Introduction We give conditions on weight functions tu so ......The proof of the above theorem is based on the following Theorem 2 (Weighted restriction theorem for the Fourier transform).

PROCEEDINGS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 112, Number 1, May 1991

UNIFORM L2-WEIGHTED SOBOLEV INEQUALITIES

FILIPPO CHIARENZA AND ALBERTO RUIZ

(Communicated by Barbara L. Keyfitz)

Abstract. We prove the inequality ( 1 ) for weights w in a class which contains

the class J , p > (n - l)/2 , introduced by C. Fefferman and D. H. Phong in

studying eigenvalues of Schrödinger operators. In our case, C is independent

of the lower order terms of P . As a consequence we prove unique continuation

theorem for solutions of A + V , V in the same class.

Introduction

We give conditions on weight functions tu so that an inequality

(1) IMIxV) < C||i»(2>)«||£W)holds for the constant coefficient operator in R"

P(D)=A + Y,aid/dxi + b,

where « > 3 and the constant C is independent of the lower order terms ai,

b in C.The weight tu must satisfy two conditions. The first is a one-dimensional

A2 Muckenhoupt's requirement, defined at the beginning of §1, which depends

only on the direction of the vector a. . The second is an estimate,

(rn Í wa(x)dx\ (r* I w~a(X)\ <cr4a,

where Br denotes a ball of radius r and C is independent of Br.

A similar, weaker condition for the particular case of the Laplace operator

was given in [CW] (see for instance (1,6)' in this reference).

Also, an inequality like (1) has been proved in [KRS] for Ü norms; more

precisely

(0) \\u\\L,<C\\P(D)u\\LP,

Received by the editors April 14, 1988 and, in revised form, June 4, 1990.1980 Mathematics Subject Classification (1985 Revision). Primary 35B66; Secondary 42B99.The first author was partially supported by the Italian Ministero della Pubblica Istruzione and

GNAFA-Consiglio Nazionale della Ricerche.The second author was partially supported by GNAFA and the Spanish Comisión Asesora de

Investigaciones y Ciencia.

©1991 American Mathematical Society0002-9939/91 $1.00+ $.25 per page

53

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

54 F. CHIARENZA AND A. RUIZ

where p and p satisfy the duality condition and the Sobolev gap l/p-l/p =

1/n.The ingredients to prove (1), as in [KRS], are a similar uniform inequality

(2) ll"lljr.V)<C||(A + r)ii||iJ(w.t)>

and a weighted restriction theorem for the Fourier transform which we state

independently, since it requires a weaker hypothesis.

As a consequence of (1), we prove a version of a result due to Chanillo

and Sawyer [CS] on unique continuation of solutions of Schrödinger equations.

More precisely, we prove that a solution u of the inequality

|Am(x)| < \V(x)u(x)\,

which is zero in an open set, must vanish everywhere. We assume some minimal

conditions on u, and assume V to be in the Fefferman-Phong class

/A = < V e Û. , such that lim sup sup r ( r~n \ Vp \ < F,P' rÔ x€K \ JBr(x) J

for any compact K c R"

for p > (n - l)/l and F some constant depending only on dimension.

Chanillo and Sawyer (see [K]) obtained a stronger unique continuation for V

in the Fefferman-Phong class for F = 0 and the same p's; their proof relies on

an L2 -* L2 estimate due to Jerison and Kenig very difficult to prove. We take

from them the idea of substituting weighted L inequalities for Lp inequalities

and also the use of maximal functions in the unique continuation context.

Our proof of (1), and hence a new proof of unique continuation property

for potential in the class / F, relies on restriction theorems for the Fourier

transform, stationary phase methods, and real interpolation.

1. Statement of results and consequences

We start with some definitions.

Let tu be a nonnegative function. We say that w is in D„, ß < 0, if there

exists a constant C > 0 such that for any ball Bt of radius /,

lt~" í w(x)dx\ Ir" Í w(x)~1\ <Ctß,

we denote by |||tu||| the infimum of such constants C.

