Weighted Hardy Spaces (Lecture Notes in Mathematics, 1381) 3540514023, 9783540514022

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1381

Jan-Olav Stromberg Alberto Tarchinsky

Weighted Hardy Spaces

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

Authors

Jan-Olav Stromberg University of Trornse, Institute of Mathematical and Physical Sciences 9001 Irornse, Norway Alberto Torchinsky Indiana University, Department of Mathematics Bloomington, IN 47405, USA

Mathematics Subject Classification (1980): 42B30

ISBN 3-540-51402-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-51402-3 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper

Preface A considerable development of harmonic analysis in the last few years has been centered around a function space shown in a new light, the functions of bounded mean oscillation, and the weighted inequalities for classical operators. The new techniques introduced by C. Fefferman and E. Stein and B. Muckenhoupt are basic in these areas; for further details the reader may consult the monographs of Garda-Cuerva and Rubio de Francia [1985] and Torchinsky [1986]. It is our purpose here to further develop some of these results in the general setting of the weighted Hardy spaces, and to discuss some applications. The origin of these notes is the announcement in Stromberg and Torchinsky [1980], and the course given by the first author at Rutgers University in the academic year 1985-1986. A word about the content of the notes. In Chapter I we introduce the notion of weighted measures in the general context of homogenous spaces; the results discussed here include the theory of A p weights. Chapter II deals with the Jones decomposition of these weights including a novel feature, namely, the control of the doubling condition. In Chapter III we discuss the properties of the sharp maximal functions as well as those of the so-called local sharp maximal functions. This is also done in the context of homogeneous spaces, and the results proved include an extension of the John-Nirenberg inequality. In Chapter IV we consider the functions defined on the upper-half space R+.+ I which are of interest to us, including the non tangential maximal function and the area function. Then, in Chapter V, we restrict our attention to a particular class of functions defined on R+.+ I , namely, the extensions of a tempered distribution on RrL to the upper-half space R+.+l by means of convolutions with the dilates of Schwartz functions. We study how the extension behaves with respect to different Schwartz functions, and an interesting result is the mean-value type inequality we show these extensions satisfy. We are now ready to introduce the weighted Hardy spaces in Chapter VI. We also describe here some of their essential properties, such as the independence of the "norm" among others. In Chapter VII we construct a dense class of functions for these spaces of distributions; this is a delicate pursuit. Chapters VIII and IX lie at the heart of the matter: in Chapter VIII we construct the atomic decomposition for these spaces, and in Chapter IX we describe an extension of the Fefferman HI duality result by means of the so-called basic inequality. In Chapters X, XI and XII we then discuss some applications. Chapter X contains the construction of the dual to the Hardy spaces, Chapter XI deals with the continuity of various singular integral and multiplier operators on these spaces,

IV

and, finally, in Chapter XII we show how the complex method of interpolation applies in these context. All in all, the essential ingredients of the theory of the weighted Hardy spaces is contained in these notes.

Contents

Preface Chapter I. Weights

III

1

Chapter II. Decomposition of Weights

18

Chapter III. Sharp Maximal Functions

30

Chapter IV. Functions in the Upper Half-Space

48

Chapter V. Extensions of Distributions

60

Chapter VI. The Hardy Spaces

85

Chapter VII. A Dense Class

103

Chapter VIII. The Atomic Decomposition

111

Chapter IX. The Basic Inequality

122

Chapter X. Duality

134

Chapter XI. Singular Integrals and Multipliers

150

Chapter XII. Complex Interpolation

177

Bibliography

189

Index

192

CHAPTER

I Weights

We begin our exposition in a general setting. Let X be a metric space endowed with a measure j.L. If the measure v is absolutely continuous with respect to j.L and if there exists a nonnegative locally integrable function w such that dv(x) = w(x)dj.L(x), we say that v is a weighted measure with respect to j.L and that w is a weight. Throughout these notes we assume that all absolutely continuous measures are weighted measures; this is the case if, for instance, j.L is a-finite. In this chapter we are mainly concerned with various relations between v and u, different conditions on w, and the continuity properties of the HardyLittlewood maximal operator. The conditions we have in mind for the weights include Muckenhoupt's A p condition, the reverse Holder condition RH r and the doubling condition Db. Aside from trivial implications these conditions are independent of each other and consequently their detailed study is justified. Most of the results presented in this chapter are well-known in the case X = R n and j.L the Lebesgue measure. The proofs we present differ methodologically from the usual ones since we don't have at our disposal tools such as the dyadic decomposition, replaced here by the notion of telescoping sequence of balls. Assume, then, that X is a metric space with measure j.L and that the class of compactly supported continuous functions is dense in the space of integrable functions L(j.L). Further, suppose there is a nonnegative real-valued function din X xX, it need not be the distance function in X, that satisfies the following properties: (i) d(x ,x) = 0 for all x in X. (ii) d(a:,y) > 0 for all x i= yin X. (iii) There is a constant Co 2: 1 such that d(x,y) cod(y,x) for all x,y in X.

2

(iv) There is a constant CI > 0 such that d(x,y) cI(d(x,z) + d(y,z)) for all x,y,z in X. (v) For each neighbourhood N of x in X there is an r > 0 such that the ball B(x,r) = {y EX: d(x,y) r} is contained in N. (vi) The balls B(x,r) are measurable for all x E X and r > O. (vii) (Doubling Db condition) There are a constant k > 0 and a number b > 0 with the property that for all x EX, t 2:: 1 and r > 0 we have

J-l(B(x,tr»

ktbJ-l(B(x,r)).

(Db)

We write J-l E Db to indicate that J-l satisfies the Db condition. The family of balls satisfies the following geometric property. Lemma 1. Let a > O. Then there is a constant Cz = ci (1 + a) + COCI a such that if B(x,r) n B(y,r')::f. 0 and r ar', then B(x,r) B(y,C2r'). Proof.

Let z E B(x,r) and

d(y,z)

Zl

E B(x,r)

n B(y,r'). Then

+ d(z,x» CI(cld(y,zI) + cld(x,ZI» + cocld(x,z) + clr + cOclr 0 let

MJ.L,nf(x) =

sup

1

)

Jl(B(y,T )

f

JB(y,r)

Ifl dp.,

We begin by showing that the mapping MJ.L,n is of weak-type (1,1) with norm bounded independently on n; once this is done the result follows by letting n tend to 00. Given A > 0, let F be the family of balls B = B(y,T), T n, such that AJl(B) < If 1dJl and let {B(Xi, Tin be the pairwise disjoint family of balls corresponding to F obtained in Lemma 2. It then follows that

IB

{},{J.Lf> A}

UB

UB(Xi,CTi) '

BEF

i

Ix

and consequently, L.i Jl(B(Xi, Ti)) A-1 If Idu. The desired estimate is obtained now since, by the doubling condition, Jl(B(Xi,CTi)) kcbJl(B(Xi,Ti))' • Let now 1/ be another measure defined on X. We are interested in studying when the Hardy-Littlewood maximal operator MJ.L is of weak-type or of type (p,p) on LP(I/). Specifically, we search for conditions on 1/ for the estimates

and Now, if XE denotes the characteristic function of a set E B, we readily see that MJ.LXE(x) Jl(E)jJl(B) for all x E B, and if (l)p holds we get

I/(E)jl/(B)

c;1(Jl(E)jJl(B))P,

Note that if Jl E Db and (3)p holds, then there is a constant C such that

1/

all E

B

0

E Dpbo Furthermore, (3)p implies that

Whence, by the type (p,p) statement in Theorem 3 for M v , we conclude that (3)p implies (l)p for characteristic functions of sets. Thus we have shown

4 Lemma 4. Condition (3)p is necessary and sufficient for the restricted weaktype (p,p) inequality for the operator MJ-L in LP(v).

Another interesting consequence of (3)p is Lenuna 5. to It.

Condition (3)p implies that v is absolutely continuous with respect

The proof of this lemma will be presented later on, as part of the implication (6) implies (2) in Theorem 15. Now, assuming for the moment that Lemma 5 has been proved, if (l)p holds, then v is a weighted measure with respect to J.L with weight 10, say. Suppose p > 1, let B be an arbitrary ball and let denote the characteristic function of the set we have {x E B :w(x) > }. Then for the function f = M f(x) > _1_ [ X 1O-1!(p-l)du J-L ­ J.L(B) iB e rr » and from (l)p it follows that v(B)

l

r::;

cP

=c

P

all x E B,

l l

1O-1/{p-l) dJ.L.

Sorting this inequality out and letting e go to 0 we obtain the so­called Ap(ll) condition for 10, to wit,

1 1 (1 1

­­ 10 dJ.L J.L(B) B

)P­l c, and the condition A 1(J.l) holds, i.e., (ess

W(X)) -1

l

wdJ-L:S; c,

all B.

When this condition is satisfied we write w E A 1(J-L). Note that if w is a nonnegative measurable function and dv = wdJ-L we have

and consequently, if wE A 1(J-L) we get MlJ.f(x):S; cMvf(x) for all x. In this case, by Theorem 3 we conclude that MJ1. is of weak-type (1,1) on L 1(v). Summing up, we have proved Lemma 7. MIJ. is of weak-type (1,1) on L1(v) if and only if v is a weighted measure with respect to J-L and the weight w E A 1(JL).

Returning to the Ap(J-L) condition, one of its basic features is Lemma 8.

If p

> 1 and w E Ap(JL), there is an E: > 0 so that wE Ap-e(JL).

Note that if w E Ap(J-L), then by Holder's inequality it follows that w E Aq(J-L) for q > p. On the other hand, Lemma 8, proved after Theorem 18, is far from trivial. It is a crucial property in the theory of weights and it implies the maximal theorem. Theorem 9. Let p > 1. Then the inequality !IMJ1.fllv>(v) :s; cpllfIlLP(v) holds for every f E LP(v) if and only if v is a weighted measure with respect to JL and the weight w E Ap(J-L). In this case, MIJ. is of weak-type (p,p) on LP(v) if and only if it is of type (p,p) on LP(v).

