ(Bc ) > ;µ(S1) where Bc = 11 n {x : ta(x) < c }]and

-- µ(^ < -cµ(i3c ) <JBC

- u(x)d1i(x) < f Iu(x)d(x).

Therefore,

Ici ^ ? Iuldu û(^) nand (5.12.9) thus implies

lao + aid <_ 2 llPIIEBV(n)l• Ilao + a l ullav (o)•µ(n)

The following is now a direct consequence of Corollary 5.11.2.

(5.12.10)

5.12.10. Theorem. Let 11 be a connected admissible domain in R" and letµ be a positive Radon rneasure such that apt p C and for some constantM>0 that

µ[$(z, r)] < Mrn -1


for all balls B(x, r) zn R". Then there exists a constant C = C(1l) suchthat for u E BV(1l) and c E med(u,p),

I u — c l l n r (n- 10 C M IDuII(0).p(n )

5.12.11. Corollary. Let SZ be a connected, admissible domain and let Aand B be disjoint Suslin subsets of fl of positive H" -1 -measure. Thenthere exists a constant C = C(f) such that for each u E BV(11) with u > 0Hn -1 -a.e. on A and u < 0 Hn -1 -a.e. on B,

IIuIIn/(n-1);f2 Ç

Proof. Lemma 5.12.6 yields measures p and Y supported by A and Brespectively such that

7(A) > p(A) > 7(A), IIpII[BV (R")). < 1,

7(B) > v(B) ? 7(8), IIÎI i BV ( R" )3 . < 1.

DefineA = p(A)v + v(B)p

and observe that 0 E med(u, A) for u E BV (S1). Since A(i) - 2p(A)v(B)and IIAIIIBv(Rn)]. < p(A) + v(B), the conclusion follows from Theorems5.12.4 and 5.12.10. ❑

5.13 Generalizations to the Case p > 1Since a BV function u is defined H" -1 -almost everywhere by means of(5.11.1), an obvious question arises whether the results of the previoussection can be extended by replacing hull"/(n-1);n that appears on the leftside of the inequalities by the appropriate LP-norm of u defined relative to ameasure that is absolutely continuous with respect to H". We will showthat this can be accomplished by establishing Poincaré-type inequalitiesthat involve Ilull n/ ( n _1) , ^, where A is a positive measure that satisfies oneof the conditions of Theorem 5.12.2.

5.13.1. Theorem. Let A be a positive Radon measure on R". The followingtwo conditions are equivalent:

(i) H" -1 (A) = 0 implies A(A) = 0 for all Borel sets A and for 1 1

271

(ii) There is a constant C 1 such that

A[B(x, r 11 < Cp rF(n-1)

for all balls B(x,r).

The ratios of the smallest constants C and Cl have upper bounds de-pending only on n.

Proof. The case p = 1 is covered by Theorem 5.12.2, so we may assumethat p > 1. Suppose (ii) holds and let f € f7' (a), f > O. Then, by Holder'sinequality,

I /p'

f (x)da < f F'da A[B(x, r)] 1 /nB (s,r)

C ill f II F• ,Arn-1 ,

Thus, the measure f a defined by

f a(E) =E f (x)da(x)

satisfies condition (ii) with p = 1. Therefore, by Theorem 5.12.2,

If uf da < MClllfllpf)u'IIBv(R")

for all u E BV (R'1 ). From the definition of Hausdorff measure, it is clearthat hfn-1 (A) = 0 implies a(A) = 0. Thus, (i) is established.

Now assume that (i) holds. For each Borei set A C Rn and each e > 0,reference to Lemma 5.12.1 supplies an open set U D A such that P(U) <ry(A) + e. Therefore, from (i) with u = XA,

A(A) 1/p < A(U) 1 /11 < CI1DXuII {Rn )

=CP(U)< C7(A) + Ce.

In view of the fact that 7[B(x, r)] = Cr"', (ii) is established.

With the help of the preceding theorem, results analogous to those ofthe Section 5.12 are easily obtained. For example, we have the following.

5.13.2. Theorem. Let Ii be a connected, admissible domain in Rn . Let pand ) be positive Radon measures supported by Sl such that

14[B(x, r)l Ç C1rn-1,

4


a[B(z, r)] < C2-P(n-i) 7 1 < p < n/(n — 1),

for all balls B(x,r). Then, there exists a constant M = M(1) such that

IJu — (i)Ilp.a 5 MCsC2 IlDull(o)u(1)

for all u E BV (12).

Proof. From Theorem 5.13.1 and Lemma 5.10.3 we have

llullp.a;n = lluollp,a^

C ll nai l e v(Rn )

^ CIIu l HV(^^.

Applying this inequality to u — û(p), we obtain

Ilu — u(1z)llp,a;s1 < Culu — fi(p)IIBV(n)< CIlu — fi(1t)IIt;i1 + cllDull(0)< CllDull(1),

Other results analogous to those in the preceding section are establishedin a similar way and are stated without proof.

5.13.3. Theorem. If c E med(u, µ), then

Ilu — cll p ,a < M GiCs IlDull (n).^(^)

Also,If u(x) = 0 on A where A is a Stalin set of positive 1-1'-measure, then

Ilullp,a < M CA llDull(fZ).

