On the differentiability of Lipschitz-Besov functions

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 303, Number 1, September 1987

ON THE DIFFERENTIABILITYOF LIPSCHITZ-BESOV FUNCTIONS

JOSÉ R. DORRONSORO

ABSTRACT. Lr and ordinary differentiability is proved for functions in the

Lipschitz-Besov spaces B%'q, 1 0, using certain

maximal operators measuring smoothness. These techniques allow also the

study of lacunary directional differentiability and of tangential convergence of

Poisson integrals.

1. Introduction. The differentiability properties of functions in the Sobolev

spaces Lpk, 1 n/p and / E Lpk, f has a.e.

an ordinary differential of order k; that is, for a.e. x E Rn f(x + y) — Pkf(y, x) =

o(\y\k), with Pkf the Taylor polynomial of / at x,

Pkf(y,*)=Y,ÍÁX)yJJ\ '\J\<k

where for each n-index J = (ji,... ,jn) E N™, fj denotes the Jth order weak partial

off.If k < n/p only Lp differentiability is possible, p* = np/n — ap; more precisely,

if / E Lpk, f has a fcth order Lp' differential (or a (p*,k) differential) a.e.; that is,for a.e. x

l-f \f(x + y-Pkf(y,x)\p'dy) =o(tk),\J\y\<t J

where fEf denotes the mean fE f dx/\E\. Finally, if k = n/p, p* = oo, and any

/ E L^/p has an (r, k) differential a.e. for all r < oo.

Of course, functions in Lpk have also lower order differentials and for them the

exceptional set becomes smaller. In fact, denoting by Ba>q the Bessel capacity

associated with Lqa (see [My] for its definition), given m E N with 0 < m < k,

any f E L\ has Bk-m^v a.e. a (p*,m) differential if k < n/p, an (r,m) differential

for any r < oo if k = n/p, and an (oo,m) differential (i.e., an mth order ordinary

differential) if k > n/p (if k — m > n/p by Bk-m,p a.e. we mean everywhere).

Starting with the work of Calderón and Zygmund [CZ], these results and the

corresponding ones for the Bessel potential spaces Lp = {/ = Jag: g E Lp},

Ja being the Bessel potential operator, a > 0, 1 < p < oo, have been studied

by several authors, among them C. P. Calderón, Fabes and Riviere [CFR] and

Received by the editors September 10, 1986.

1980 Mathematics Subject Classification (1985 Revision). Primary 46E35, 26A16; Secondary

42B25.

Supported by C.A.I.C.Y.T. 2805-83.

©1987 American Mathematical Society0002-9947/87 $1.00 + $.25 per page

229License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

230 J. R. DORRONSORO

T. Bagby, Deignan, Federer and Ziemer [BZ, DZ, FZ]. The purpose of this paper

is to deal with these questions in the context of the Lipschitz-Besov spaces Bpq,

a > 0, 1 < p < oo, 1 < q < oo, of those Lp functions / such that

/ \v\-n-aq\\Akyf(-)\\ldv = \f\a,P,q,

where Ayf(x) = f(x + y) - f(x) and k — [a] + 1; with the norm ||/||a,p,9 =

||/||p + \f\a,p,q, Bpq becomes a Banach space (see [St] or [T] for more properties

of Bpq). For an integer A;, we have the embeddings Bkq C Lk, with 1 < q 2 [St, Chapter V] which inmediately give fcth orderdifferentiability results for these ranges of q. In general, these results cannot be

improved: for instance, the one variable function

f(x) = g(x) I ¿2-"n-1/2 cos 2nx ]

with g E Cq™, g = 1 if |x| < 1, g = 0 if |x| > 2, belongs to Bpq, p > 2, q > 2 but is

differentiable only in a zero measure set [Z, pp. 47 and 206].