Let y in R", and let us write for any y in Rn , y = ty + y , where y is

in some hyperplane II of R" . We will say that tu is in A (y) if the function

w ,: R —> R+, defined by tu A) = w(y) is an A weight in R for almost

every y in II, with A norm uniformly bounded.


UNIFORM L2-WEIGHTED SOBOLEV INEQUALITIES 55

Theorem 1. Let P(D) = A + X)ctjd/dXj + b, ajt b in C. Then there exists a

direction y in Rn , depending only on the direction of the vector (Re a A, such

that if w is in A2(y) and wa is in D„ for some a > (n - l)/2, ß = -4a, the

following inequality ( 1 )

II"HlA) * ciiKHi'îiPd))«!!^-,)

holds with an absolute constant C depending only on the A2(y) constant of w .

The proof of the above theorem is based on the following

Theorem 2 (Weighted restriction theorem for the Fourier transform). Let do be

the uniform measure on the sphere S"~ in Rn and (do)~ its Fourier transform;

let w be a doubling weight tu > 0, i.e. there exists a constant C' such that

f w <C' [ w.Jß2r Jb,

Then for wa e D„ for 1 < a < ( 1 - « - ß)/l and n > 3, there exists a constant

C independent of |||tua||| such that

IKÂ/ll^^CHKlll'^ll/ll^-,

for every /eÇ.

Theorem 3 (Weighted L restriction theorem for the resolvent). Let w be as

in Theorem 1, then there exists a constant C such that

I1\ II II ^ /~i\\\ Qll|l/al I — 1 — (â/4d) ,, , . , ||

(3) ll"llL'(li,) < C|||u; HI ' 1*1 '||(-A + z)M||i2(url)

for any u in C^ and z in C.

The following method gives weights which satisfy the hypothesis of Theorem

1, with A2(y) constant depending only on the dimension.

Take yx = y, y2, ... ,yn, an orthogonal basis of R" and define the strong

maximal function

Mf(x) = sun¡Rf1 i \f(y)\dy,R JR

where the sup is taken over all the rectangles with edges in the directions yx =

"A > Y2 ' ■ ■ ■ > y„ > which contain x .

Let us define Maf = (M(fa))l/a . Then we have the following lemma due

to Chanillo and Sawyer:

Lemma 1. Let f be in J F for p > (n - l)/l, and 1 < a < p; then there

exists a constant Ca, depending only on the dimension and a, such that Maf

is in JpF>,for F' = CaF.

Now we are in position to construct the appropriate weights. Following [R],

let us takeoo

s/=£(2crxAj=o



Then if V e J^F , define

(4) w(x) = S(VXbr).

The following properties hold:

(a) Maw(x) < Cw(x) for some C > 0; this is a A*(yx, y2, ... , yn) con-

dition for tua , hence tu satisfies the same property.

(b) tu is in J F,, if a < p , p > (n - I)/I ;

(c) wa is in D„ for ß = -4a and |||tua|||1/a < C F ;\ / ¡j i- ill ill — a '

(d) tu is in A2(y).

Conditions (a) and (b) are consequences of the results in [GR, page 433]

and Lemma 1.

Condition (c) is a consequence of Jensen's inequality.

Condition (d) follows from Theorem 6.2 in [GR].

As a consequence of Theorem 1, we prove the following unique continuation

property:

Corollary. There exists F > 0, depending only on the dimension, such that for

any solution of the differential inequality

(5) \Au(x)\<V(x)\u(x)\,

where V is in JpcF for some p > (n - l)/l and u is in HX(¿.2, if u is zero in

an open set, u must be zero everywhere.

//l0¿ denotes the Sobolev space of functions in LXoc with derivatives in

Aoc-

Proof of the corollary. When tu satisfies the hypothesis of Theorem 1, the fol-

lowing Carleman inequality holds, with constant C independent of v e Rn

and I e R :

f£\ II Av'X h . .'-'Ill Q i ■ i 1 /t» 11 IWX . ||

(6) \\e u\\L2{w) < C\\\w HI \\e A«||lV-v

In fact (6) can be reduced, by changing v = eXv'xu to

(0 IMIlV) ^ C\\\wa\\t,a\\P(D)v\\û{w.,y

where P(D) = J2\Dj + i^vj\ > v — (*j)i-i n > wriich is a particular case of

(ABy a reflection argument (see [KRS]) we may assume that u = 0 out of a

compact set, and hence (by translation and dilation) to the case u = 0 out of

Bx = {xinRn, x2 + --- + x2n_x+(xn + l)2< I};

it suffices to prove that u = 0 in a neighborhood of the origin.