5.14 The Trace Defined in Terms of IntegralAverages

For u E BV (I0), recall the following facts established in Theorem 5.9.6:

(i) E = {x : \(x) < µ(x)} is countably (n — 1)-rectifiable,

(ii) —oo < A(x) < p(x) < oo for fin - I-almost all x E Rn ,

(iii) For Hn - '-almost every z E E, there exists a unit vector y such thatn(z, A.) = y whenever a(z) < s < µ(z).

5.14. The Trace Defined in Terms of Integral Averages 273

Although our convention (5.11.1) of setting u(x) = z[A,(x) + µu(x))allows a meaningful pointwise definition of u at fin - '-almost all points,the simple example of u as the characteristic function of a ball shows thatit is not possible to define u in terms of its Lebesgue points Hn -1 -almosteverywhere. This is merely one of the ways that the BV theory differs fromthe Sobolev theory developed in Chapters 3 and 4. However, in this sectionwe will show that a slightly weaker result holds:

lim u(y)dy = 1 1A„(37)2 + Pu(x)J

B(x,r)

for Hn - '-almost all x E SZ. If S2 is admissible, then a similar result will beshown to hold for the trace u`, the only difference being that the ball B(x, r)in the above expression will be replaced by B(x, r) n SZ. Briefly stated then,a BV function can be defined pointwise H" - '-almost everywhere on SZ asthe limit of its integral averages.

5.14.1. Remark. If u E BV(R) is bounded, then it is easily seen that(with convention (5.11.1) in force),

lim !u(r) - u(xo)rdx = 0 (5.14.1)r^0 B(zo,r)

for x o E and that for all z E E for which (iii) above holds,

lim lu(s) - Au (z) I° dz = 0, (5.14.2)r--+0 9+(sir)

where

lim Iu(z) — Au (z)I ° dz = 0,r^0 _(x,r)

(5.14.3)

B+ (z,r) = B(z, r) n (y: (y - z )•v > 0),B- (z,r)= B(z,r) n {y: (y z) < 0),

and

lb verify (5.14.1), use Remark 5.9.2 to conclude that there is a Lebesguemeasurable set A such that /3(A, x0) = 1 and

lim = u(xo).x ^xoxEA

Then

lim r- " lu(x) - u(x o )I°dx = lim r -n lu(x) - u(xo )rdxrô f (zo,r) 7-00 B(xfo,r)nA

+ li O r — " J Ili(r) — 4L(x0)I° GIX.B(xo ,r)f1A


The first term tends to 0 by the continuity of u I A. The second term alsotends to 0 because u is bounded and D(A, xo ) = 0, where A = R" - A.

Now consider (5.14.3), the proof of (5.14.2) being similar. from the def-inition of (h(z), we have that D(A = , z) = 0 for each t > p(z). From (iii)above, D(A, fl {y : (y - z) • v < O}) = 0 for A(z) < s < µ(z). Thus, if e > 0,t- s<e, and s<µ(z)<t,we have

r—.0 f (z,r) r—+0 fa' (z,r)fl(A, —Ai)-n I u(x) - Au (z)I° dx < lim sup r -n e

r—+0 8- (z,r) r—+0 (z,r)fl(A, —Ai)

+ sup lu(x) -„(z)]° 'limsupIB-(z,r) n

n( AtU)4,)J .zE R^ 1--00 T

The last term is zero since u is bounded and therefore the conclusion followssince e is arbitrary.

Our task now is to prove (5.14.1), (5.14.2), and (5.14.3) without theassumption that u is bounded. For this we need the following lemma.

5.14.2. Lemma. If u E BV(Rn) and A (x o ) = µ,,(x 0 ), then there is aconstant C = C(n) such that

i/Qlimsup Iu(x) - u(xo)I°dx < Climsupr l- °MujI[B(xo,r)].

•—•0 iB(=o,r) T • 0

Proof. For each r > 0, consider the median of u in B(x o , r),

tr = inf f t : IB(x o , r) n {x : u(x) > t}I < IB(x o , r)1},

and apply Theorem 5.12.10 and Minkowski's inequality to conclude

limsup Iu(x) - u(xo )I°dxr•—■0 8(zo,r)

< C lim sup r i-n IIbuII[B(xo , r)1 + C'I tr - u(so)i.r-00

Moreover, t r -. u(x 0 ) since au (xo ) = µ4 (x0 ).

5.14.3. Theorem. If u E BV(R"), then (5.14.1) holds for tfn - '-almostall xo E R" - E, whereas (5.14.2) and (5.14.3) hold for H' -almost allxa E E.