Therefore, we will concentrate in the study of lower order differentials, for which

the full range 1 < q < oo can be considered. In this context partial results have

been given by D. Adams [Adl], Neugebauer [NI, N2], and Stocke [Sto], and all

follow the pattern set by the Lp cases. In fact, if a > n/p one should expect ordinary

differentiability to hold, whereas if a < n/p, because of Herz's imbedding theorem,

only Lr differentiability, r < p*, is to be expected in the full range 1 < q < oo,

while Lp differentiability should hold for restricted values of q. Also, we can

consider a capacity type set function Aa¡p¡q associated to Bpq (more precisely, a

family A* p q, 0 < s < a, of set functions, all having the same zero sets; see §2),

and the exceptional sets should be measured in terms of the corresponding Besov

capacities. We postpone their definition until §2, and state now our main results.

THEOREM 1. Given f E Bpq, a > 0, I g a.e.;

(ii) if a > n/p, f has an (oo,b) differential Aa_6iP,Q o.e. (ifa — b > n/p,Aa-btP,qa.e. is to be interpreted as e.e.)

(For nonintegral 6, we mean by (r, b) differentiability that

(/ \f(x + y)-Pbf(y,x)\rdy) =o(t»),\J\y\<t J

where Pbf denotes the [6]th order Taylor polynomial of / at x; for (oo, b) differentials

just change the Lr norm to an L°° norm).

As mentioned above, although this result is quite satisfactory if a > n/p, one

should have p* differentiability for certain g's when a < n/p. This is indeed the

case if 1 < q < p, but the measure of the exceptional set is then less precise, because

of the following fact: if q > p any set of zero Hn~ap Hausdorff measure has also

zero AaiPi9 capacity [Adl], but this is not presently known to hold of 1 < q < p

(see however [Ad2]); we are thus forced to introduce another set function, theLicense or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

LIPSCHITZ-BESOV FUNCTIONS 231

bracket \Hn~ap, Aa,p,q] (see [Adl, Nl] and §2); as we shall see, if Hn~ap(Ei) =

Aa,p,q(E2) = 0, then [Hn-ap,Aa,p,q](Ei U E2) = 0. We now have the following

theorem.

THEOREM 2. If f E Bpq, 1 < p < oo, a < n/p, l<q<p and b with 0 < b < a

is given, f has a (p*,b) differential [//"«-(a-(')P) Aa-b,p,q] a.e.

Note that any / E Bpp has then a (p*,b) differential Aa_f,iPiP a.e. (and hence

Ba-b,p a.e.; see [Ad 2]).

The proofs of these theorems rely on estimates for certain maximal operators first

introduced by A. P. Calderón and Scott [CS] and extensively studied by DeVore and

Sharpley [DVS]. They can also be used to study lacunar directional differentiability

of Lipschitz functions. More precisely, given a point u E Sn-i, the unit sphere in

Rra, we will say that / has lacunar differential of order b in the direction u at a

point x E Rn if

lim 2kb\f(x + 2~ku) - Pbf(2~ku, x)\=0.k—*oo

If / E Lp, C. Calderón [CC2] has recently proved the existence Ba-6,p a.e. of

a lacunary differential of order b in the direction of each u E Sn-i (note that such

an / can be essentially unbounded in any arbitrarily small ball). In the Lipschitz

case we have

THEOREM 3. If f E Bpq, a < n/p, I < p < oo,0 <b < a, and u E Sn-i, fhas Aa-b,p,q a.e. a lacunar differential of order b in the direction of u.

Another application of these techniques is to study tangential convergence of

Poisson integrals of functions in Bpq. For potential spaces these type of results are

due to Nagel, Rudin, Shapiro and Stein [NRS, NS], and Mizuta [Mz] has studied

the case of Bpp. In [Do] it is shown how tangential convergence can be related to

the regularity of the functions involved, and here we have

THEOREM 4. If x E Rn, c > 0 and 1 < r < oo, consider the set Dc,r(x) =

{(x, y) : z E Rn, y > 0, \x - z\ < y1-"/"-}. Then:

(i) If I < p < oo, 1 < q < oo, a < n/p and b is such that 0 < b < a, the Poisson

integral u(z,y) of any f E Bpq converges to f(x) inside Db,s(x) for Aa_¡,,Pi9 a.a.

x E Rn and any s < p.