The above transformations preserve the spaces / . Also, by adding to V

the function e|x|-2 , we obtain a new potential in (5), which is in JlpocF+e and

is bounded below in compact sets.


UNIFORM AWEIGHTED SOBOLEV INEQUALITIES 57

We still call V the resulting potential of applying rotations, inversions, and

the above perturbation to V in (5).

Take now tu as (4) after Lemma 1, with y given by Theorem 1 when a. =

2i/.. From Conditions (a)-(d), tu satisfies the hypothesis of this theorem.

Let us choose n e C™(BS), n = 1 in \x\ < à/1, S > 0 to be fixed, then (6)with v = (0, ... , 0, 1 ) gives for S small

We^gW^^lCFWeÂgW^-^,

where g = un,

< lCF\\eXx°uV\\L2[w-ídx + C'\\eXx"g\\LHw-ldx^

but

^«^»C-'ä.jW - (jB \eXx'u(xfv\x)w-\x)dx\

' \JBtll

\e "u(x)\ w(x)dx

1/2

since V(x) < S(VxB ) — w(x) for ô small enough.

Assuming 2CF < 1/4, we obtain

H^HlW,*^) ^ 2C'\\eXX"AghAw->dx,B{\Bi/2)

uniformly in k ; then there exists £ > 0 such that g = 0 on xn > -Ç, if we

prove that the right-hand side is finite.

WtehHw-'dx.B^)

< HCA'/MIl'o«-1,/*.*,) + Hv?/ • VuWl\w-Ab6) + Wi^WlHw-Ab,)

^ C\\u\\l}(dx,Bt) + C\WUh>{dx,Bt) + CWVuhAw-Ax,Bs)

since tu-1 is bounded in Bs, and u is in /f¿¿ . To bound the last term

consider that

Í \Vu\2w~l dx< f\u\2Vdx<C f\Vu\2dx,

since V is in the Fefferman-Phong class with p > 1, and Sobolev estimates

hold [FP].

3. The proofs

Proof of Theorem 1. As in [KRS], by rotation, density, and replacing u by

ue~'x'c for some c in Rn , we may reduce to the case

P(D) = A + o + t(d/dxn + ie).

We have to prove (1) with C independent of ct, t, e in i?\{0} .



Take

m(Ç) = (-\A\2 + o- + h(Çn + e))-1;

we have to prove

1/2

|(m«)/«))Ä||1»(w)^C|||u»tt|ir/a||/||LJ(w..).

By inserting a cutoff function x such that x(t) = I on 1 < |?| < 2, we may

write ^

(m«)/«l))Ä(*) = [E**(«/«)) (*).

where «^(¿;) = ;^(¿;)m(<í;) and ^(¿;) = x(lk(Z„ + e)). Then (7) is bounded by

I f 2 1/2E / K^/HAI w(x)dx

sa«,»(Ç/lM,,H.,.-Wand then, by the above argument, bounded by

Ci, i a 1111/a m /*n

III™ III ll/lliV1)"Only (8) remains to be proved; it may be reduced to the boundedness of the

Fourier multiplier

ml(Ç)=xk(Z)(-\Ç\2 + e + n2-krl,

which is bounded by Theorem 3 with z = a + ixl~ .

Hence only the difference mk-m*k has to be bounded as a Fourier multiplier;

it is given by

(Aff (0 = Xk(tn)r(t)ir(t„ + e - rk)

x (-\A\2 + o + it(C„ + e)f\-\A\2 + o + ixTkyx.