Proof. For each positive integer i, let

i if u(x) > iui (x) = u(z) if fu(x)1 0, let

IID(u - u;)Ii[B(xo , r )]Z; = { xo : lim sup <_ Fr-*Q c((n - 1)rn-1

and refer to Lemma 3.2.1 to obtain

EHn-1(U - Z1) s IID(u - us)11(U)

whenever U C R't is open. By Theorem 5.4.4,00

I I Du(ü - % )II(U) <_ P[{x : (u - us )(x) > s }, Ulds0

+ P[{x : (u - u;) (x) < s }, U]da00

00

= P[{x : u(x) > i + 8}, U]d9Q —

o+

0

P[{x : u(x) < -i + s }, U]ds0

(5.14.5)

(5.14.6)

If U is bounded, then

- P[{x : u(x) > s }, U)ds.1.1>i

(5.14.7)

P[{x : u(x) > a }, U]ds = IIDuII(U) < 00,

and therefore the last integral in (5.14.7) tends to zero as i --0 oo. Hence,we obtain from (5.14.6) that Hn - 1 (U - Z; ) -. 0 as i -. oc. Then,

00

H"-1[00

n (Rn -Z; ) = 0,

)=1i-j

and

1=1

j=1i=j

Hn— r (5.14.8)


To prove (5.14.1), it suffices by (5.14.4) to consider xo E (U°__ 1 W;) — E.Because u — u ; has 0 as an approximate limit at each such point 10 of W;,it follows from Lemma 5.14.2 that

1/s

lirn ju(x) — u, (s) ^ ° dzr^0 B(so,r)

< C lim supr l-nIlD(u — ui)111B(xo, r)1. (5.14.9)r-60

From (5.14.8) we may as well assume that zo E n71 1 U i Z.. For i suffi-ciently large, reference to Remark 5.14.1 yields

tro

urn 114(X) — u(x)l ° dxr

-^ o (4(20,0= 0.

From (5.14.9), there exists i sufficiently large such thatva

lien iu(x) — ui (x)rdx < Ce.r—.0 IB(zo,r)

Consequently, (5.14.1) follows for unbounded u E BV (Rn) by Minkowski'sinequality and the fact that e is arbitrary.

Essentially the same argument establishes (5.14.2) and (5.14.3). D

As an immediate consequence of the above result, we obtain the follow-ing.

5.14.4. Theorem. If iZ C Rn is open and u E BV (0), then

Hm l u(x) — u(xo)I nl(n-1)dx = 0r•p B(xo,r)

for H" -1 -almost all xo E SZ — E, and

lirn u(x)dx = u(x o )r—.0 B(xo.r)

for Hn -1 -almost all xo E fi. If SI is admissible, then the trace te satisfies

11în lu(x) — u'(xo )Inl (n --1) dx = 0r—.00 B(so ,r)Ilrt

for 11' 1 -almost every xo E OR.

Proof. The statements concerning the integral averages of u follow imme-diately from (5.14.2) and (5.14.3). Also, referring to Remark 5.10.6 leadsto the last part of the theorem. ❑

Exercises 277

Exercises

5.1. Suppose u is a function of a single variable and that u E BV(a, b),a < b. Prove that IIDuiR(a, b) = ess V'(u). Hint: Use regularizers of u.

5.2. Suppose Sl C Rn is an open set and u E BV (f1). Prove that thereexists a sequence of polyhedral regions {Pk} invading Sl and piecewiselinear maps Lk : Pk —+ R l such that

li M ILk —ul=o,

klit^ IDLk I = IIûÎ(^)•Pe

Hint: By Theorem 5.3.3, it suffices to consider the case when u ECO°42) n BV(fl). Let P1 C P2 C ... C f2 be polyhedral regions suchthat IS1 — Pk1 - O. Choose each Pk as a simplicial complex, so thatit is composed of n-dimensional simplices. Since IjDuII i ;o < oo, wemay choose k so as to make fil_p IDul arbitrarily small. Moreover,we may assume without loss of generality that each simplex, a, inPk has its diameter small enough to ensure that the oscillation of luiand 'Dui over o is small, uniformly with respect to all a. Suppose ais spanned by the unit vectors v l , 1)2 , ... , vn . Define the linear mapLk on a so that it agrees with u at the n + 1 vertices of a. Clearly,IILkJI^,Pi IIuIIi,n. To see that the L 4 -norm of the gradients alsoconverge, note that

'Dui = IDu(p)I Iois

for some p E a. On each of the edges of a determined by the vectors v^,i = 1, 2, ... , n, there is a point p, such that [Du(p i ) — DLk (p.)] • v, = 0(by the Mean Value theorem). But DLk(p,) = DLk(p) since Lk islinear and IDu(p i )I is close to IDu(p)I because of the small oscillationof IDuI over a. Therefore, I Du(p)—DLk(p)I is small and consequently,

IDLk I is close to fa 'Dui, uniformly wiLh respect to all a.

5.3. if E C R" is a measurable set, let us say that E is open in the densitytopology if D(E, x) = 1 for each s E E. Prove that the sets open inthis sense actually produce a topology. In order to show this, it mustbe established that an uncountable union of density open sets is open;in particular, it must be shown that it is measurable. Hint: Use theVitali covering theorem. If we agree to call the exterior of E all pointsx such that D(E, x) = 0, we see that am E is the boundary of E inthe density topology.

Ja


5.4. In the setting of metric spaces, there is an inequality, called the Eden-berg Inequality, that vaguely has the form of the co-area formula. Itstates that if X is a separable metric space and u : X -' R" is aLipschitz map, then for any E C X and all integers 0 < k < n

f• n k 1 k na(n — k )a(n) k n" H - (E n u- (y))dH (y) < 2 a(n) H (E).