6

Proof. It only remains to prove the sufficiency. Since M J1.f( x) eM,J( x )l/(p-e) when wE Ap-e(J.L) and since p/(p - E) > 1, it follows from Theorem 3. • A close relation to Ap(J.L) is the reverse Holder condition RHr(J.L) , r requiring that for some constant e > 0 we have

When this happens we write w E RHr(J.L). Note that if J.L and v are mutually absolutely continuous and w(x)dJ.L(x), then the assumption w E RHr(J.L) is equivalent to (1/w) where (r - 1)(p - 1) = 1. Now for the case p = 00 and the condition Aoo(J.L). For each ball B the median value WB,p. of the weight w with respect to the measure expression where and

tl = sup{t > 0 :J.L({x E B :w(x) < t})

J.L(B)/2}

tz = inf{t > 0 :p({x E B: w(x) > t})

p(B)/2}.

> 1,

dv( x) = E Ap(J.L) , we define J.L by the

Observe that 'for any real number a we have

We say that w E A(X)(J.L) if

l

ui

dp.

eWB,p.,

all B.

A computation using the above observations gives that if 0

l

w E A oo (J.L), then

wadJ.L '" (WB,J1.t,


0 and an index p 1 such that v(E)/v(B) c(J.L(E)/J.L(B))l/p for each ball B and each measurable set E B. (3) v is a weighted measure with respect to J.L and the weight w satisfies the condition RHr(J.L) for some r > l. (4) v is a weighted measure with respect to r and there is a constant c such that WB,J.L cv(B)/J.L(B) for all balls B.

8

is a weighted measure with respect to It and there is a constant e such that. WB,v eWB,It for all balls B, (6) 1/ is a weighted measure with respect to It and there are a constant e a.nd an index IJ 1 so that I/(E)jv(B) e(lt(E)!Jt(B»P for each ball Band each measurable set E B. (7) v is a weighted measure with respect to It and the weight w satisfies the condition Ap(/t) for some P 1. (8) v is a weighted measure with respect to Jl and v(B)/Jl(B) for all balls B. First some observations. Note that condition (1) is symmetric in v and Jl, and that (2) becomes (6) if we exchange v and u, Also, if v and Jl are mutually absolutely continuous, then (5) is symmetric in Jl and v, and if we exchange Jl and v (3) becomes (7) and (4) becomes (8). Now, without assuming a doubling condition on either JL or v, we have (5)

1/

Lernma 12.

The following diagram is true:

(1)

(2)

(3)

(6) \

(4)

(5)

(7)

(8)

T ,,1

r

/

Also, (2) together with (6) imply both (7) and (8), while (4) together with (8) imply (5). Note that (2) together with (6) imply every other condition and that (1) is implied by every other condition. However, there is no implication emanating from (1). The crucial step in proving the equivalence of the above conditions is (1) implies (2). In this direction we have Theorem 13. Suppose that Jl and v satisfy a doubling condition. Then (1) implies (2) and, by symmetry, (1) implies (6). Corollary 14. Under the assumption that Jl and v satisfy a doubling condition, (1)-(8) are equivalent. On the other hand, if we only assume the doubling condition for u, (6) implies a doubling condition for v, and by the (6) - t (1) and Theorem 13 it also follows that (6) -+ (2). This makes it possible to complete the results of Lemma 12 and obtain

9

Theorem 15. diagram holds:

Suppose Jt satisfies a doubling condition. Then the following

(3)

(7)

(2)

-+

i

{6)

(4)

-+

(1)

-+

(8)

I -+

(5)

r

By means of an example we show below that without further assumptions the above diagram cannot be improved. Nevertheless, if X = R'"; Jt is doubling and the balls are the dyadic cubes of H", the doubling condition on v is not required in Theorem 13 for the implication (1) -+ (2). Indeed, we show that the following holds. Theorem 16. Let X = R", and suppose Jt satisfies a doubling condition and the B's are the dyadic cubes of R": Then the following implications are true:

(7)

(6)

-+

(5)

(8)

-+

(4)

(3)

To see that this result cannot be improved, let X intervals (k2 i,(k + 1)2i ) and put

poex)

0

={ 1

if 0 < x < 1 otherwise,

1/2

Pl(X)= { -1/2

o

(2)

=

(1)

R with the dyadic if 0

x

1

otherwise.

Now, if dJt(x) = (1 + Pl(x»dx and dv(x) = po(x)dx, then (1)-(4) are satisfied, but (5)-(8) are not. If, on the other hand, dJt(x) = (1 - Pl(x»dx, then (1)-(5) and (8) are satisfied, but not (6) or (7). It is possible to avoid assuming that v is doubling in Theorem 13, provided some restrictions are impossed on the family of balls. Specifically, we have Theorem 17. Suppose that Jt satisfies a doubling condition and that the function Jt(B(x,r» increases continuously with r for each x E X. Then, condition (1) implies that v also satisfies a doubling condition. From Theorems 13 and 17 we conclude the following result. 'I'heo rern 18. Suppose that Jt satisfies a doubling condition and that Jt(B(x,r» increases continuously with r for each x E X. Then conditions (1)-(8) are equivalent and Jt and v are mutually absolutely continuous. The last statement of Theorem 17 follows directly from (2) and (6). We pass now to prove Lemma 12; the proofs of Theorems 13 and 17 are given after Lemma 24.

10 Proof of Lemma 12. (2) -+ (1) is immediate. (6) -+ (1) is also immediate once we observe that (1) is symmetric in fL and u, (3) -+ (2) and (7) -+ (6) are a straightforward application of Holder's inequality to the expressions IB XE w dfL and IB XEwl/PW-I/PdfL respectively. As for the implication (2) -+ (3), the statement concerning the absolute continuity is immediate. Let now E).. = {x E B: w(x) > A}. By Chebychev's inequality and (2) it follows that AfL(E)..) :s; v(E)..) v(B)(fL(E)..)/fL(B))I/p. Thus fL(E)..) :s; min(fL(B), v(Bi /(AfL(B)I/pi, oo which, substituted into the expression IB wr dfL = r Jo Ar - I fL(E>.) gives (3) with r < 1 + l/p. (8) -+ (1). Let = 1/4. If fL(E) :s; fL(B)/4, then it follows that

s

o;

fL( {x E B \ E : w(x) Thus IB\E

ui

dp. > 4WB,jJ.fL(B)

WB,Il})

CfL(B) .

v(B)/e, and so veE) < (1 - (l/e))v(B).

(4) -+ (l).Observethatv(E):S; v(B)/2+v({x E E:w(x) < WB,Il}):S; v(B)(1/2+ (fL( E) / fL( B))). (2) -+ (4). The absolute continuity is immediate. Let now E = {x E B :w(x) > WB,Il}; then we have v(B) 2v(E) 2WB'llfL(E). But by (2), fL(E)/ fL(B) > (v(E)/v(B))P [c > 0, and so WB,jJ.fL(B) < ev(B). (5) -+ (4). Let E = {x E B: w(x) WB,Il}' Then we get that fL(B) :s; 2fL(E) :s; 2 IE WdfL/WB,jJ. :s; fL(B)/WB,v, from which (4) follows. (5) -+ (8). Let E = {x E B:w(x):S; WB,v}' Then v(B):S; 2v(E) = 2IEwdfL:S; 2fL(E)WB'1l < efL(B)wB'1l from which (8) follows. That (4) and (8) imply (5) is immediate. We now show that (2) and (6) imply (7). From (2) we conclude that v is absolutely continuous with respect to fL, and from (6) that fL is absolutely continuous with respect to t/, Thus, dfL(X) = (l/w(x))dv(x) and, by (6), (l/w) satisfies a reverse Holder inequality with respect to u, From (3) -+ (2) it now follows that (7) holds. Finally we show that (2) and (6) imply (8). As above we get that J1 and v are mutually absolutely continuous and that that dfL(X) = (l/w(x))dv(x). Also, (6) implies that (2) holds with fL and v exchanged. Since (l/w)B,1l = l/WB,jJ.' (8) follows from the implication (2) -+ (4). • Lemmas 5 and 8 follow from Theorem 15. Indeed, Lemma 5 follows from (6) -+ (2) in that theorem, and we now indicate how (7) -+ (2) there gives Lemma 8. First note that if wE Ap(fL), then w-I/(p-I) E Ap1(fL) for some PI > 1. Thus, w-I/(p-I) E RHr(fL) for some r > 1, i.e.,

(_1_ [

_(1 f w-r/(p-l)dfL < e w-I/(P-I)dfL)r, fL B) JB fL(B) JB

all B.

From this inequality and the fact that wE Ap(fL) we conclude that W E Ap- 1 for which wE Ap(Jl) is an open interval (Pw, 00) or the empty set, i.e., Pw = 00. The number Pw is called the critical index of the weight w for the Ap(Jl) condition. Also, if w E Ap(Jl)nRHr(Jl), P, r > 1, from the observation that w r E ApJJl) for some PI > 1 it readily follows that w E Ap-e(Jl) n RHr+e(Jl) for some E > O. Thus the set of values of r > 1 for which w E RHr(Jl) is an open interval (1, rw) for some number r w > 1 provided that w is in some Ap(Jl) class. The number r-» is called the critical index of the weight w for the RHr(Jl) condition. By the way, the condition w E RHr(Jl) alone does not necessarily imply that w E RHr+e(Jl) for some E > 0 unless we are under the assumptions of Theorem 18. In view of Lemmas 10 and 11 we formulate the above remarks as follows: Lemma 19. Let w be a locally integrable weight, and let I denote the set of those real numbers a such that

Then one of the following three conditions holds: (i) I is empty, (ii) I is an interval with 0 as an endpoint, (iii) I is an open interval (a_,a+) with a : < 0 and a+ > O. In this case a : = -(Pw - 1)-1 and a+ = rw, where Pw and r-» are the critical indices of w. Furthermore, note that under the assumptions of Theorem 18, (ii) cannot occur. Before we proceed to prove Theorems 13 and 17 we introduce a "reverse" doubling condition. We say that Jl satisfies the reverse doubling condition RD d , and we write Jl E RDd, if there are a nonnegative constant k 1 S; 1 and a number d > 0 such that

Jl(B(x,tr» > k 1 t d Jl(B (x ,r» ,

all t

1 and

B(x,r).

In case X is compact we only require (RD d ) for tr S; c. Lemma 20. Assume that there is a constant C3 > 1 such that for all x in X and r > 0, or 0 < r < c if X is compact, we have B(X,C3r) \ B(x,r) # 0. If Jl E D d for some d 1, then also Jl E RDdl for some d l < l. Proof. The conclussion is apparent if there are constants al > 1 and 0 < a2 < 1 such that Jl(B(x,r» S; a2Jl(B(x, air», all B(x,r).