(ii) // moreover, 1 < q < p, u(z,y) converges to f(x) inside Db,p(x) for

[Hn-(«-»)p,A„_6,p,,] a.a. 1ER".

(Observe that if a > n/p, any / E Bpq is continuous; also, if a = n/p, 1 < q < p,

convergence inside regions with exponential contact holds for functions in Bpq; see

[Mz, Do].)

This paper is organized as follows. §2 contains the definition and some properties

of Besov capacities, and the Calderón-Scott-DeVore-Sharpley maximal operators

are discussed in §3. Theorems 1 and 2 are proved in §4 and Theorems 3 and 4 in

§§5 and 6 respectively.

2. Besov capacities. It is well known (see [St, p. 153]) that the Bessel poten-

tial operator J(, is an isomorphism between Bpq and 0^+6' 1 < P, 9 < oo, 0 < a,6.

Following [Nl, Sto], this isomorphism allows us to define capacity type set func-

tions: given a > 0, 1 < p, q < oo, for any s with 0 < s < a and E C R™,License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use


we set

AaaiPig(E) = inf {»SUS,,,,,: g > 0, Ja-Sg > lE} ,

where Ie stands for the characteristic function of E. We first show that the o,p, q-

Besov capacity of a set is essentially independent of s.

LEMMA. If 0 < s < s' < a, there is a constant C = Ca>3i, such that for anyEcRn

K,p,q(E)/C < <p,,(£) < CAsa¡pjE).

PROOF. If gE B°<q,g > Oand Ja-Sg >lE,h = J3*-Sg is also > 0, Ja-S>h > lE

and ||ä||s.,Pi, < C||o||s,p,g; thus Asapa(E) < Aaa^q(E), Conversely, if 0 < e <

min(l,s) and h E Bpq, h > 0, Ja-S'h > Ie, consider g = Ja_e(|v|), where v is

such that JS'-ev = h\ then g > 0, Ja-s9 = Ja-s'(Js'-e(\v\)) > Ja-s'h > lE and

\\g\\s,p,q < C\\h\\a,p,q. Therefore, <„,,(£) < CAsa[p,q(E).

As a consequence, all the Asa capacities have the same zero sets; in this sit-

uation we will just write AatP^q(E) = 0 or Aa,p,g a.e. without specifying any s

index.

For a given E C Rn we define

[Hn~ap, AlpJ (E) = inf{Hn-ap(Ei) + A^JE*) :E = E1UE2};

as a consequence of the lemma all the [Hn~ap, Asa ] are equivalent set functions

and in particular all have the same zero sets. In this case we will also write

[Hn~ap, AaiPtq](E) = 0, without specifying the s parameter.

If 1 0, imply that for some

C,

Ba.e,p(E)/C < AlpJE) < CBa+e,p(E);

in particular, if Aa,p^q(E) — 0, E has n — ap Hausdorff dimension. When p = 1,

the embeddings [DVS] B^q C B\l_e C Lra_n/r,_2e, with r < n/n-a, imply [Adl]

that H^-a+e(E) < CA3a¡hq(E) for any e > 0, where if 0 < b < n, 0 < d < oo,

H%-b(E) =infl^2\Ql\1-b/n: E c[JQi,Qi cubes, side Q% <d

moreover, if a < n, an easy homogeneity argument [Sto] gives AsaXq(Q(x,r)) <

Crn-aAsaXq(Q(l,0)) (Q(x,r) = cube of center x and side 2r), if 0 < r < 1; this

and the countable subadditivity of Asa x q imply that AsaXq(E) < CHx~a(E).