Here, L is the Lipschitz constant of u and f • denotes the upperLebesgue integral. Also, Hin denotes m-dimensional Hausdorff mea-sure which has a meaningful definition in a metric space.STEP 1. By the definition of H", for every integer s > 0 there existsa countable covering of E in X by sets E; . , i = 1, 2, ..., such thatdiamE;. <11s and

00Hn ( E) > ^n ) ^ (d iam E;,. )" — 8 .

i-1

Hence,

a tà o0H" -k (E n u- '(y)} <( "_k

—k ) li rn inf [diam (E;,„ n u-1( y )jn--k

2 . ôo i=1

STEP 2. Consider the characteristic function of the set u(A), X(u(A)),where u(A) denotes the closure of u(A). Then

Idiam (A n u- 1(y ))]" - k < (diam A)"—kX(u(A), y).

Hence, from Step 1,

Kr" (E {l u -1 (y)) < a( 2 " kk) liminf E (diarn E;, a ) n—k X(u(E), y).• 00

i=i

STEP 3. Apply Fatou's lemma (which is valid with the upper integral)to obtain

a(n — k) 00H"-k(E n u -1 (11))dH"(y) ^ 2"_k firn ^f E (diarn E;,,) "-k

R" i=1

•

R X(u(Ei,.),EI)dH"(y)•

R.However,

X(u(Ei,a), y)dH" (tJ) = Hn (u(Ei,a)) < a(n)[diarn u(E,,.)]".

00

Now use Step 1 to reach the desired conclusion.

Exercises 279

5.5. Let u E BV(Il) where Si C R" is an open set. For each real numbert, let A t = {x : u(x) > t}. As usual, let XA denote the characteristicfunction of the set A. For any Borel set E, prove that

+00Du(E) =f DXA, (E)dt.

^

5.6. Prove the following version of the Gauss-Green theorem for BV vectorfields. Let II be a bounded Lipschitz domain and suppose u : ft - R"is a vector field such that each of its components is an element ofBV (0). Then the trace, u', of u on 810 is defined and

div u(S2) =u* (x)v(x, SZ)dHn-1 (X).f

For this it is sufficient to prove that

Diu(1l) = u'(x)vi (x, Çl)dH" -1 (Z)fn

where u E BV(1l), 1 < i < n, and D i = ô/axi . With the notation ofthe previous exercise, observe that

DiXA t (n) = - vi(x, Af )dHn-1 (x)/Ana- A i

= - vi(x, At)dH" -1 (x)La. A,(since Dix Ar

(Rn ) = O)_ - vi (z, SZ)dHn -1(x)fannO•A e

= - yi(z, 11)dHT1 `t (z )A •

= D;Xn(ô' At ).

With the help of the previous exercise, conclude that+oo

D i u(it) = DiXA, (ft)dt-^

= D; xn (a'At )dt

o + ^= Dtxn ( ôÀ# )dt +DiXn(^7À t )dt

-^ o ro

= D;Xn(â'At)dt - J DXn(ô'S2 - ô'A l )dtf0 +00

co

280 5. Functions of Bonded Variation

co 0=D;Xn({x . u' > t })dt — D,Xc^({x : u' < t })dt

o - co

nf u'dDixS2

=u'(x)v,(s,Si)dHn -1 (x).8A

5.7. Give a description of the result in the previous exercise on the line,i.e., when n = 1.

5.8. Under the conditions of Corollary 5.2.4, prove that {Du i } -^ Duweakly as measures as i --, oc.

5.9. Suppose ) is an open subset of R" and that u: SZ --0 R 1 is differ-entiable at xo E 1 in the sense defined by (2.2.2). Also, let M ={(x, u(x)) : x E SZ}. Show that there exists a vector, u, that satisfies(5.6.10) at (xo , u(xa)).

5.10. Let SZ C R/z be an open set with the property that 11I has a tangentplane at xo E asp. That is, assume for each e > 0, that

C(c)nO1 n B(xo,T ) = 0

for all small r > 0, where C(c) is introduced in Definition 5.6.3.Assume also that

lim sup in n B(xo, r) I > 0r—.0 I $(x0 , r) I

andUm sup I(R" — SZ) (1 B(xa, r)I > 0.

r—U IB(Xo, r)IProve that xo E 0'0.

Historical Notes5.1. BV functions were employed in several areas such as area theory andthe calculus of variations before the formal introduction of distributions,cf., [CE], [TO]. However, the definition employed at that time was in thespirit of Theorem 5.3.5.

5.3. Theorem 5.3.3 is a result adopted from Krickerberg [KKj. This resultis analogous to the one obtained by Meyers and Serrin [MSE] for Soholevfunctions, Theorem 2.3.2. Serrin [SE] and Hughs [HS] independently dis-covered Theorem 5.3.5.