In case X is compact we only need this inequality for 0 < r < c. As for the inequality itself, it follows from D d provided we can find a constant C4 > 1 with

12 the following property: For each B(x,r) there exists a ball B(y,rl) B(x,alr) such that B(x,r) n B(y,r) = 0 and B(x,r) B(y,c4rl). To see that this is the case, put a = 1 in Lemma 1, pick C4 > COC2, and let y E B(X,C3c4r)\B(x,C4r) i= 0. It then follows that C4r < d(x,y) ::; C3c4r, and, by (iii), that (C4r/cO) < d(y,x) ::; COC3C4r. Whence d(y,x) > C2r, and by Lemma 1 we have, on the one hand, that B(x,r) n B(y,r) = 0, and, on the other hand, that B(x,r) B(y,alr), where al = COC2 C3 C4· • It is important to note that there is no implication from RDd l to D«. Also, contrary to the Ap(JL) condition, the set of values of d for which JL E Dd can either be a closed interval {dw , (0) or an open interval (d w , (0) or the empty set, i.e., d w = 00. d w is called the critical value of JL for the doubling condition Dd. A close relation to doubling is the condition B>. for a measure JL. We say that J.t E B>. provided that

L 2->'kJL(B(x,2kr)) ::; cJL(B(x,r)) , 00

all B(x,r).

(B>.)

k=O

Note that if JL E D>.-e for some 0 < e < .x, then JL E B>.. Also, if JL E B>., then JL E D>.-e for some e > O. Furthermore, we have Lemma 21. JL E B>.-e.

If JL E B>. for some

.x >

0, then there is 0 < e


. we get

L ak+k 00

k=O By taking next a sum over k 1

;::::

00

1 ::;

cak1

•

k 2 we get 00

L L a k+ kl=O

k=O

kl

+k2

::;

2

c a k2

•

Repeating this argument m times it follows that

L

k,k1,···,k

ak+k1 +..+k m

::;

cm+1ao .

Tn

Now, given an integer f, let Nm(f) denote the number of (m + I)-tuples of nonnegative integers k, k l , ... ,k m such that k + k1 + ... + k m = f. The above estimate may then be rewritten Nm(f)al ::; cmao.

L e

13

Let 0 < over m

E

< lie. Multiplying the above inequality through by

Em

and summing

0 it follows that

L LE m m

But clearly Nm(l)

eao/(l - sc) .

e

em1m! and so we obtain

L 00

Em Nm(e)

ed

.

m=O

Whence we conclude that

L-J ed ae < _ e I aO .

e

From this we get that, as asserted, JL E B>"-t: for some

E

> o. •

Given two measures JL and v on X, we consider next the relative doubling condition

where e land d 1 are constants independent of the balls, and the relative reverse doubling condition

where e > 0 and 0 < d 1 < 1 are constants, also independent of the balls. If JL(B(x,r)} rv r, the relative doubling condition essentially reduces to t/ E Dd and the relative reverse doubling inequality to v E RDd1" We now show that under some additional assumptions on X and JL, it is possible to find a normalized distance function d*(x,y) with the property that the collection {B*(x,r )}r>O of balls relative to d* coincides with {B( x,r )}r>O for each x E X but with the corresponding radii relabeled and so that JL(B*(x,r)) rv r. For this purpose we construct for each x E X a function r x : [0,00) [0,00) that satisfies the following five properties: (i') rx(O) = O. (ii') r x is continuous. (iii') r x is strictly increasing. 00 as t 00. (iv') If X is not compact, then rx(t) (v') There is a constant e such that rxCt)/e :5: JL(B(x,t)) :5: crx(t). If such a function is available, then d* (x,y) is plainly defined to be r xC d(x,y)). In this direction we have

14

Lemma 22. Suppose jl is a doubling measure defined on X. With r x and d* (x ,y) defined as above, d*( x ,y) satisfies the conditions (i )-( vii) of the function d(x,y) introduced at the beginning of the chapter, the collection of balls {B*(x,r)}r>O coincides with {B(x,r)}r>O, and jl(B*(x,r)) rv r. Proof.

Clearly (i') implies (i) and (iii') implies (ii). As for (iii), first note that

d*(x,y)

= rAd(x,y))

rx(cod(y,x))

cjl(B(x,cod(y,x))) ckcgjl(B(x,d(y,x))). 2d(y,x), then B(x,r) n B(y,r) =1= 0, and by Lemma 1 B(x,r)

Further, if r = B(y,czr). Thus we may continue with the above estimate and obtain

d*(x,y)

ckc&jl(B(y,czd(y,x)))

ckZ(cocz)bjl(B(y,d(y,x)))

(ck)z(cocz)bry{d(y,x)) = Next observe that B(x,t) = B*(x,s) whenever s = rx(t), and that the inverse function r;I is defined on [0,00), or on [O,c x ) in case X is compact. Now, in the compact case, if s 2:: cx, there is a t such that B(x,t) = X = B*(x,rxCt)) for rAt) < C x s, and consequently, B*(x,s) = X = B(x,t). With these remarks

out of the way, properties (iv) and (v) are immediate. To verify (vi) observe that either d(x,y) 2czd(x,z) or d(x,y) czd(y,z). In the former case, by the doubling condition, jl(B(x,d(x,y))) cjl(B(x,d(x,z))) and since jl(B(x,t)) rv rx(t) it follows that d*(x,y) cd*(x,z). In the latter case, by (iii) we have d(y,z) ::; cod(x,y) coczd(y,z) and we conclude that d*(y,x) cd*(y,z); thus d*(x,y) cd*(y,z). As for (vii), it follows at once from the relation

jl(B(x,t))

rv

rx(t). •

Our next step is to construct the function r x . Lemma 23. Assume that jl is a nonidentically zero measure which satisfies both a D d and an RD d 1 condition. Then

rx(s)

= e-(l+s)-l

t'

jl(B(x,st))dt, s

Jl/Z satisfies the conditions (i')-(v') given above.

=1=

0,

rx(O)

= 0,

°

Proof. The continuity of r x away from the origin is obvious and at s = follows since, by the RD d 1 condition, limt-+o jl(B(x,t)) = o. This condition also implies in the noncompact case that (iv') holds. r x is strictly increasing because jl(B(x,t)) is nondecreasing as a function of t and the factor e-(l+s)-l is increasing. As for (v'), it holds since, by the doubling condition, we have jl(B(x,t)) rv jl(B(x,st)) whenever 1/2 s ::; 1. • The only possible implication between the Ap(jl) and Bq{jl) conditions is given by

15

Lemma 24. Suppose v is a weighted measure with respect to J.L with wE Ap(J.L), 1 < p, and that function d is normalized so that J.L(B(x,r» "-' r. Then v E Bp. Proof.

v E DP t/

E Dp •

Since w E Ap(J.L) implies that w E Ap-E(J.L) for some e > 0, and since implies that v E B p, it is enough to show that w E Ap(J.L) implies Let B I B 2 • By Holder's inequality we get E

Thus v E D p • • We remark that given 1 :::; a :::; b :::; 00, it is possible to construct in X = R endowed with the Lebesgue measure J.L, a weighted measure v with respect to J.L such that dv( x) = w( x )dJ.L(x) and the critical indices Pw = band dw = a. Finally we present the postponed proofs of Theorems 13 and 17; first some notations and definitions. If B = B(x,r), let BI = B(x,cr) where the constant c C2. We say that a sequence {Bk} is a telescoping chain of balls if we have

B}

...

Bl ...

We also say that {Fk} is a telescoping sequence of collections of balls, Fk {Bi,kh, k = 1,2, ... , provided that 1. Bi,k n B j,k = 0, i "I i, for each k. 2. For each Bi,k = Bi; E Fk there is a telescoping chain of balls Bl Bl+I ... such that B j E Fj for all i k. Proof of Theorem 17. Suppose that J.L is a doubling measure with constant c in the doubling condition Dd, that J.L(B(x,r» is a continuous function of r, and that (1) holds with constants 0 < c,8 < 1. We claim that for t > 1 we have

J.L(B( x,r»/J.L(B( x,rt»

1/ etd

(1 - c)k,

k large enough.

In fact, k = e' + e"ln t, with e' "-' In c] In(I/(1 - » and e" = d] In(I/(1 - e) will do. Then from the continuity of J.L(B(x,r» it follows that there is a telescoping chain {Bj} consisting of k + 1 balls so that B o = B(x,r), B k = B(x,rt) and J.L(B j-I)/ fl( B j) (1 - s) for j = 1, ... ,k. Note that for each j this last property of the balls implies that J.L(B j \ B j_ I) :::; cfl( B j), which in turn, by 1., implies that v(Bj \ Bj-I) :::; (1 - 8)v(Bj), or V(Bj_I)/v(Bj) 8. Thus v(Bo)8- k V(Bk), or, as we wanted to show, v(B(x,rt» :::; Mtd1v(B(x,r». •

16 Proof of Theorem 13. To prove the assertion we construct a telescoping sequence of collection of balls Fk' k = 1,2, ... , k o, with the following properties: If B l for k 1, s, = B Eo = E, and Ek =

U

U

ee»,

BErk

(a) Once such a telescoping sequence of collection of balls has been constructed the conclusion obtains since, with c the doubling constant for v, it follows that

v(Ek-d

s V(Ek-l n E k) + v(E

E k) (1- (3)v(E k) + V(Ek) - V(Ek) k \

(1- ((3jC))V(Ek).