Another way of defining Besov capacities is by setting for a compact K

A'a^q(K) = inf{\\f\\pa^q:fECox',f>lK}

and extending it as an outer capacity to general sets in Rn (see [Adl, Sto]; re-

placing the Bpq norm with the Lp norm we obtain the usual Bessel capacity). It

can then be proved [Sto, Lemma 1] that Asapq(E) = 0 implies A'apq(E) = 0,

1 < p, q < oo.

We finally note that functions in Bpq are well defined AaiPi, a.e.; in fact, if

0 < s < min(l,a) and g E Bpq, the integral f Ja-S(x — y)g(y)dy is finite for

Aa,p,q a.a. x E R" [Nl, Sto], and as a consequence of the dominated convergence

theorem, f. ,<t Ja-S(y)g(x + y) dy tends to 0 with t for Aa>Piq a.a. x.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use


3. Maximal operators. If a > 0, m = [a], / E Lp, 1 < p < oo, and Q is a cube

in R", we denote by Pqf the unique polynomial in Pm = {polynomials of degree

< m) such that for any multi-index J = (ji,... ,jn) E Nra with \J\ = ji-\-\-jn <

m,fQ(f(y)-PQf(y))yJdy = 0.Pqf is a best approximation polynomial in the sense that [DVS, p. 17]

inf j^ \f-P\dy:PEPm}~ j \f-PQf\dy

(A ~ B means A/C 0

Ef(x, t) = sup j / |/ - PQf\ :xEQ, side Q = t\,

Cp is then defined [DVS] as the space of those f E Lp such that Gaf(x) =

sup{raEf(x,t): t > 0} is also in Lp; with the norm ||/||a,p = ||/||p + ||Ga/||p,

Cp becomes a Banach space.

Now, if x E Q' C Q, with Q, Q' having side lengths t, t', and we write

PQf(v)= E Cj(Q)(y-x)J/J\\J\<m

and Pç'f(y) in a similar manner, we have for any \J\ < m [Do]

ft fjo

(2) \Cj(Q)-Cj(Q')\<C Ef(x,s)8-W-;Jv s

there estimates can be further refined.

PROPOSITION 1. With x, Q, Q', t, t' as before, if Gaf E Lp and a - \J\ < n,

(3) \Cj(Q)-Cj(Q')\<G(tr-lJl-n f Gaf(x + z,t)dzJ\z\<t'

■+C f \z\a-^-nGaf(x + z,t)dz,Jt'<\z\<t

where Gaf(x, t) = sup{s~aEf(x, s) : 0 < s < t}; also, if a — \J\ = n,

(4) \Cj(Q)-Cj(Q')\<Clogt- f Gaf(x + z,t)dz1 J\z\<f

+ C f Gaf(x + z,t)log-{-dz.Jt'<\z\<t \z\

Finally, if a — \J\ > n/p, and s is such that a — \J\ > n/s > n/p,

(5) \Cj(Q')-Cj(Q)\<Cta-\J\-n's[ f Gaf(x-+z,t)sdz)\J\z\<t )

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PROOF. Denoting by B(x, s) the ball centered at x with radius, (2) implies

\Cj(Q') - Cj(Q)\ <C f I -f Ef(z, s) dz) S-IJI -Jt1 \Jb{x,s) J s

< C f sa-lJl-" f Gaf(z,t)dzdsJt1 Jb{x,s)

<C f sa-^-n I fSr71'1 f Gaf(x + ru,t)dudr\ -

and (3) and (4) now follow by Fubini's theorem. To prove (5), just note that (2)

implies

1/Sdr

r

i/s

\Cj(Q) - Cj(Q')\ <C f ra~\J\ ( / Gaf(z,r)°)Jo \JB{x,r) J

<Cta-\J$[ Gafrz,tydz

\JB(x,t)

We observe next that Ga can be seen as a derivative.

PROPOSITION 2. If f E Bpq, a > 0, I < p,q < oo and 0 < e < min(l, a), there

is anFE BPq with \\F\\e<p,q < \\f\\a,p,q such that Ga - ef(x, 1) < CF(x).