5.4. The theory of sets of finite perimeter was initiated by Caccioppoli [C]and DeGiorgi [DG1], [GD2) and subsequently developed by many contrib-utors including [1{K), (FL), [F1), and [F2). Sets of finite perimeter can beregarded as n-dimensional integral currents in R" and therefore they canbe developed within the context of geometric measure theory. The isoperi-metric inequality with sets of finite perimeter is due to DeGiorgi [Dl], [D2)and the co-area formula for BV functions, Theorem 5.4.4 was proved byFleming and Rishel [FRI.

5.5. The notion of generalized exterior normal is due to DeGiorgi 11)2] andis basic to the development of sets of finite perimeter. Essentially all of theresults in this section are adapted from DeGiorgi's theory.

5.6. The concept of the measure-theoretic normal was introduced by Fed-erer [F1) who proved [F2) that it was essentially the same as the generalizedexterior normal of DeGiorgi.

Definition 5.6.3 implicitly invokes the notion of an approximate tangentplane to an arbitrary set which is of fundamental importance in geometricmeasure theory, cf. [F4]. Theorem 5.6.5 states that the plane orthogonal tothe generalized exterior normal at a point xo is the approximate tangentplane to the reduced boundary at x 0 .

5.7. Countably k-rectifiable sets and approximate tangent planes are closelyrelated concepts. Indeed, from the definition and Rademacher's theorem,it follows that a countably k-rectificable set has an approximate tangentk-plane at He almost all of its points. Lemma 5.7.2 is one of the importantresults of the theory developed in [F4). Theorem 5.7.3 is due to DeGiorgi[D1], [D2], although his formulation and proof are not the same.

5.8. In his earlier work Federer, [F1), was able to establish a version of theGauss-Green theorem which employs the measure-theoretic normal for allevery open subset of Rn whose boundary has finite H" -1 -measure. AfterDeGiorgi had established the regularity of the reduced boundary (Theorems5.7.3 and 5.8.1), Federer proved Theorem 5.8.7 [F2].

5.9. A different version of Lemma 5.9.3 was first proved by William Gustin[GU]. This version is due to Federer and appears in [F4, Section 4.5.41.Theorem 5.9.6 is only a part of the development of BV functions thatappears in [F4, Section 4.5.9]. Other contributors to the pointwise behaviorof BV functions include Goffman [GO] and Vol'pert [VO]. In particular,Vol'pert proved that the measure-theoretic boundary r3M E is equivalent tothe reduced boundary.

5.10-5.11. The trace of a BV function on the boundary of a regular domainas developed in this section is taken from [MZ]. Alternate developments canbe found in [CI1] and [MA3]. This treatment of Poincaré- type inequalitieswas developed in [MZ].


5.12. Lemma 5.12.3 was first proved by Fleming !FL]. The proof of thelemma depends critically on Lemma 5.9.3. Fleming publicly conjecturedthat the claim of the lemma (his statement had a slightly different form)was true and Gustin proved it in [GU]. Theorems 5.12.2, 5.12.8, and thematerial in Section 5.13 were proved in NZ].

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List of Symbols

1.1

0 empty set 1X E characteristic function of E 2Rn Euclidean n-space 1x z = (xi, ... , xn ), point in Rn 1x • y inner product 1Ix' norm of r 1S closure of S 1aS boundary of S 1spt u support of u 1d(x, E) distance from r to E 1diam(E) diameter of E 1B(x, r) open ball center x, radius r 2B(x, r) closed ball center x, radius r 2a(n) volume of unit ball 2a a = (nl, • - - , an ), multi-index 2lal length of multi-index 2xQ s' = xr • xr • .. Zen 2Z na! a! = a i !a2' • .. an ! 2D; = a/ax ; ith partial derivative 2D° b°=Dj"...Dnn_

a=ê^â=Rn 2

Du gradient of u 2Dk u Dk41 = {Dau}1a1=k 2C ° (S2) space of continuous functions on Sl 2Ck(SZ) Ck(1) _ {u : Dau E C° (11)} 2Co (0) C1(0) = Ck (0) n {u : spt u compact, spt u C SZ) 2Ck (SZ) {u : Daft uniformly continuous on 11, lal < lc} 2Ck (S2; Re`) Rte`-valued functions 2C°.0 (fi) Holder continuous functions on 12 3Ck • a (SZ) Ck'°(S2) {u : Dpu E C° 'a (n), 0 < IQI < k} 3

1.2

JEl Lebesgue measure of E 3

distance between sets 4d(A, B)

1.3

ball concentric about B with 5 times its radius 7

h (x,r) f dµ integral average 14

1.4

7-dimensional Hausdorff measure 15

1.5

V3 (0) functions p th -power integrable on S2 18L(Q) u E LP(K) for each compact K C S2 18IIUII p;n LP-norm on S2 18hullo, fl sup norm on 52 18IP (S2;µ) underlying measure is p 18L ôc (S2;µ) underlying measure is p 18Iluli p;o 3h underlying measure is p 19Builoo;n;ls underlying measure is µ 19p' p'= -1 ifp>1,p'=oo,if p= 1 20

1.6

(PE regularizer (or mollifier) 21u, uE = cp, * u, convolution of u with (pc 21

1.7

T Schwartz distribution 24

1.8

E; E; ={x : If (x)I > a) 26a f(a) off(s) =1E; 1 26f '

non-increasing rearrangement of f 26f.. f"(r)