In fact, this is the only place in the proof where the doubling condition is invoked. When X = H" and the balls are the dyadic cubes, then B l = Band the doubling condition on v is not required. Returning to the proof, from the above inequality we get

v(E)jv(B)

cv(E)jv(B)

(1 - ({3jc))k O ,

and the desired result follows from this estimate because, as we show below, k o is of order (In(J.L(E)jJ.L(B))jln(cjc)). So, it only remains to construct the Fk'S and to estimate ko. First observe that there is r > 0 such that for x E B, B ( x, r) B 1 and J.L(B) c3J.L(B(x,r)). Thus, with C4 a constant to be determined, and if E B has measure J.L(E) cJ.L(B)j(C3C4) and x E E, then we can find a ball B(x,r) such that

By the density property of the measure J.L, which holds since by assumption functions in Lloc(J.L) differentiate the integral u-e.e., for each x E E except possibly r with the for a subset of J.L measure zero, we can assign a ball B(x,r x ) with r x property that for some constant Cs > 0

Let now F l = {B j} be a pairwise disjoint subfamily of these balls such that each B(x,rk) is contained in B} for some B j E Fl. We now proceed recursively: Having selected Fj for j = 1, ... , k - 1, if J.L(E k - 1 ) cJ.L(B)j(C3C4), we construct F k in the same way we constructed F 1 but replacing E above by Ek-l. This process

17

goes on as long as the above inequality holds. That is, if Fko is the last collection we construct in this fashion, then we have

We verify that {Fk} is a telescoping sequence of collections of balls. First, by construction the balls in Fk are pairwise disjoint, k = 1,2, ... , ko. Also, if B(X,T) E Fk-l, there is a ball B(X,Tx ) with Tx T so that (a) holds with E replaced by Ek-l. Since B(X,Tx ) was competing in the covering argument used to select F k, we have B(X,Tx ) for some Bk E Fk; to conclude that B1(x,T) we need to know whether Tx > COT. That this is the case follows from the doubling condition since

Bl

Bl

provided that we pick C4 large enough. Since J.l and (b) we conclude that (a) holds.

1/

satisfy condition (1), from

Finally we estimate ko. Since the balls in Fk satisfy (b) with E replaced by E k- 1 there, we get that J.l(Ek) ::; J.l(E k- 1 )C4CS/E. Thus, by the doubling condition,

ko

(In(J.l(B)/J.l(E)))/ln(M C4/E). •

Sources and Remarks. That the condition A p is relevant in the study of the weighted inequalities for the Hardy-Littlewood maximal function is due to B. Muckenhoupt [1972]. The basic properties of the A p weights were established by B. Muckenhoupt [1974], F. Gehring [1973] and R. Coifman and C. Fefferman [1974]. In the generality given here the theory was considered by A. P. Calderon [1976], whose approach we have adopted, and by R. Coifman and G. Weiss [1977]. Lemma 4 is due to R. Kerman, cf. R. Kerman and A. Torchinsky [1982], and Lemma 10 was established independently by R. Wheeden. Examples showing that aside from the implication in Lemma 24 the various conditions discussed are independent from each other were constructed by B. Muckenhoupt and C. Fefferman [1974] and J.-O. Stromberg [197gb].

CHAPTER

II Decomposition of Weights

This chapter is devoted to proving a factorization of Ap{fL) weights with control on the doubling condition they satisfy. We assume that X, fL and dare as in Chapter I and that fL(B(x,r)) '" r. In addition we assume that there are nontrivial a-Lipschitz functions defined on X; we make this condition precise. Whenever B denotes the ball B(x,r), we let B k denote the ball B(x,ckr) where the constant c is so large that every ball with radius rl ::; 2r that intersects B is contained in B I . Further, we assume that c is so large that there are a function a(x) with support contained in B I such that 0 ::; a(x) ::; 1, a(x) = 1 for x E B, and a constant M so that la(x) - a(y)1 ::; M(d(x,y)/ry\ a > O. a is called an a- Lipschitz fun ction. As for the decomposition, it is given in Assume v is a weighted measure with respect to fL with weight > 1. Then we can write W = where WI and W2 are AI(fL) weights. Furthermore, if v E RD d 1 n D d2 , 0 < d l < 1 < d2, and if e > 0, and dVI(X) = wI(x)dfL(X) and dV2(X) = w2{x)dfL(X), then VI E RD d 1 - E and V2 E Dd2 + E Theorem 1.

wE Ap(fL) n RHr(fL) for some p, r

We remark that if WI, W2 E Al (fL), then by Holder's inequality it follows that E Ap(fL). Theorem 1 is a strong converse to this observation. The proof of Theorem 1 is quite intricate and is achieved in a number of steps. First we decompose W in a large fixed ball B o, say, and then the general result follows by an easy limiting argument. Also, it is convinient to work with = -. We WI

19

build up the functions ¢+ and ¢- from nonnegative, respectively nonpositive, Lipschitz functions supported on certain collections of "red" and "blue" balls. This is how it goes. Given an arbitrary real-valued function ¢, let (e >'(2m + I)} n B o fm(x)= { 1 J-L-a.e.on E;;" = {Y:¢(Y)-¢B o .(2m-l)}nBo. This observation is a consequence of the following lemma and the corresponding blue version. Lemma 4. Suppose Xo is a Lebesgue point of the set {y: ¢(y) - ¢B o 2: O}. Then there is an integer j(xo) such that Xo rt. B],m for all Bj,m E Bm with j 2: j(xo). Moreover, if Xo is also a Lebesgue point of the set E;t, then j(xo) can be chosen so that j(xo) jo for some Bjo,m E R m with Xo E Blo,m'

This lemma will be proved later on. Now set

f= (fo+ m#O

= ( L

=

f;t

r + I>,

+ LU;t m'. Finally we set

>. + ( L

i: +

m>O

x

= r +f- +9+¢Bo ' From the above properties

Verification of the Properties of ¢+ and ¢-

Since the verification can be carried out in a similar fashion for both functions we only discuss ¢+ here. First we consider (a) and (c).

23 Let B be a fixed ball contained in B o; we may assume that B is much smaller than B o. Note that + = A aj,m .

I: I: m

'R",

We split the balls in R rn \ {Bd} into three families, to wit: 1. Those Bj,rn's such that BJ,rn n B = 0. Since aj,rn = 0 on B these balls can be disregarded. 2. Those B j,m 's with radius r j,m > r = radius of B; we call these the large balls. 3. Those Bj,rn's contained in BJ; we call these the small balls. Thus we may further write +(x) = !(x) + }(x), where ! and } are the sums of the aj,rn's extended to the large and the small balls, respectively. The following two estimates hold:

I!(x) - !(y)!

CA

all

(d)

x,y E B,

and

(e)

(a) and (c) then follow from these estimates, with f3I replaced by f3I - c. Here are the lemmas needed for this purpose, they are proved at the end of the chapter. Lemma 5. If B I = B(x, rl) is a red ball nested with a ball B = B(y, r), then rl cre- 2A")'1 ; if it is a blue ball, then rl cre- 2 A")'z . Also Bf B 3 • Lemma 6. Suppose n, E R m , B; E R rn 1 and s, 1: BJ 1: B;. Then, (i) If m 1: ml, one of the balls is nested with the other or Bl n B; = 0. (ii) If m = ml, one of the balls is nested with the other or B; n B j = 0". From Lemmas 5 and 6 we get Corollary 7. Suppose {Bj} is a family of balls in Um tc; \ {BJ} such that r radius of Bj 2r for all j. Then the balls are pairwise disjoint and L: j XB; c. One of the basic estimates is given by Lemma 8. If {B j } is a collection of pairwise disjoint balls in everyone of which is nested with a ball B, then

I: J1.( B j) j

ce- Afh J1.( B) .

Um tc.; \ {BJ},

24

Lemma 9. Suppose {Bj,rn} C F m is a family of balls of the same color, none of which is nested with another. Then Lj laj,m(x)1 2. Lemma 10.

If rj,rn denotes the radius of Bj,rn E F m , then for all x,y we have

Assuming for the moment that the above results have been proved, (d) is obtained as follows: If x, Y E B = B( x ,r), and if E denotes the collection of large balls in Um R m , then

I'}

{x E B:

t

m

I

2, from the properties

El+t.

Thus, if>' is sufficiently large, (e) follows from (J). This concludes the verification of (a) and (c); we pass now to that of (b). Let B 1 = B(X,Tl) B 2 = B(y,T2) be two balls in B o. As before we only discuss +. First note that from (a) and (c) it follows that

+(x) 2:: and since rl

1>t -

c,

all

x E B2,

cr2 we get

So it remains to show that

+ - 1>B + 2:: - ----=-s(ln 1 1>B r2 -In rl ) - C. 2 1 /2

For this purpose we split the balls in Um R m \ {Bn into four classes, to wit: (1) Balls Bj,m such that B;,m n B 1 = 0. Since the aj,m's may only increase but they vanish on Bl, these balls can be disregarded. (2) Balls which are large relative to B 2 • If 1>! denotes the corresponding partial sum, the estimate

11>!(x) -1>!(y)1

c,

all

x,y E B 2,

is obtained as before. Whence, these partial sums can be neglected. (3) Balls which are small relative to B 1 and which are contained in B}. If denotes the corresponding partial sum, as above we get the estimate

JL({x E

:s

> t})

ce- t({3H/2)JL(Bt}.

Thus c, and since 2:: 0 it follows that 2:: 0, and consequently, 2:: -c. (4) All the remaining balls, namely those balls of intermediate size, larger than B 1 but smaller than B2; let t denote the corresponding partial sum. We show below that

26

n: To- tprovefollows. the assertion in case (4), we split the intermediate size balls into From this estimate, and those for the cases (1)-(3), the desired estimate for 1

collections (ft., f = 1,2, ... of nonnested balls such that each ball in ge, f 2: 2, is nested with a ball in ge-l. First we count how many such nonempty collections can occur. If r(f) is the minimum of the radii of the balls in ge, by Lemma 5 we have

r(f) 2: r(£ - 1)!e2 "Yl A , c

and since rl

r(l) and r(£)

r2 whenever 9f

(e 27 1A/c)f Thus f

(

+ ¢>M(x)

0, we get

cr2/rl'

1

\ ) (In rz - In rl) - c . 21'1/\ - c

From Lemmas 6 and 9 as above we get that 0

o

f:.