PROOF. Writing a' = a — e and / = Ja'9, g E BPq, and k = [a1], we then have

[Do, Theorem 4]

dsx,s) —

s

/oo sa'-k-1Eg(

I , f°° , T7S< Cta mg(x) + Cta / sa-k-1l3*g(x)—,

Ji s

where m denotes the "local" maximal function

mg(x) = sup < -f \g\dz: x EQ, \Q\ < 1 >

and ls(z) = s-nlQ{0,i)(z/s)-

Writing now g — Je-¿h, h E Bv¿ ', and taking into account that for side Q < 1,

fQJd(x + z)dz < CJd(x) [AS, p. 418], it follows that mg < CJe-d(\h$. Also,

Ay(l8 * <7)(x) = ls* (Ayg)(x), and

a°° , ds\ II r°° ,8a-k-1la*g(-)-jj sy s^-^WUÂygW,

<C\\Ayg\\p;

thus, taking e < 1 , it follows that f1 sa k 1ls * g(x) ds/s is in BPq.

We can derive now the existence of Taylor polynomials of degree [b] of a given

/ E Bpq for any b with 0 f E Lp; thus, if |J| < b and

a — \J\ < n/p, (3) and the local estimates for Bessel potentials imply

(6) \Cj(Q)-Cj(Q')\<C f Ja,_lJ\(z)F(x + z)dz,J\z\<t

with F the BPq function of Proposition 2. Therefore, as t tends to 0, Cj(Q)

converges Aa_|j|iPig a.e. to a finite limit fj(x) satisfying

(7) \fj(x)-Cj(Q)\<C f Ja,_lJl(z)F(x + z)dz.J\z\<t

Moreover, if o — \J\ > n/p and e, s are chosen so that a' — \J\ > n/s > n/p, (5)

implies that as t tends to 0 Cj(Q) converges e.e. to a finite limit fj(x) satisfying

(8) \fj(x) - Cj(Q)\ < Cta'-W ( / F(x + z)s ds\J\z\<t

If \J\ = 0, Co(Q) = Pqf(x) tends to f(x) a.e. [DVS, p. 9], and thus, redefining/ in a measure 0 set, (7) and (8) can be rewritten as the AaiPig a.e. estimates

(9) \f(x)-PQf(x)\<C f Ja,(z)F(x + z)dz,J\z\<t

(10) |/(a:) - Pqf(x)\ < Cta'-n's If F(x + z)s dz\J\z\<t

according to whether a' < n/p or a' > n/s > n/p.

Finally, it follows from the above that if 0 < b < a the Taylor polynomial of

degree < b of an / 6 Bpq,

\j\<b

is defined for Aa-b,p,q a.a. x E R" (in fact, fj coincides with the Jth order weak

partial of /).

4. Differentiability of Lipschitz functions. We now prove Theorem 1. If b

is such that 0 0, we have

t~b\f(x + y)- Pbf(y, z)\ < t~b\f(x + y)- PQf(x + y)\

+ crb J2 \fj(x) - Cj(Q)\tW\j\ b, suppose first a — b < n/p;

using either (7) or (8) depending on whether a — \J\ < n/p or a — \J\ > n/p,

1/75

i/775



we estimate II as

II(M)<C J2 tW-bta-\J\-e I -f F(x + z)sdz)\J\n/p

+ C £ t^~b f Ja_e_lJl(z)F(x + z)dzI n^t, J\z\<t\J\<b

a — \J\<n/p

(the first sum may be empty; for instance, if a < n/p); thus

II(a:,i!) <C\ta-b- -f F(x + z)sdz

J\z\<t

1/s

+ / Ja-e-b(z)F(z + z)dz )J\z\<t J

<C Ja-e-b(z)ms(F)(x + z) dz,J\z\<t

where ms(F) — (m(\F\a))l/s, which if 1 < s < p is again a function in BPq [Nl],

and if p = s = 1, just as we did in Proposition 2, can be bounded by CG, where G

is another BPq function. In either case II(x,i) tends to 0 Aa_(,iPiQ a.e. as i! goes to

0. When a — b > n/p, the situation is simpler: we just use (6) to obtain

II(x,í)<C J2 tw-bta-\J\-e-n'p\\F\\p = C\\F\\pta-b-e-n/p,

\j\ n/p, tends to 0 e.e.