= t fo f '(r)dr 28

L(p, q) Lorentz spaces 281 f IIp,q Lorentz norm 28L(p, cc) weak IY 28

2.1

Wk,p(S1) Sobolev space on i2 43II! IIk,p;l Sobolev norm on i2 43Wô'p (12) norm closure of C°(l) 43BV (S2) BV functions on 0 C R" 43u+ u+ = max { u, 0} 46u - uJ = min{u,0} 46

2.2

S1' C C f2 f2 C S2, SI--1

compact 52

2.4

w(n — 1) area of unit sphere in Rn 58p' Sobolev exponent, p' = np/(n — kp) 58

2.6

Jr. Riesz kernel of order o 64w hf 2 2Û ri z )

7a Riesz constant rya = e( n _ Q) 64

go Bessel kernel of order a 65a(s) ti b(x) a(x)/b(x) bounded away from 0 and ago 65L„ iD space of Bessel potentials 66Bo, ,p Bessel capacity 66Ra , p Riesz capacity 667« , p variational capacity, Exer. 2.8 104b bn B„ ,v 71u,r, —d,p

2.7

div V divergence of the vector field V 77

uuu

2.8

Mtn(x) maximal function of f at x 84

3.4

p(m) Taylor polynomial of degree m at x 126

3.5Tk(E) bounded kth-order difference quotients on E 130tk (E) formal Taylor expansions of degree k on E 131Tk,p(x) bounded kth-order difference quotients in V' 132tkiP(x) kth-order differentiability in L" 132

3.10

Mp ,Ru(x) L" maximal function 154

4.1

(Wm'p(1))' dual of Wm"p(Il) 179

4.7

M k µ(x) fractional maximal operator 204ek,p (µ) (k, p) -energy of to 204

5.1

jjuilBv norm of 13V function 221IlDull total variation measure of BV function u 221

5.3

ess V6 (u) essential variation of u on [a, II] 227

a+ %04 V 1 1Jy I I I 1,04L.1n

5.4

JU1

P ( E, S2 ) perimeter of E in SZ 229

5.5

ô- Ev(x, E)

reduced boundary of Egeneralized exterior normal of E at x

233233

5.6

C(x,e,v) = Rn f1 {y: 1(y— x).vi > Ely --xI}measure-theoretic normal to E at x8 E = {x : n(x, E) exists}

240240240

5.8

aM E measure-theoretic boundary of E 249

5.9

D(E,x)D(E,x)b(E,x)ap !im sup u(y)

y -* x

ap lim sup u(y)g-* x

upper density of E at xlower density of E at xdensity of E at xupper approximate limit of u at x

lower approximate limit of u at x

250250250250

250

5.12

7(E)

13V-capacity of E

262

Index

Page numbers are enclosed in parentheses; section numbers precede pagenumbers.

AAbsolute continuity on lines, 2.1

(44)Absolutely continuous measures,

1.3(15)Adams, David, 2.9(92), Historical

Notes 4(218)Admissible domain, 5.10(256)Approximate limit

lower, 5.9(250)upper, 5.9(250)

Approximate tangent plane, His-torical Notes 5(218)

Approximately continuous, 5.9(250)Aronszajn--Smith, Historical Notes

2(108)

BBV function

extension of, 5.10(257)trace of, 5.10(258)

BV functions, 2.1(43), 5.1(220)approximation by smooth func-

tions, 5.3(225)and capacity, 5.12(262)and co-area formula, 5.4(231)compactness in L' of unit ball,

5.3(227)and Poincaré inequality, 5.11

(261)BV norm, 5.1(221)BV', and measures, 5.12(266)Banach Indicatrix formula, 2.7(76)

Banach space, 1.5(21)Beppo Levi, Historical Notes 2(108)Besicovitch, A.S., Historical Notes

1(40)Besicovitch covering theorem, 1.3.6

(12)Besicovitch differentiation theorem,

5.8(248)Bessel capacitability, Suslin sets,

2.6(71)Bessel capacity, 2.6(66), 4.5(194)

and Hausdorff measure, 2.6(75)inner regularity, 2.6(72)and maximal functions, 3.10

(154)metric properties, 2.6(73)as outer measure, 2.6(67)

Bessel kernel, 2.6(65)Bessel potentials, and Sobolev func-

tions, 2.6(66)Blow-up technique, 5.6(237)Borel measure, 1.2(6)Boundary, of a set, 1.1(1)

measure-theoretic, 5.8(249)reduced, 5.5(233)

Bounded variation, on lines, 5.3(227)

Boxing inequality, 4.9(218)Brelot, 2.6(68)Brezis-Lieb, Historical Notes 2(110)Brezis-Wainger, 2.9(91), Histori-

cal Notes 2(111)

JV *

Burago-Zalgaller, Historical Notes1(41)

CCaccioppoli, R., Historical Notes

5(281)Calderon, A.P., Historical Notes

3(175)lemma, 1.8(36)