Take polar coordinates Ç = p£ , by Minkowski's inequality

L2(w{x)dx)

p dproo il /• ,

/ / (r/HKV* "</*«')Jo WJs"-*

/»oo

= / \\(dcrr-ç(m/)(Çn)f(pOf(px)\\L2{w)p"-ldp,J 0

where

,)'Xk(pCH)it{pCH + e-2k)

x I - p + a + ir(pCn + e)\~ \ - p + a + ixl~ |~/•OO

= / \$dcT • ç(mp(Çn)f(pQr(y)\\û{wiy/p)dy)p"/2-[ dp

/*oo

< / \\(rnp(i:n)f(pQr(y)\\u{w(ylpy,dy/l2-l-ßl2adp\\\wa,. A/a


UNIFORM AwEIGHTED SOBOLEV INEQUALITIES 59

by Theorem 2, and

\\\wa(x/p)\\\2 = p-ß\\\wa(x)\\\2,

\\{p~nmp(p~lrH)f(Q}-(yip)\\^{w{yjfirldy)p'/ nr^-"^ /Ar \/vru~f„/„mi ^"/2_1_'8/2aiiu„aiii1/a

JO

■U . <„ ,. ,„ „-*\, 2 . , . .„ ..-1, 2 , , . „-*,-!

10

But since

mp(P Cn) = Xk(C„)ñ(C„ + e-2 )\-P +(r + h(Cn + e)\ -p+a + hl

is an L2(w~1) —> L2(w~l) operator with norm

Ti''

(p2-a)2 + (xl-k)2

where C depends only on the A2(y) constant of tu , we have

iir/iiLV)<ciilTO«in"^ (;_j);;yf)2ii/nw,

roo « —1— /?/2a j „-m an|l/a -/f/4a-l / £__djp -

= C|I|U;111 " io (^ _1)2+A211/11^-V

with A = t2_ /rj ; if we take —ß/4a- 1 = 0, the integral is uniformly bounded

with respect to X and the theorem is proved.

Proof of Theorem 1. It is known that

(da)" = \x\ ^(n/2)-i(M) ' wbere Jt denotes the Bessel

function of order t.

' J(n/2)-\If |x| > 1 , J,nm-\(\x$ is asymptotically like ei]xl\x\ 1/2.

Let us take a cutoff function ip(\x\) suchthat supp \p is contained in [5,2]

and00

E V(l~ks) = 1 for |s| > 1 ;

fe=i

write da = ¿2T=oFk(x) = AZT=o ¥k(x)dâ(x), where \pk(x) = y(T \x\).Take Tk the operator of convolution with Fk . We are going to estimate its

L (w~ ) —y L (w ) mapping norm:

For 0 = 0 we use P. Tomas's estimate (see [T]),

||rfc/||2<C2*||/||2.

For 6 = a > 1 ,

\Fk(x)\<CW(Tk\x\)eiM\x\-{n~m\

<crk("-l)/2x{X[ w<2».}(|*|).



Hence

$Tkf)(x)\ < crk{n-l)/2 [ k\f(y)\dy

_. ~—k(n-$l2 , , r, \~,nk< 1 M2kf(x)l ,

where Mtf(x) = sup|B,>in fB \f(x - y)\dy is the truncated Hardy-Littlewood

maximal function.

We invoke the following lemma (see, for instance, [GR]):

Lemma 2. Let V be a measure in Rn+i, and u a doubling measure in Rn,

such that for any cube Ih of side length h > 0, there exists a constant K > 0

such that

V(Ihx[0,h])<Ku(Ih);

then

\ r 1/2 / r \1/2/ \MJ(x)\2dV(x,s) <Kl,2( \f(x)\2du) .

\JrI+> \Jr" )

We take dV(x, t) = wa ®ôt, where ôt is the singular measure on j = í in

Rn , and du(x) = w~a(x) dx ; in this case the Carleson constant K is bounded

by |||tua|||V, in fact:

wa®ôt(Ihx[0,h]) = ôt([0,h])wa(Ih)

= Xh>t(h)wa(Ih)= I wa(x)dxxit „Ah)Ji„

< [ w-a(x)dx\\\wa\\\2hßx{tt0o)(h)A

<w-a(Ih)\\\wa\\\2tß (ß<0).