< 2:!h aj,m(x)

2. Whence,

1 ) (,1- (c/2,X))(lnrz -InrI - c,

which finally gives the desired estimate provided that ,X is chosen large enough. To complete the proof of Theorem 1 it only remains to prove Lemmas 2-6 and 8-10; we begin with some preliminary observations. Note that, by (a) or (b), if two balls intersect and the difference ofthe median values is large, then one must be much larger than the other. Thus, for instance,

(g) Whence, if Bb is m-blue and B; is m-red, and if n B; = 0, then one ball is much larger than the other. More precisely, if B b is the larger ball, by (b) it follows that (radius of B b ) 2: ce Z7 2\radius of B r ) . From this remark we conclude the first part of Lemma 5, the second part is then easy. By Lemma 5 it follows that the balls in the sequence {B j } in Lemma 2 which are roughly of the same size cannot be nested. Therefore these balls must be pairwise disjoint, and since they are all contained in BJ there can only be a finite number of them. Lemma 2 now follows easily. As for Lemma 3, just consider two cases, namely, when the m-blue ball which makes B 1 nested with B 2 is larger or smaller than B», Proof of Lemma 4. The first half follows readily from Lemma 2 and the definitions of Lebesgue point of density and of median value. As for the last part,

27

let j(xo) be chosen so that B;(xo) is the last rn-blue ball in f m that contains Xo; if no such ball exists put j(xo) = o. By the Lebesgue density theorem there is a smaller m-red ball B; containing it. If B; E R m, then B; = Bj with j > j(xo). On the other hand, if B; f/: R m , by property (ii) in the definition of R m there exists Bj E R m such that B; B], but it is not nested with it. Thus Bj(xo) is larger than Bj and j > j(xo). • Before we proceed with the proof of Lemma 6 we show that if B i E R m and

s, f; BJ, then

(2m

+ l)A -

Cz

< ¢Bj

- ¢B o

< (2m+ l)A - cz/2,

provided that C2 > 0 is large enough. The left-hand side of the above inequality holds by definition. Also, since by Lemma 5 B] cannot be nested by B j, from property (iii) of R m we conclude that BJ is not m-red. Whence by (g) above, the right-hand side inequality is also true. Proof of Lemma 6. We do (i) first. We may assume that Bi is larger than Bj, and begin by considering the case m > ml. By the above inequality B, is m-blue. Thus, if B[ nB; f; 0, the balls are nested. On the other hand, if m < ml, then the ball B[ is m-red but it does not satisfy the right-hand side of the above inequality; thus Br f/: R m • If B[ n B; f; 0, then B} Br, and by the property (iii) ofRm we conclude that Bj is nested with Br. Also, since the involved m-blue Bj is nested with Bi. As for (ii), it is just property ball is much smaller than (i) of R m . • Proof of Lemma 9. Fix x.1f Bj,m, Bk,m E R m are not nested, and if aj,m f; 0 and ak,m f; 0, then at,m(x) 0 for all j i k. We then conclude that k

k

L lai,m(x)1 i=j Proof of Lemma 10.

=L

i=j

ai,m(x)

= Ak,m(x) -

Aj,m(x)

2. •

Observe that the truncations are also contractions. Thus m

IAj,m(x) - Aj,m(y)1

Llaj,m(x) - ai,m(Y)I, i=l

and if rj,m denotes the radius of B j,m and r

= ri,m, by Corollary 7 we get

28

s 2L

laj,m(x) - aj,m(y)1

j=:l 1

s 2 Ld(x,yjrj,mY" (XBl,m (x) + XB;.m (y)) j=:l

m kr)-a 2d(x,y)a L(2 k=:l m

d(x,y)a L(2 kr)-a

(XB7,m (x) + XB;,m (y))

L rj

r

= c(d(x,y)jr)a.

•

k=:l

It now only remains to prove Lemma 8; this is the basic estimate in the construction. As a first step we invoke (a) to prove

If {Bi} is a collection of pairwise disjoint balls contained in a fixed

Lemma 11. ball B, then

and

Proof. Since both inequalities are proved in a similar fashion we only prove the first. By (a) it readily follows that

s 1

e-fh{x}pE'P IJ

P,

and if f is merely measurable and 0 < function Mg::.sf(x) equal sup inf P,((B)) inf{A B

B::>{x}pE'P IJ

> 0: p,({y

E

8

l

B

lf - PIrd P,

)l /r

,

< 1 we let the local sharp maximal

B: If(y) - p(y)1 > A}) $ 8p,(B)}.

31

When v = p, we drop the subscript v in the notation. Likewise, when P is the collection of constants, we drop the superscript P in the notation; these conventions apply to all other concepts introduced in this chapter as well. We study in detail, keeping the applications we make of these concepts in later chapters in mind, three questions, namely, how do the sharp maximal functions behave when: (1) We vary the class P, (2) We consider different values of T and s, and, (3) We have v i- u: We begin by introducing another maximal function which is useful in the sequel. Given 0 < s < 1 and a measurable function i, let

=

sup (inf{A

B:J{x}

> O:p,({y E B: If(y)1 > A})

sp,(B)}).

We then have Assume p, is a doubling measure, and let

2a})

SJ-l(Bi,i-l),

we also have

Next we claim that if If(x) - fBi,i-11 2a, then If(x) - fB o1 > 2a. Indeed, since fey) - fBo = (f(y) - fBi,;_I) + (fBi,;_1 - fB) + (fB - fBo)' it follows that

If(y) - fBol 2: -2a + A - czka - (A - cz(k

+ 1)a) 2: cza > 2a,

provided that Cz 2: 4. Thus,

I::J-l( {x

E

Bi,i-I: If(x) - fBid-11

2a})

i

J-l({x E B o : If(x) - fBol > 2a})

sJ-l(Bo) .

(9)

Whence, combining (7), (8), and (9), it readily follows that J-l(U i BL-I) (cs/(1- s))J-l(Bo). Thus, if = {Bi,i-I E Fi-I : Bi,i-I B o},

:rJ-I

Now, provided that S is small enough, we have csj(1 - s) = quently,

f3k oJ-l(B I)

J-l(E) Pick now

CI

so that

and we are done.

f31/c2 rv


A})

c I::(cIe- C,6)kJ-l({MJ,s!(x) > A/2 kf3}), k=O

A> O.

36

First we determine the constant cfo If X is compact, there is ro > 0 such that B = B(x,ro) = X for all x E X; set then cf = [e- Otherwise, if X is not compact, let x E X and r > 1. Then Jl(B(x,r» c(x)re 1 for some £1 > 0, and by an argument similar to the proof of Theorem 2 we have IfB(x,2r) -

fB(x,r) 1

in(f

yEB x,r)

:::; c

MJ,sf(y)

(I IMo,sf Il U

p

L P( Il)

/

Jl ( B x,r

») lip :::; cf,x r- e Ip 1

•

It thus follows that fB(x,2k) converges to a constant cAx) as k tends to 00; since from the above estimate it also follows that cf(x) is independent of x, this is our choice for C f We estimate the measure of the set E = {y: If(y) - cfl > A} by constructing a telescoping sequence of collections f"k of balls which differs from the previous constructions in that there is a remainder set Fk . More precisely, let f"k and Fk , k = 1,2 ... , satisfy the following three properties: 1. The balls in each of the f"k's are pairwise disjoint. 2. For each ball s, E f"k-1, n, g; Fk, there is a ball s, E r, such that Bf 3. Although there may be infinitely many collections f"k' each ball Bi E f"k is contained in a finite telescoping chain of balls 0

BJ.

BJ

BJ,k+l'

Bi,k'

Bf,k+1 ,

where Bi,k' E f"k', Bi,ko Fko' and k o = ko(B i) depends on s; As usual, let Ek = U B• EF k Bi, and Ek = UBiEFk Bf. Now, suppose that this construction can be carried out for each B j in f"k so that E E 1 and if f"L1 = {B i E r.., :B[ e.;», g; Fk-d, we have

(10) We claim then that the following estimate holds: 00

Jl(E)

L1 k -

1Jl(F k) .

(11)

k=l

Indeed, for each B i E f"k let the "stopping time" t( B j) be the smallest integer ko such that there is a chain B],k B],k+l ... B],ko Fko' where Bj,k E f"k for each k :::; k l :::; k o. Now, for each m 2, put f"k,m = {B j E f"k :t(B j) = m}, and note that we can use (10) to get

Jl( UB;EFl,m Bj) :::; 1Jl( U

B;EF2.m

:::; 1 m - 2 Jl(Fm )

.

Bj) :::;

,(m-2 Jl( UB;EFm.m

B j)

37

Whence summing over m it follows that

completing the proof of (11). It thus remains to construct the collections :Fk and the remainder sets F k . Let Fk = {MJ sf(y) > >./2 kf3 }, where f3 is a large positive constant yet to be chosen. When X is not compact we may have that for an integer k o, F k = X for all k k o and Fk -:I X for k < k o. If this is not the case, i.e., if Fk -:I X for all k we set k o = 00. Let {B(X,Tx)}XEE be a collection of balls such that IfB(x,r,,) - ell> >.; by a Lebesgue density argument such balls exist for x j.l-a.e. on E. By a covering argument we can find a pairwise disjoint sub collection {B j} = :F1 of these balls such that E Ue, EFl B} . Now suppose that :Fk-1 has been constructed and that k < k o. For each B j E :Fk-1 which is not· contained in :Fk-1 we have inf xEBJ MLf(x) >./2 k- 1 f3. We also assume that every Bj E :Fk-l satisfies (12) As in the proof of Theorem 2 it is now possible to find for each B; in :Fk-l a ball such that

B:

provided that B; is not contained in F k - 1 • By a covering argument we get a such that each B: is pairwise disjoint sub collection :Fk, say, of those balls contained in B}, B j E :Fk. It then follows that if B, E :Fk-l, either B, F k- 1 or B, B} for some Bj in :Fk. Furthermore, the estimate (12) holds with k - 2 there replaced by k - 1. Whence from (12), (13), and Theorem 2 we conclude that for each B j E :Fk we have

B:

From this estimate we get (10) and the desired estimate for j.l(E).

•

We continue now with the consideration of the second item in our list at the beginnig of the chapter, i.e., the relations for different values of T and s. Combining the comments following Lemma 1 with Holder's inequality it readily follows that

MJ:: f(x)

f(x)

f(x),

0 < r < rl, 0 < s < 1.

38

It is interesting to point out that the local sharp maximal function controls, in the norm sense, the sharp maximal function. This result follows from the "factorization" inequality of in terms of which increases functions, and which decreases functions, proved in Proposition 4 below. 'First an observation: The argument given in the proof of Theorem 3 also holds locally. More precisely, there is a constant 0 < 81 < 1 such that for each ball B and all measurable functions f, there is apE P such that

MJ.Ll

MJ':,

JL( {x E B : IJ(x ) - p(x ) I > ,\}) 00

{x E B:

c

MJ:: f(x) > '\/2 k(3})

k=O

for all ,\

> 0 provided (3 is a sufficiently large positive constant and 0 < 8

Proposition 4.