Calderon-Stein, 2.5(64)Calderon Zygmund, Historical Notes

3(175)Catkin, H., Historical Notes 2(108)Capacity

and BV functions, 5.12(262)and Hausdorff measure, 5.12

(265)inner regularity, 5.12(263)regularity of, 5.12 (267)

Carathéodory, C., Historical Notes1(41)

Cartan, H., Historical Notes 2(109)Cauchy's inequality, 1.5(19)Cesari, L., Historical Notes 5(280)Chain rule, Sobolev functions, 2.1

(48)Change of variable, Sobolev func-

tions, 2.1(52)Characteristic function of a set,

1.1(2)Choquet, G., 2.6(70)Closed hall, 1.1(2)Co-area formula, 2.7(76), 4.4(192),

4.9(210)for BV functions, 5.4(231)general formulation, 2.7(81)

Continuity of Bessel capacity, 2.6(69)

Continuous functions, space of, 1.1(2)Convolution

of distributions. 1.7(25)of functions, 1.6(21)

Countably rectifiable sets, 5.7(243)and Cl sub-manifolds, 5.7(243)and Lipschitz maps, 5.7(243)

DDeGiorgi, E., Historical Notes 5

(281)DeGuzman, M., Historical Notes

1(41)Deny, J., Historical Notes 2(109)Difference quotients, hounded in

I7, 3.6(137)Distance function, 1.1(1)

smooth approximant, 3.6(136)Distance to a set, 1.1(1)Distribution function, 1.8(26)Distribution

derivative of, 1.7(25)integrable function, 1.7(24)multiplied by a function, 1.7

(25)Radon measure, 1.7(24)Schwartz distribution, 1.7(24)

Dual of Sobolev spaces, 4.3(185)Dual of Wm.P, representation, 4.3

(185)Dual of Ka , repri sentation, 4.3

(187)

Ee - 6 domains, 2.5(64)Essential variation, 5.3(227)Exterior normal, 5.5(233)

generalized, 5.5(233)measure-theoretic, 5.6(240)

FFederer, H., Historical Notes 5(281)Finely continuous function, 3.3 (123)Finely covered set, 1.3(8)Finite perimeter

containing smoothly !imitatedsets, 5.4(229)

atd countably rectifiable sets,5.7(243)

and relative i ioperinwtric: in-equality, 5.4 (230)

sets of, 5.4 (229)

I I IVGA

sets of and Gauss-Green the-orem, 5.8(248)

sets of and Lipschitz domains,5.8(248)

Fleming -Federer, Historical Notes2(110)

Fleming Rishel, Historical Notes2(110), Historical Notes5(281)

Frehse, J., Historical Notes 2(109)

GGagliardo, E., Historical Notes 2

(109)Gauss-Green theorem, 5.5(234)

and sets of finite perimeter,5.8(248)

Gelfand, I.M., Historical Notes 1(41)Generalized derivative, 2.1(42)Giusti, E., Historical Notes 5(281)Coffman, C., Historical Notes 5(281)Gradient of a function, 1.1(2)Gradient measure

continuity of, 5.2(223)lower semicontinuity of, 5.2(223)

Gust in, W., Historical Notes 5(281)

HHahn-Banach theorem, 4.3(186)Hardt, Robert, Historical Notes

1(40)Hardy's inequality, 1.8(35)Hardy Litt]ewood-Wiener maxi-

mal theorem, 2.8(84)Hausdorff dimension, 1.4(18), Ex-

ercise 1.2(37)Hausdorff maximal principle, 1.3(7)Hausdorff measure, 1.4(15)Hausdorff measure, as an element

of (Wm'p)•, 4.4(191)Hedberg, Lars, Historical Notes 2

(111)Hedberg Wolff, Historical Notes

4(218)

Holder continuous functions, spaceof, 1.1(3)

Holder's inequality, 1.5(20)Hunt, R., Historical Notes 1(41)

iInner product, 1.1(1)Integral currents, historical Notes

5(281)Isodiametric inequality, 1.4(17)Isoperimetric inequality, 2.7(81)Isoperimetric inequality, relative,

5.4(230)

JJensen's inequality, 1.5(20)Jessen- Marcin.kiewicz Zygrmind,

Historical Notes 1(40)Jones, Peter, 2.5(64)

KK-quasiconformal mapping, His-

torical Notes 2(108)Konclrachov, V., Historical Notes

2(109)(K, p) -extension domain, 2.5(63)Krickerberg, K., Historical Notes

5(280)Kronrod, A.S., Historical Notes 2

(110)

LLebesgue measurable set, 1.2(3)Lebesgue measure, 1.2(3)

as an element of (Wm`.p)', 4.4(188)

Lebesgue pointand approximate continuity,

4.4(190)arbitrary measures, 1.3(14)for higher order Sobolev func-

tions, 3.3(122)ofa Sobolev function, 4.4(190)for Sobolev functions, 3.3(118),

3.10(157)Lebesgue, H., Historical Notes 1(39)

V V V

Leibniz formula, Exercise 1.5, (37)Length of multi-index, 1.1(2)Lipschitz domain, 2.5(64), 4.4(191)Lorentz space, norm, 1.8(28)Lorentz spaces, 1.8(28), 2.10(96)Lorentz, G., Historical Notes 1(41)LP

at all points, 3.9(152)derivative, 3.4 (129)difference quotients, 2.1(46)norm, 1.5(18)related to distributional deriva-

tives, 3.9(147)spaces, 1.5(18)