Hence

1/2 / r x 1/2

and

U\Mtf(x)\2dwa(x)\ < \\\wa\\\tßl2 (^j\f(x)\2w-a(x)dx^j

.k(l-l/a)~(-k(n-\)/2+nk+kß/2)l/c,,, a,,A/a

¡k\\L2(w-')^L2(w) ̂ ^¿ L HI™ HI

We apply interpolation and

||7, i, . r,~k(l-l/a)~(-k(n-l)/2+nk+kß/2)\/a,,l a,,,\/a\\Jkh2(w-l)-*L2(w) ̂ C2 ¿ HI™ HI

ŒC2*<l-<l-«-A/<2-»|,,w.|||l/«f

and the sum is convergent for 1 < la < (I - n - ß).

Proof of Theorem 3. We follow along the lines of the proof of Theorem 2

Take m(A) = (|¿;|2 4

by a density argument

Take m(¿;) = (|<¡;|2-r-z) ' as a Fourier multiplier; we may assume ImzÔ


UNIFORM A WEIGHTED SOBOLEV INEQUALITIES 61

The Fourier transform of m is given by

, v I/2(*/2-l)

K{X) = C\ixT2) Kn/2_x((z\x\2)1'2),

where Kn/2_x is a modified Bessel function (see [KRS] and [GS]) whose behav-

ior is

l*B/2-i(0l < C\tf/2+l, for \t\ < 1, Reí > 0,

Kn/2_x(t) = b(t)t~l/2e~', for |i| > 1, Rei>0,

where b(t) satisfies

$d/dp)jb(pt/\t$\<Cj\p\-j.2 1/2

We take a determination of (z|x| ) with positive real part. Now,

|ä"(jc)| < C|xf"+2 for |z1/2|x|| < 1,

with C independent of z, and we may consider the convolution operator with

y/k(r\x\)K(x). One can see that the worst case occurs when Imz is small. In

this case the operator behaves like the restriction operator dilated by |z|. This

is the idea in the following calculation:

(a) Tk: L —yL2 with norm 2 /|z|. In fact, if we denote s = \x\ and1/2

z ' = r cos a + ir sen a, the multiplier is

z(n-2)ß / ^(rkDJc^^^z'^lxDIzlxl2!-^2^-^^J R

(n—2)/2 f , ,,, 1/2 ,, 2,—(n-l)/4 — irssena—rscosa= z / w,(rs)b(z s)(zs ) e' L , Wk(rs)b(zl/2s)(zs2)~

J2k-><rs<2k

I iswl j , \ n—\ i• I e do(w)s ds

A-1

(n—2)/2 f , ,,, 1/2 ,, 2,—in—1)/4 —irs sena—rs cosa= zK " w.(rs)b(z s)(zs ) " eL t ipk(rs)b(zl/2s)(zs2)'

J2k~{<rs<2k

■a(s\i\)(s\i\)-{H-l)/2emsn-ids,

where a is a function with the same type of estimates as b .

Since cosa > 0, if cosa > n > 0 the above integral is bounded byk

Cexp(-2 nr). Hence the worst case is when 0 < cosa < n ; we use the

stationary phase method and obtain the bound

, ,(n-2)/2, ,-(n-\)/2~k -1 „Jt -2\z\ |rsena| ' 1 r «C2 r .

(b) 7;:L2(tAa)^L2(tua) with norm r-2-^/2)2^"+1+^^2|||îjt;Q|||, by a rep-

etition of the argument in Theorem 2.

Using interpolation we obtain the range of convergence 2<2a<(l-/A«)

and the operator norm L2(tu_1) —► L2(tu) of Tk

^ -2-(ßl2a).., a..AlaCr ' \\\w HI ' .

This is independent of r for ß = -4a.



Proof of Lemma 1. We may assume, without any restriction, that the y's are in

the coordinates' directions.

Define

(\ i/Q

sup r— / \f(xx,... ,xj_x,t,xj+x,...,xn)\adt) .a<xj<b°-a J[a,b) J J J

We see that Maf(x) < Mx a---Mn af(x) ; hence if we prove that for any

j = I, ... , n , there exists a constant C'a such that for any g e J F , M- ag e

Jp F,, for F' = C'aF, a recurrent application of this fact would prove our

statement with C = C' " .a a

Let Q be a cube centered at z = (zx, ... , zn) of side length S ; in order to

evaluate the average of Mx a in Q, we may assume that / is supported on

the strip

S = {x in Rn, there exists t with (t, x2, ... , xn) in Q}.