T> 0,

There is a constant 0
r(r + ro - l)lro. Then,

42

Proof. Let B = {B j: Theorem 2 also gives

s,

BI and A't(f, Bj) > a}. Note that the proof of

p({y E B:I!(y) - PB(Y)! >,X}

\U

Bi E S

Bj)

s ce-c>.!ap(B) ,

all 'x,a > O.

Now fix a ball B; we want to estimate

in terms of

A

r = V(BI) 1 r ( u.p )r lBI MO.l/.J dv.

By a covering argument we may assume that the B/s are pairwise disjoint. Since we also have that MJ::'s!(x) > ap(Bj)/v(Bj) for all x in Bj, we conclude that with al a number to be chosen shortly,

j

c LP({Bj: v(Bj)/p,(Bj)

>

-.» +

c LP({Bj: v(Bj)/p,(Bj) > ad)

L(p,(Bj)/v(Bj))r1v(Bj)

+

Since w E RHro(p,) it follows that

p,({x E B:w(x) > ad) > p,(B)/2,

whenever

v(B)/p,(B) > al'

Thus, the first sum above is dominated by

cp( {x E B I : w(x) > cad)

cal -ro

s

r

w(x rOdp, lBI cal -ro p,(BI)(v(B I)/ p(B I

It then follows that

Whence by choosing

al we get

= (a/Ard(ro+r-I)(v(B)/p,(B))-(ro-I)!(ro+r-l) Adp,(B)

c(Av(B)/v(B)ar1rO!(r+ro-I).

n

ro

.

43

For

E

> 0 a small number, we set

Q

= AI-EAv(B)/j.L(BY; and we note that

j.L( {y E B: If(y) - pn(y)1 > A}) provided that T2 < rIr/(r

c(Av(B)/ Aj.L(B)r2 fl(B) ,

+ 1). Thus, if in addition r < r2, it

Whence (fl(B)/v(B))A-:(J,B)

cA, and we are done.

follows that

•

Next we consider the dependence of the sharp maximal functions on P; this result corresponds to Lemma 5. In the proof of that lemma there is a constant I > lo(fl), where if A denotes the Lebesgue measure we have lO(A) = o. The proof of the general result follows along similar lines, except that now we must handle terms involving fl(B)/v(B) in our estimates. Thus setting B(x,2 j ) = B j , (16) in that lemma must be replaced by the following estimate: If q.k = (v(Bo)/fl(Bo))((fl(Bk)/v(Bk))A-: and \Ilk = then we have sup

B(x,2 0, emanating from x in the direction of y totally contained in Qt, corresponds to those values of t such that 1. t 1, because y E Qo, and

2. Iy + t(x - y)1

R/2, which implies that t because this is the least favourable case.

(R/lx - yl) + (2r/lx - yl),

c«R/r) + 1) (1/2)(IQtl/IQol)t/n = a, say, the At any rate, for values of t segment described above is totally contained in Qt. Set now pet) = p(x+t(y-x)). Then p(l) = p(y), and sup Ip(t)1 sup I p(y)l·

s yEQl

O sUPO 0, 1

!> 0,

::I 0

= o.

We consider, then, for the values of a and ,\ indicated above and for 0

N>. a p(F,x) • ,

=

(

V

Jr

lh

JR"+l

+

(!>>.

'

a(x - y,t)F(y,t))P(at)-ndy dt

t )

where v denotes the volume of the unit ball in R"; and

N>.,a, blq, or

p

< q and)" > b[p,

Theorem 5. Suppose that v E Db and wEAn 1 conclusion of Theorem 4 holds provided that

ni p ).. > { nip +

(1 -

(b - n)plq

Theorem 6. Suppose v E Db, and that 0 following weak-type inequality holds:


0 and t 1. Given an open

set 0 in R"; we associate with it the set U defined by

Then

(i) fa(lr) = UXEUC fa(x) fl(OC) (ii) If (y,t) E f a(UC), then v(B(y,t)) If (y,t) E I' a(UC), there is x B(y,at). Thus,

Proof.

x

E

v(B(y,t) n O)lv(B(y,t))

= UXEOc fleX). 2v(B(y,t) n OC). E UC with [z - yl
0 let = {f > s} and = {g > s}. If g E 0 < q < p, and

e,

veErs) then

f

Proof.

Cl-( 1)

rs

P

I'

Jo

+

r,s> 0

E Lrn(Rn), and

Since

IIflllq.

IIfllh,

= q JoOO(rs)q-1v(E rs) d(rs), we have

q l°O(rs)q-l ((CI!(rs)P) is tP-1Y(EDdt) d(rs)

= 1+ J,

52

say. Moreover, since J

= c2rqllglllq. , and

the conclusion follows upon setting r

= (Cd C2)1/P(p _ q)-l/P.

•

Proof of Theorem 1. Let O, be the open set of finite measure O, = {Sl,p(F) > s}, and let Us be associated to O, as in Lemma 7. Further let = {Sa,p(F) > s}. Now, for all r,s > 0 we have v(Us) + Note that from Lemma 8 it follows that

f Sa,p(F,x)Pw(x)dx Ju. cab-n(sr)-P f Sl,p(F,x)pw(x)dx e

Joe.

cab-n(sr)-Pp

1 s

tP-lv(Ot) dt.

Also, since M v is of weak-type (1,1) on Ll(v), it follows that

We combine these estimates and invoke Lemma 9 with to get

IISa,p(F)lIh

= cab- n and C2 = cab

ca(b-n)q/Pab(l-q/P)(p - q)-q/PIlSl,p(F)llh

= cab-nq/p(p Finally, when p

Cl

= q we have

q)-q/PIISl,p(F)llh·

53 Proof of Theorem 2. Since q > p, to estimate IISa,p(F)lIh we use the converse to Holder's inequality and compare the integrals

1= [

JRn

Sa,p(F,x)Pg(x)w(x)dx,

where 9 E

and

J= [

JRn

Sl,p(F,x)PMvg(x)w(x)dx,

(Rn) has norm less than or equal to 1. Since

Mg(z)

v

(B/ » [ g(x)w(x)dx, y,at JB(y,at)

all

z E B(y,t)

we have

[ g(x)w(x) dx JB(y,at)

cab

v B y,at

[ g(x)w(x) dx JB(y,at)

cab [ Mvg(x )w(x) dx, JB(y,t) and consequently,

Moreover, since v is doubling, by Theorem 1 in Chapter I, c.

Combining these estimates we get

which gives the conclusion.

•

Proof of Theorem 3. Since it follows along the lines to that of Theorem 2, we only sketch it. Suppose first that q/p < r and set dve(x) = w(xYdx, where o < e < p/ q will be chosen shortly. Since

(v(B)/IBI)e '" ve(B)/IBI

all balls B,

it follows that V e E De(b-n)+n whenever v E Db. Let 0 = (c-p/q)(q/p), and dvs(x) = w(x)Sdx. For 9 E less than equal to 1, we have

[ Sa,p(F,x)Pg(x)w(xYdx JRn

(R n ) with norm

cae(b-n) [ Sl,p(F,x)P(x)Mv.g(x)w(x)edx JRn IIMv.gIlL(q/p)'. '"

wi

54

We choose now

e=

(1 _(qJp - 1») (pJq) , (r 1)

and observe that if wEAr, then (w 5 Jwt:) E A(q/p),(vt:). Whence, from the maximal theorem we get IIM".gIlL(9/p)' cllgIl L ( 9 / p )' c, and the conclusion ",6

follows in this case. When qJp argument. • Proof of Theorem 4.

r, we choose

We do the case q

U/&

E

=

0 and repeat the above

p first. Note that

00

Ih,A(x - y,t)P

fV

L 2- kAPx(lx - yIJ2kt). k=O

Whence, multiplying this relation through by IF(y,t)IP(vi" )-1 and integrating over with respect to it follows that

s Since qJp

00

C

L2-k(AP-n)S2k,p(F,x)p. k=O

1, from Theorem 1 we get that

s

C

(JR"r

r

(

w(x)dx

)1/

q

s c (k"

r

=

C

2- k(AP-n)(q/p)II S2k,p(F)lIhy/q

s

C

2- k(Ap-n)(q/p)2k(b-nq/p)

II S1,p(F)IIL:'.

This sum is finite provided that >.q > p, which is our assumption. The proof of the case q > p, being similar to that of Theorem 5 is left to the reader. • Proof of Theorem 5. consequently, I

=

We only sketch it. We now have that qJp > 1, and

= IIgl,p(F)PIIL':!p

C

L 2- k(AP-n)II S 2

55

By Theorem 3 it then follows that

I:S c( L

if q/p?: r ,

and, in case q/p < r, that

I:S c( L 2-k(AP-n)2 k(1-«q/P)-1)/(r-l»(b-n)(p/q») II S l ,p(F )lI h · In both instances our assumptions imply that the sums converge. Proof of Theorem 6. 0 let

For s

> 0 let

e = {MvC Sl,p( F)q)

•

> sq/2Cd}, and for

k

o, = {Sl,p(F) > 2bk/ qs},

and Uk = {MVXo k > (2Cd2kb)-1}.

Observe that Uk £ for all k. Indeed, if x E Uk, then there is a ball B(y,r), say, which contains x and such that v(B(y,r) n Ok) > (2Cd2kb)-lv(B(y,r)). Thus

f

JB(y,r)

f Sl,p(F,z)qw(z)dz lB(y,r)nok ?: 2kbsqv(B(y,r) n Ok) ?: (2Cd)-lsqv(B(y,r» ,

Sl,p(F,z)qw(z)dz?:

and x E E, We also have

I

= sPv({g;,p( F) :S

C

> s} n £C):S

f lee

F, x )pw(x) dx

L 2- k(AP- n) 1S2k'P(F,x)Pw(x)dx e-

:S

C

L 2- k(AP- n)1 S2k'P(F,x)Pw(x)dx

:S CL2- k(AP-n)+k(b-n) :S c

1

Sl,p(F,x)Pw(x)dx

f JRa

Let now h be the least nonnegative integer such that x E 0%. Then the above sum is of order L:k>h 2- k(Ap-b) 2- h(Ap-b). From the definition of h we have "J

Sl,p(F, x) :S 2bh/ qs,

consequently,

2- h(Ap-b) :S c(Sl,p(F,x)/srq(AP-b)/b.