Turning our attention to III, we write for T > t Q(x,T) as Q*; then

/ \

iii(i,0 <c E + E1 |J|<f> |J|<6

\a — \J\>n/p a — \J\<n/pJ

tlJl-b\Cj(Q)-Cj(Q*

+ CJ2 t\J\-b\Cj(Q*)\;\J\>b

by (1), the second term is bounded by

C\\f\\p E t^-bT-\J\-n'p,

\j\>b

whereas in the first one the first sum is empty if a — b < n/p, and in any case, using

either (5) or (4) and Proposition 2, it can be bounded by

i/s

Cta-b-e ( / Flzydz

+ C7 E «'■"-'/J\6

= A(x,t) + B(x,T) + C(t,T);

here A and 5 tend to 0 with T Aa-b,p,q a.e. if a — b < n/p and e.e. if a — b > n/p;

thus, given any r5 > 0, if x is such a point, we can find a T¿ such that for all t < Tg

A(x, t) + B(x,Ts) < 6, and we can find now a ts small enough so that C(t, T$) < 6

for all t < ts- Hence III(-x, t) also tends to 0 Aa-b,p,q a.e.

Finally we deal with I. If a < n/p, we can write any r < np/n — op as 1/r =

1/s -f 1/u — 1, with s < p and u < n/n — a; choosing e small enough so that

u < n/n — (a — e), (9) and Young's inequality imply

(/ I(y,X,t)rdy) < Ct-b-n/rta-e-n+n/u ( Í F(x + z)"dz\

\J\y\<t J \J\z\<2t J

(L< C / Ja-e-b(z)msF(x + z) dz,

J\z\<2t

X 1/73

= Cta-e-b ( ^ F(x + Z)sdz)

\z\<2t I

'\z\<2t

which tends to 0 Aa-b,P,q a.e. Next, if a > n/p it follows from (10) that if \y\ < t,

i/s

l(y, x, t) < Cta-b-e ( I F(x + z)s dz )(f\J\z\<2t

provided a — e > n/s > n/p, and it also tends to 0 Aa_{,iPig a.e.; the theorem is

thus proved.

PROOF OF THEOREM 2. If 1 < q < p we have the embedding Bpq C Cp [DVS,

p. 58] and, as a consequence, if f E Bpq, Gaf E Lp; thus estimating as before

t-b\f(x + y)-Pbf(y,x)\<I(y,x,t)+ll(x,t)+m(x,t)

and writing for a < n/p, p* = np/n — ap, (9) and Sobolev's inequality if p > 1 give

( / l(y, x, t)p' dy) < Cta~b ( -f Gaf(x + z)p dz)\J\v\<t J \J\z\<2t J

which as it is well known tends to 0 //"-(«-^p a.e. When p = 1 the same argument

works once we use instead of (9) the estimate

\f(x) - PQf(x)\r <C f \zr~nGaf(x + z)r dz,J\z\<t

0 < r < 1, proved in [Do, Theorem 5]. If we estimate II(x,i) and lll(x,t) as before

it follows that / has a (p*,b) derivative \Hn~â~b"IP, Aa-b,P,q] a.e.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use


It may be worth noting that if / E Cp the reasoning in Theorems 1 and 2

can be easily modified to obtain (p*,b), (r,b), r < oo, or (oo,¿>) differentiability

(depending on whether a < n/p, = n/p or > n/p) for / 7?a_¡,.p a.e. if b is an integer

and r7"-(a-6)p a.e. for nonintegral b. Since Lp C Cp this implies the Lp results

mentioned in the Introduction, and also differentiability for functions in the Triebel-

Lizorkin spaces Fpq, 1 < p, q < oo (see [Tr] for their definition). We can associate

with these spaces certain capacity type set functions and one could guess that the

above differentiability results should be given with respect to them. This is not

the case however, for D. Adams has recently shown [Ad2] that the Fpq capacities,

1 < p, q < oo, are all equivalent with Ba>p (for Cp capacities this is proved in [Do]).