Lusin's theoremfor Sobolev functions, 3.10(159),

3.11(166)

MMarcinkiewicz interpolation the-

orem, 4.7(199)Marcinkiewicz space, 1.8(28)Maximal function, 2.8(84)

fractional, 4.7(204)Maz'ya, V., Historical Notes 2(109),

Historical Notes 5(281)Measure densities, 3.2(116)Measure, with finite K, p-energy,

4.7(204)Median, of a function, 5.12(269)Meyers, Norman, Historical Notes

4(217)Meyers-Serrin, Historical Notes 2

(109)Meyers-Ziemer, Historical Notes

4(218), Historical Notes5(281)

Michael-Ziemer, Historical Notes3(176)

Minimax theorem, 2,6(72), 5.12(268)Minkowski content, 2.7(83)Minkowski's inequality, 1.5(20)Mollification, 1.6(21)Morrey, C.B., Historical Notes 2

(109)

Morse, A.P., Historical Notes 1(40)Morse-Sand theorem, 2.7(81), 4.4

(192)Moser, J., 2.9(91)p-measurable set, 1.2(6)Multi-index, 1.1(2)

NNirenberg, L., Historical Notes 2

(109)Non-linear potential, 4.7(205)Non-tangential limits, 3.9(147)Norm of vector, 1.1(1)

0O'Neil, R., Historical Notes 1(41)Open ball, 1.1(2)Outer measure, 1.2(6)Outer regularity of Bessel capac-

ity, 2.6(69)

PPartial derivative, as a measure,

5.1(220)Partial derivative operators, 1.1(2)Partition of unity, 2.3(53)Poincaré inequality

abstract version, 4.1(179)and BV functions, 5.11(261)with Bessel capacity, 4.5(195)extended version with Bessel

capacity, 4.6(198)general version for Sobolev

functions, 4.2(183)with Hausdorff measure, 4.4

(191)indirect proof, 4.1(177)with Lebesgue measure, 4.4

(189)and measures in (BV)+, 5.12

(266)and median of functions, 5.13

(269)Polya-Szego, Historical Notes 1(41)Projection mapping, 4.1(178)

iiiaex .wr

RRademacher, H., Historical Notes

2(108)Rademacher's theorem, 2.2(50)

in LP, 3.8(145)Radon, Historical Notes 1(39)Radon measure, 1.2(6)Radon-Nikodym derivative, 1.3(15)

and differentiation of measures,5.5(234)

Rearrangement, non-increasing, 1.8(26)

Reduced boundary, 5.5(233)Regularization, 1.6(21)

of BV functions, 5.3(224)of Sobolev functions, 2.1(43)

Rellich-Kondrachov compactnesstheorem, 2.5(62)

Rellich-Kondrachov imbedding the-orem, 4.2(182)

R.esetn)ak, S., Historical Notes 2(110)Riesz capacity, 2.6(66)Riesz composition formula, 2.8(88)Riesz potential, 2.6(64)Riesz representation theorem, 4.3

(186)

SSchwartz, L., Historical Notes 1(41)Sigma algebra, 1.2(3)Sobolev, Historical Notes 2(109)Sobolev function, 2.1(43)

approximate continuity of, 3.3(122)

fine continuity of, 3.3(124)integral averages of, 3.1(115)LP approximation by Taylor

polynomial, 3.4(127)represented as a Bessel po-

tential, 4.5(194)Sobolev inequality, 2.4(56), 2.7(81)

critical indices, 2.9(89)critical indices in Lorentz spaces,

2.10(98)

involving a general measure,4.7(199)

relative to sub-manifolds, 4.7(203)

for Riesz potentials, 2.8(86)with general measure and p =

1, 4.9(209)Steiner, J., Historical Notes 1(41)Strichartz, R., Historical Notes 2

(111)Strong-type operator, 4.7(199)Suslin set, 1.2(6)

TTalenti, G., 2.7(82)Taylor polynomial, 3.4(126)Thin sets, 3.3(123)Tonelli, L., Historical Notes 5(280),

Historical Notes 2(108)Total differentiability, Lipschitz func-

tions, 2.2(50)Total variation measure, 5.1(221)Total variation, for functions, 2.7(76)Totally bounded set, 2.5(62)Trace of a Sobolev function, 4.4(190)

of 13V function in terms of in-tegral averages, 5.14(276)

Trudinger, N., Historical Notes 2(111)

Truncation of Sobolev functions,2.1(47)

VVitali, G., Historical Notes 1(40)Vol'pert, A., Historical Notes 5

(281)

WWeak derivative, 2.1(42)Weak LP, 1.8(28)Weak-type operator, 4.7(199)Whitney extension theorem, 3.5

(131)LP version, 3.6(141)

Whitney, H., Historical Notes 1(40)

(W k.p)', measures contained with-in, 4.7(205)

YYoung's inequality, 1.5(20)

in Lorentz spaces, 2.10(96)

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