56

Thus, since p - q(>..p - b)/b

= q when>" = bl q, it follows that

r Sl,p(F,x)P(Sl,P(F,x)/srq(>,p-b)/bw(x)dx =csq(>'p-b)/b r Sl,p(F,x)qw(x)dx. JRn

I::;

c

JRn

Since also

> s} n £) ::; v(£) ::; cs- qIlSl,p(F)1I19w ,

v( the conclusion follows.

•

The results for p = 00 and 0 ::; a < 00 require different techniques of proof and are somewhat simpler. Let Noo,a,oo(F,x) = Ma(F,x) denote the nontangential maximal function, N oo,o,oo( F, x) = M o( F, x) denote the radial, and N>.,l,oo(F,x) = N>.(F,x) denote the tangential maximal function, respectively, associated to F. The following three results then hold. Theorem 10. Let v E Db, 0 < al < az < 00, and suppose that Mal(F) belongs to L'!v(Rn). Then also M a2(F) E L'!v(Rn) and there is a constant c independent of F, al and az such that

.

::; c(az/adb/qIIMal

Theorem 11. Let v E Db, 0 < q < 00, and suppose A > bl q. Then if M1(F) is in L'!v(Rn), also N>.(F) E and there is a constant c independent of F such that

::;

.

Moreover, if>" = b/q the following weak-type estimate holds:

veiN>. > s})::; cs- QIlM1(F)1I19w ,

C

= c(>..) .

Theorem 12. Let t/ E Db, and suppose F(x,t) is continuously differentiable with respect to the x variables in t > O. Let G(x,t) = IV' F(x,t)l, where V' F(;,t)

=

(0:1 F(x,t), ... ,

F(x,t)) ,

is the gradient of F with respect to the x variables. Further, suppose that Mo(F),Ma2(G) E 0 < allaZ < 00, and 0 < p < 00. Then there is a constant c depending only on aI, az, band p such that

II Mal (F) ilL:;'

::;

w

..

if IIMo(F)IILl: ::; IIMa2(G)IIL:;" and

IIMat(F)IIr.::, ::;

ell Mo(F) II L:;' ,

We begin by discussing a preliminary result.

otherwise.

57

Lemma 13. Let v E Db and assume that 0 < al < a2 < constant c independent of F, aI, a2 such that for all s > 0,

Then there is a

00.

Proof. Let x E O2 = {Ma2(F) > s}. Then there is (y,t) such that Ix - yl < a2t and IF(y,t)1 > s; consequently, B(y,at) {Mal (F) > S} = 0 1. We claim that

O2

U

= {MVX0

1

(x) > cd- 1(aI!(al

+ a2»b}.

Indeed, for x E O 2 we have

Whence, by the weak-type (1,1) of M v on L 1 (v ), it follows that

Theorem 10 follows at once from Lemma 13. As for Theorem 11, we note that

Consequently, {N>.(F) > s}

> c2Aks}, and by Lemma 13,

v({N>.(F) > s})

LV({M2k(F) > c2Aks}) c L2 kbv({M1(F) > c2>'ks}).

Suppose now that>' = bjq. We then have

2>.kQs q v( {M1(F) > c2Aks})

c2 k .\ s

u:-, sQ-l v( {M

C [

Whence, substituing in the above sum, and since >.q

v({N>.(F) > s})

s-Q L

s2k.\ s

1(F)

> s}) ds.

= b, we get

[ sQ-lv({M1(F) > s})ds }c2(k-l).\s

cs- Q Il M1 (F )lI h .

58

If instead A > b/q, we have

1

00

q

o

c( L 2

sq-ll/( {NA(F) > s}) ds

k(b-Aq»)

1

00

q

1/( {M1(F) > s}) ds

0

= C IIMl (F)lIiq

w

,

which is the desired conclusion. Proof of Theorem 12. Without loss of generality we may assume that and az = 2. Given 0 < r < land s > 0, consider the set

U

= {M1(F) > s} n {Mz(G)

al

= 1

r-1/ps}.

We claim that for a constant c independent of F, G, r, s, band p we have

I/(U)

cr-b/p//({Mo(F) > s/2}).

If this is the case, we have

1/( {M1(F) > s})

cr-b/p//({Mo(F) > s/2}) + 1/( {Mz(G) > r- 1/Ps}).

Thus, multiplying through by

pSp-l

and integrating we get

+ If I = 2:: J = we set r = 1 above and replace J by I. If, on the other hand, I J, we put r = (I/ J)p2 /(b+p) and obtain the desired conclusion in this case as well. So, it only remains to estimate I/(U). Let MfCF,x) = M1(X[O,kIF,x). Since MfCF, x) increases to M 1 (F, x) as k -+ 00, it suffices to work with Mf(F) instead. Suppose, then, that Mf(F, x) > s. Then there is (y, t) such that Ix - yl nand IF(y, t)1 > s, If in addition M z( G, x) r- 1 / p s, then we have IG(z, t)1 r- 1 / p s for Ix-zl 2t. From the mean value theorem it follows that IF(y', t)1 > s/2 whenever Iy - y'l r 1/Pt/2, and consequently, Mo(F, y') > s/2 for y' in B(y,r 1/Pt/2). Let {B(Yi, ti)} be a pairwise disjoint collection of these balls so that U Ui B 1(Yi, ti). Now, note that for any x E B(Yi, ti) we have

B(x, 2ti(l + r 1/

/2» ;2 B(Yi, tiel + r1/p/2».

p

Select Xi E B(Yi, ti) such that Mo(F, y') > s/2 for Y' in B(xi, r 1/Pti/2). With this choice of Xi we get

59

Since all the constants above are independent of k, we are done.

•

Sources and Remarks. The results in this chapter, as well as those in the rest of these notes, hold, with straightforward modifications in the proofs, in the socalled nonisotropic case. Specifically, let At, t > 0, A ts = AtA s' be a continuous group of affine transformations of H" leaving the origin fixed. If P denotes the infinitesimal generator of At, we have

We further assume that if Ixl denotes the Euclidean norm, we have

and (Px,x) 2 (x,x). Under these conditions, given x ERn, there exists a unique t > 0 such that IAt1xl 1. We then define p(x) to be this value oft, and observe that p defines a translation invariant, nonisotropic if P =J I, distance p(x ­ y). Further details about this metric, and the general outline of this chapter, consult the paper by A. Calderon and A. Torchinsky [1975]. Theorem 6 is due to N. Aguilera and C. Segovia [1977]. As for the the tent spaces, d. R.Coifman, Y. Meyer and E. Stein [1983]. In the case when F( x ,t) = f * ptex) is the convolution of a tempered distribution f with the Poisson kernel Pt of the upper half­space, some of these results are analogous to those of D. Burkholder and R. Gundy {1972], and C. Fefferman and E. Stein [1972]. The reader should consult these authors for further references concerning the unweighted case, i.e., w(x) = 1. For instance, in that setting Theorem 6 is due to C. Fefferman [1970], and was extended to the weighted case by B. Muckenhoupt and R. Wheeden [1974J. A weaker version of the result, in the context of our presentation, is due to A. Torchinsky [1979].

=

CHAPTER

v Extensions of Distributions

In this chapter we consider tempered distributions f E steRn) on R" and their extensions to the upper half-space Rt.+I. Let , ¢ E S(Rn), and suppose that has nonvanishing integral. Then, for all p, 0 < p < 00, and every a > 0 (small) and N > 0 (large), there is a constant c which depends only on , ¢, p, a, N such that for all tempered we have distributions f and each (x,t) E

In applications it is important to relax the assumptions on . A way to go about this is the following: Let k 1, and suppose = (1P) ,... ,(k») is a k-tuple of functions in S(Rn) that satisfy

"" k

sup L.,.,I t>o i=l

.

c> 0,

(1)

With the notation

we have Theorem 2. (a) Let = ((1), ... ,(k») , k 1, be a k-tuple of Schwartz functions that satisfies (1) above and suppose that ¢ E S is such that =

2:7=1 f(

near the origin for some f E e OO ( Rn). Then, for every a, N > 0, there is a constant C depending only on , ¢, p, a, N such that for all tempered distributions f and each (x, t) in R+.+ 1 we have i)

In fact, if ¢ vanishes in a neighbourhood of the origin, the second integral above may be ommited.

62

(b) If now the assumption on 'l/J is replaced by the weaker condition: All Taylor polynomials of -J; at the origin are contained in the ideal of polynomials generated by the Taylor polynomials of ¢ at the origin, then there is a constant C which depends only on ¢>, 'l/J, p, a, N such that for all tempered distributions f and (x, t) in Rn+l

+ '

The following result is also important in applications, d. Chapter VI. Theorem 3. Let ¢>, 'l/J be Schwartz functions on Rn, suppose that ¢> has a non= vanishing integral, and let r denote the cone r = U E R'": = (6,··· (6,e) and lei < Then, for all p, 0 < P < 00, and each N > 0 there is a constant C depending only on ¢>,'l/J,p,N, such that if f E S'(R n) has support contained in r, then for (x, t) E R+.+l we have

In applications we also often deal with functions

f E Vo; in that case we have

Theorem 4. Suppose ¢> = (¢>(1), ... , ¢>(k») is a k vector of Schwartz functions that satisfy condition (1), and let 'l/J E S'(Rn) be a tempered distribution such that DOt-J; coincides locally in Rn \ {O} with an Lfoc(Rn) function, 1 S; q S; 2, for lad S; m, and

for all r > 0, lal S; m. Then, for all p, 0 < p < 00, 0 S; A' < A, 0 S; B' < Band m' S; m - n( - ;)+, there is a constant C depending only on

¢>, v,», A, A', B, B', m, m', such that for any f

is dominated by

C

1 1 00

o

n»

'

E

Do and (x, t)

E

R+.+l, 1F",(x, t)IP

ds \Fet>(y,s)\P(l + (Ix - yl/s))-m s-n min«tls) A' ,(sit) B' )Pdy-. s

Note that we can take m' = m in the above inequality provided that p S; q. Also, we could of course allow the range 2 < q S; 00 and use Holder's inequality in the condition on DOt-J; to reduce it to the case q = 2. This would then require that p S; 2 and that m' < m - n(l/q - lip).