Decompose S in pair of rectangles Rk U R_k in Q, where

RkUR_k = {yeR", \yj-zj\<ô,j = l,...,n, lkâ < \yx -yn\ < lk+lS},

k=l,l,....

Let us denote

fk = JXRk > /o = fXQ > f-k = JXR_k ■

Then

t I'" jrWuafkA ■k=—oo y J

For k = 1, 1, ... , and x in Q

l la

M, ■**w -{ik L^, M°d,)°--hence since a < p ,

UP

\ff)

where Qk is a cube of side length 2 +lô


UNIFORM AwEIGHTED SOBOLEV INEQUALITIES 63

The above expression is bounded by 2 t"-1^-2 p. por fQ> since p/a > i

and Mx is bounded in Lp/a , we have

S2 (|Qf ' ¡\Muj/)llP =S2\Q\-l/p (KjQ\Mx\f,\a\Pla)llP

< CÔ2\Q\-l/p(\\f0\\p/a)1'2

= cô2 (1/101 j i/r) '<cQ/-.

For /c = -1, -2, ... , we obtain similar bounds (with \k\ in the exponent).

The sum is convergent if (« - l)/2 < p .

1 0Remark and open questions. The condition u in H ' in the corollary is chosen

to avoid a more complicated condition, which states that Vu e L2(w~l) and

u e L (tu), for the Ax weights constructed in the proof. In [KRS] the similar

requirement is u e H 'p , p = ln/(n +1).

In [KRS] the inequality (0) is proved for the operator d2/dt2 - An+ lower

order; our proof involves the geometry of the level sets of the kernel; in the

Klein-Gordon case this geometry is more complicated. We wonder if there is

an appropriate class of potentials V adapted to this case.

Acknowledgment

We have learned during the elaboration of this note that S. Chanillo and E.

Sawyer have obtained similar results.

We want to thank S. Chanillo, J. Garcia-Cuerva, and the referee for their

clarifying comments.

References

[CS] S. Chanillo and E. Sawyer, Unique continuation for A + V and the C. Fefferman-Phong

class, preprint.

[CW] S. Chanillo and R. L. Wheeden, iA estimates for fractional integrals and Sobolev inequal-

ities with applications to Schrodinger operators, Comm. Partial Differential Equations 10

(1985), 1077-1116.

[F] C. Fefferman, A note on spherical summation multipliers, Israel J. Math.

[FP] C. Fefferman and D. H. Phong, Lower bounds for Schrodinger equations, J. Eqs. aux dérivées

partielle. Saint Jean de Monts, 1982, pp. 129-205.

[GR] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norms inequalities and related topics,

North-Holland, Amsterdam, 1985.

[GS] I. M. Gelfand and G. E. Shilov, Generalized functions, vol. 1, New York, Academic Press,

1964.

[K] C. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique

continuation, Harmonic Analysis and PDES (Proceedings, el Escorial, 1987), Lecture Notesin Math., vol. 1384, Springer-Verlag, New York, 1989.

[KRS] C. Kenig, A. Ruiz, and C. Sogge, Uniform Sobolev inequalities and unique continuation for

second order constant coefficients differential operators, Duke Math. J. 55 (1987), 329-347.



[KW] D. Kurtz and R. L. Wheeden, Results on weighted norm inequalities for multipliers, Trans.

Amer. Math. Soc. 255 (1979), 343-362.

[R] J. L. Rubio de Francia, Factorization theory and A weights, Amer. J. Math. 106 (1984),

533-543.

[T] P. Tomas, y! restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975),

477-478.

Dipartamento di Matemáticas, Università di Catania, Víale A. Doria 6, Catania, Italy

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid,

Spain


Introduction We give conditions on weight functions tu so ......The proof of the above theorem is based on the following Theorem 2 (Weighted restriction theorem for the Fourier transform).

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