5. Lacunar differentiability. Here we prove Theorem 3. With u E 5n-i, we

estimate as before 2kb\f(x + 2~ku) - Pbf(2~ku, x)| by 1(2"*«, x, 2~k) + ll(x, 2~k) +

III(a;, 2~~k), and we consider only the first one, the other two going to 0 for Aa-b,p,q

a.a. x. With F E BPq related to / as before, e small enough, we have again

l(2-ku,x,2-k)<C2kb / \y-2-ku\a-e-nF(x + y)dy

J [y-2ku\<2-k

= C2kb F(x + 2-ku + z)\z\a-»-e-n ( / \z-y\b-ndy) dz

J\z\<2~k \./|z|/4<|z-¡,|<|z|/2 /

<C2kb F(x + 2-ku + z)\z\a-b-e-n ( \z-y\b-ndy\ dz

J\z\<2-2~k \./3|ï|/4<|it|<3|z|/2 /

<C2kb / F(x + 2-ku + z)\z\a-b-e-n\z-y\b-ndzdy

J\y\<4-2-k A|!/|/3<|z|<4|W|/3

<C2kb \y\a~b~e~n i F(x + 2-ku + z)\z-y\b-ndzdy

J\y\<4-2-k J\y-z\<3\y\

<C j \y\a-b-e-nT(F)(x + y) dy,

J\y\<1-2-k

where if h > 0,

771(c) = sup 2kb f \v\b-nh(ç + 2~ku + v) dvk>0 J\v\<12-2-k

= sup 2kb f \w- 2~ku\b-nh(ç + w) dw./|ti)-2-'cu|<12-2-t

= sup 2*" f K(2kw)h(ç + w)dw

with K(z) = \z — u\b~n if \z - u\ < 12, and zero otherwise. The kernel K belongs

to L log+ L on its support and therefore Lemmas 1.3 and 1.4 of [CCI] imply that

T is bounded in Lp, p > 1, and, in fact, in BPq, for we have \Tf(x + y)- Tf(x)\ <

T(Ayf)(x). Therefore,

7(2" V x, 2~k) <C f \y\a-b-e-nT(F)(x + y) dyJ\y\<4-2~k

which tends to 0 as k goes to 00 Aa_(,,p,9 a.e.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use


6. Tangential boundary values. We recall first that the complement of the

Lebesgue set of a function in Bpq has zero Aa,P:<? capacity. Thus, since nontangential

convergence holds on each point of the Lebesgue set, the case b — 0 of Theorem 4

is proved.

Fix next an / E Bpq, b such that 0 2Y

since \z — x\ < Y, \z — u\ > \x — u\ — \x — z\ > \x - u\/2, and

II < j \f(u) - f(x)\Py (j^j du = 2" j \f(u) - f(x)\P2y(x - u) du

which tends to 0 for all x in the Lebesgue set of /, that is, Aa^p^q a.e. Next, if

r = ns/n — bs,

l< f \f(u)-Pbf(u-x,x)\Py(u-x)duJ\x-u\<2Y

+ C E \fj(x)\ f \x-u\^Py(x-u)duaî ri^t J\x-u\<2Y0<\J\<b

<C||P,,||r' ( / \f(u)-Pbf(u-X,x)\rdu\\J\x-u\<2Y J

+ C ¿2 \fj(x)\Ym-0<\J\<b

Since fj(x), 0 < \J\ < b, is finite for Aa-b,p,q a.a. x, the second term tends to 0

Aa-fc,p,g a.e. The first term can be estimated by

Cy-n/ryb+n/rY-b I f |y^ + ^ _ p^j^ ^r j

\J\u\<2Y J

< CY~b '(/ \f(x + u)-Pbf(u,x)\r\J\u\<2Y j

for Y — yslT, and it tends to 0 Aa-b,p,q a.e. by Theorem 1. This proves Theorem

4 in the range l<g<oo;ifl<g<pwe proceed in just the same way, using

now Theorem 2 to obtain convergence inside Db,p(x) for [Hn~â~b">p, Aa^b,p,q\ a.a.x€Rn.

ACKNOWLEDGMENT. The author wants to thank Professor D. Adams for letting

him see unpublished work.

References

[Adl] D. Adams, Lectures on Lp potential theory, Umeá Univ. Reports, No. 2, 1981.

[Ad2] -, The classification problem for the capacities associated with the Besov and Triebel-Lizorkin

spaces (preprint).



[AS] N. Aronszajn and K. T. Smith, Theory of Bessel potentials. I, Ann. Inst. Fourier (Grenoble)

11 (1961), 385-475.

[BZ] T. Bagby and W. Ziemer, Point-wise differentiability and absolute continuity, Trans. Amer.

Math. Soc. 191 (1974), 129-148.[CS] A. P. Calderón and R. Scott, Sobolev type inequalities for p > 0, Studia Math. 62 (1978),

75-92.

[CZ] A. P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential

equations, Studia Math. 20 (1961), 171-225.[CCI] C. Calderón, Lacunary spherical means, Illinois J. Math. 23 (1979), 476-486.

[CC2] _, Lacunary differentiability of functions in R", J. Approx. Theory 40 (1984), 148-154.

[CFR] C. P. Calderón, E. Fabes and N. Riviere, Maximal smoothing operators, Indiana Univ. Math.

J. 23 (1974), 889-897.[DZ] D. Deignan and W. Ziemer, Strong differentiability properties of Bessel potentials, Trans. Amer.

Math. Soc. 225 (1977), 113-122.[DVS] R. Devore and R. Sharpley, Maximal operators and smoothness, Mem. Amer. Math. Soc.

No. 293, 1984.

[Do] J. Dorronsoro, Poisson integrals of regular functions, Trans. Amer. Math. Soc. 297 (1986),

669-685.[FZ] H. Fédérer and W. Ziemer, The Lebesgue set of a function whose distribution derivatives are

p-th power summable, Indiana Univ. Math. J. 22 (1972), 139-158.

[My] N. G. Meyers, A theory of capacities for potentials of functions in Lebesgue spaces, Math. Scand.

26 (1970), 255-292.

[Mz] Y. Mizuta, On the boundary limits of harmonic ¡unctions with gradient in Lp, Ann. Inst. Fourier

(Grenoble) 34 (1984), 99-109.[NRS] A. Nagel, W. Rudin and J. Shapiro, Tangential boundary behavior of functions in Dirichlet

type spaces, Ann. of Math. (2) 116 (1982), 331-360.[NS] A. Nagel and E. M. Stein, On certain maximal functions and approach regions, Adv. Math. 54

(1984), 83-106.[Nl] C. Neugebauer, Strong differentiability of Lipschitz functions, Trans. Amer. Math. Soc. 240

(1978), 295-306.[N2] _, Smoothness of Bessel potentials and Lipschitz functions, Indiana Univ. Math. J. 26

(1977), 585-591.[St] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press,

Princeton, N.J., 1970.

[Sto] B. Stocke, Differentiability properties of Bessel potentials and Besov functions, Ark. Math. 22

(1984), 269-286.[T] M. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean n-space. I, J. Math.

Mech. 13 (1964), 407-479.

[Tr] H. Triebel, Theory of function spaces, Birkhäuser, 1983.

[Z] A. Zygmund, Trigonometric series, Cambridge Univ. Press, 1959.

División de Matemáticas, Facultad de Ciencias, Universidad Autónoma,28049 Madrid, Spain

On the differentiability of Lipschitz-Besov functions

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