63

Remarks. Theorems 1-4 admit a formulation with weights. Namely, if we replace the Lebesgue measure dy in the integrals above by a weighted measure dv(y) = w(y)dy with respect to the Lebesgue measure and replace the left-hand side by IF",(x,tWv(B(x, t))t- n , then Theorems 1,2 and 4 are true provided that u is doubling, and Theorem 3 is true if w E A=. Of course the constant C now depends on w as well. Also, the integral with respect to the measure ds j s in Theorems 1,2 and 4 can be replaced by a sum at the points s, = ,fJit provided fJ < 1 is close enough to 1 (the choice depending on ¢), and, is any positive number in the interval (13,1], except that in Theorem 2 we must pick, = 1. The index i in the sum runs over those integers such that the Si'S lie in an interval corresponding to the integration interval of the measure dsj S in those theorems. We refer to this instance as the "discrete version" , while the original formulation is referred to as the "continuous version." Finally, the important particular instance of the Poisson kernel P(x) = cn (1 + IxI2)(n+t)/2

in place of ¢ E S(Rn) in Theorems 1,2 and 4 holds as well provided that the extension Fp(x, t) is well defined. As for Theorem 3, it is also true if we replace lim inf by lim sup in the conclusion. Moreover, if '\l P( x) denotes the (space) gradient of P, '\l P( x) = (n

+ l)c n (1 + IxI 2) - ( n-I)/2( Xl, ... , x n ),

note that we may also replace ¢ by '\l P in Theorems 2 and 4. Also, if 'I/J = P, i = 1, ... , n, and 'I/J = P - 7] with 7] E S with integral equal to 1, then 'I/J satisfies the assumptions of Theorem 4 with A = 1 for any B > 0 and m o. Before describing the applications of these results we proceed to prove a somewhat simpler estimate and then, by means of a partition of unity argument in the Fourier transform side (Lemma 6), we carryover the arguments to prove Theorems 1-4. We begin by proving Theorem 5. Let ¢, 'I/J E S(Rn), and assume that 4> is compactly supported and that -J; = f¢ for some r E C=(R n ) . Then for all p, 0 < p < 00, all doubling weighted measures dv(y) = w(y)dy with respect to the Lebesgue measure on E" and every N > 0 there is a constant C depending only on ¢, 'I/J, w, p, N such that for all f E SteRn) and (x,t) E R,++t we have

If * 'I/J(x )IPv(B(x, 1)) :::; C

[ If * ¢s(yW(1 + Ix JRn

yl)-Npw(y) dy.

We may assume that f has compact support and consequently, Ir(y)1 :::; CN(1 + Iyl)-N for all N > o. Therefore, Proof.

If * 'I/J(x)1 = Ir * (J * ¢)(x)1 :::; CN

Ln If * ¢(y)l(l +

Ix - yl)-N dy

(2)

64

for all N > 0, which gives the result in the unweighted case when p = 1. By Holder's inequality we also get the result, in the unweighted case, for p > 1. When 0 < p < 1, and in the general case, we first consider ¢ = '1/;. Observe that since 4> has compact support we can write 4> = f4> with rED. Whence from (2) it follows that

If * ¢(x)1

CN

r If * ¢(y)l(l + Ix inn

yl)-N dy,

N

> O.

(3)

Now, for each N > 0 let

MN,(J,X)

= MN(X) = sup If * ¢(y)l(l + [z v

_ yl)-N

j

for each fixed N, MN(x) is finite for all x or identically infinite. Assume that the former occurs, and note that

If * ¢(y)1

(1 + Ix - yl)N MN(X) ,

all x, yin R n

.

Let 7J(x, y, z) = (1 + Ix - yl)/«1 + Ix - zl)(1 + Iy - zl)). Since 1 + Ix - yl 1 + [z - zl + Iy - zl (1 + Ix - zl)(1 + Iy - zl), we get that sup 7J( x, y, z) Now, from (3) it follows that when 0

(4)

l.

z


(x)[P :::; CN,! [

yl)-Np dy,

x E Rn

,

(6)

with a constant CN,f that depends on N,j, but not on x. From (6) it follows that MN(x) is finite, for

(If * 4>(z)I(1+lx - zl)-N)p

JRn If * 4>(y)IP(l + Ix -

zl)(1 + Iy - zl)-Np dy

JRn If * 4>(y)IP(1 + Ix -

yl)-Np dy.

:::; CN,! [ :::; CN,! [

What this implies is that we obtain (5) for all N > 0 provided the right-hand side of (5) is finite; if it is infinite there is nothing to prove. Note that the constant CN in (5) is independent of f. Now, for an arbitrary 1f; E S(R n ) , from (5), (2) and the definition of MN(X) we get

If * 1f;(x)1 :::; CN' [ MN(X)(1 + Ix - yl)N-N' dy:::; cCN,MN(X)

JRn

:::; cCNlcljP

(in

If * 4>(y)IP(1 + Ix -

yl)-Np dY) liP,

provided we pick NT > N + n. This completes the proof in the unweighted case. As for the weighted case, note that, similarly to (3), we have

1\7(1 * 4»(x)1 :::; CN

in

If * 4>(y)I(1 + Ix - yl)-n dy

(7)

which is also a direct consequence of (2) since we can write (fJ4>/fJXi)(X) = Ti4>, i = 1,... ,n with Ti E D. Let MN(X) be defined as before, and set

mN, (1, x)

= mN(x) = sup 1\7(1 * 4»(y)I(1 + Ix -

From (7) with NT > N we get

y

yl)-N.

66

Thus, mN(x) that If Iy -

cNMN(X). Assume now that MN(X) is finite and let Xo be such 1

If * 4>(xo)1

"2MN(x)(l + Ix - xol)N.

xol < 1 we see that IV'(J * 4»(y)1

mN(x)(l + [z - yl)N cNMN(x)(l + Ix - xol)N 2NCNMN(X)(1 + [z - xol)N.

Thus there is a ball B(xo, TN), 0 < Tn < 1 such that 1

If * 4>(y)1

4MN(x)(1

+ Ix -

xol)N,

Y E B(xo, TN),

and consequently, it follows that

[ If * 4>(y)IP(l + [z - xol)Np(l + Ix - yl)-Nopw(y) dy

JRn

(MN(x)j4)P [

JB(xo ,rN)

(1 + Ix - xol)Np(l

+ [z -

yl)-Nopw(y) dy

CMN(X)P(l + Ix - Xo I)(N-No)p v(B(xo , TN)) x v(B(x, l))jv(B(xo, 1 + Ix - xol)) , where in the last step we used that B(x, 1) for some b, we have

B(xo, 1 + Ix - xol). Since v E Db

V(B(XO,TN))jv(B(xo, 1 + Ix - xol)) Hence if No

c(l

+ Ix -

xol)-b.

N - b[p we get

1f*4>(x)IP

MN(X)P CN

1))

kn If *

4>(y)IP(l + Ix - yl)-NoPw(y) dy,

(8)

provided MN(X) is finite. As before this estimate gives

If * 4>(x)IP

CNo,v

for x ERn. Whence, if N dominated by

1))

kn If *

4>(y)IP(l

+ [z -

yl)-Nopw(y) dy,

= Nob/p, it follows that (If * 4>(z)l(l + Ix -

zl)-N)p is

CNo,/(1 + Ix - zI)Nqv(B(z, 1))

x

f If * 4>(y)IP«l + Iz -

JR"

CNo,f

1))

In If *

yl)(l + Ix - zl))-NoPw(y) dy 4>(y)IP(1 + Ix - yl)-Nopw(y) dy.

67

We conclude then that MN(X) is finite whenever the left-hand side of (8) is finite and N = No + b/p. Thus (8) holds with N = No + b/p for all No > 0, and the constant CN is independent of f. The estimates for an arbitrary 'ljJ E S(R n ) can be obtained exactly as in the unweighted case (p < 1), and is therefore left to the reader. • Proof of Theorems 1, 2 and 4. The proof of Theorem 5 was based on the simple identity ¢ = f¢. Now we will use a more complicated identity involving (4)tf( (4)(1), ... ,4>(k»), k 1, is a k-tuple of Schwartz functions which satisfies condition (1) above. Then, there are numbers (30,Rl,Rz, o < (30 < 1, 0 < R 1 < Rz, such that for every (3 E [(30,1), there is a k-tuple 'T/ = (1P),·.·, r/ k ») of Schv.:rartz functions with supp r](i) { has nonvanishing integral, (30 and Rz can be chosen arbitrarily small. Let sr:' = E R" : = I} denote the unit sphere in R", By (1) and the continuity of ¢, to each 0 and i j = 1, ... , N, there corresponds an integer h such that f3 t E I h. Let R2 > R 1 > 0 hi (2Rt, Rz/2], and let a E 1), a 0, be such that a = 1 be such that on (2Rt, R2/2] and vanishes off (Rt, R2]. Then we get k

L

L Iii)

00

i=-oo

=

c>

o.

i=l

It is readily seen that £.1 E COO(R n ) is homogeneous of degree zero, i.e., £.1(f30 = ")

,(i)

Set now fJ ( t (0 = a(Orf> (0/£.1(0, 1 i k. We leave to the reader to verify that the k-tuple TJ = (TJ(1), .• • ,TJ(k) satisfies the desired properties. • Proof of Theorem 1.

,

By the continuous version of Lemma 6 we see that

:::: 1 -

l

a

ds ,

,

o

s

< a < 00,

0

is a function in, Vj also, by letting R 2 in that lemma be small enoue;h, w!! get that supp ( 1rf>(aU2)1 c} for some c > o. It also follows that E 1) for any r E (a/2,a]. Let iJ E Cg='([a/2,a]) be a nonnegative function so that 00 iJ(s) ds] s = 1, and set

10

= ¢(O (fJ(sO + iJ(s)( xi)/4>(sO) .

f(s ,s) Then we have the identity

'I/J( ) :::: A

l

a

o

ds f(s ,s)rf>(sO , A

s

which is the substitute to the identity o < s a and a multi-index a we have c a ,N( 1 +

+

¢

=

f4> in

(9)

(10)

Theorem 5. Then, for every

s-laIXIRl,R21(1 1) +

Ca,NsN-laIXIO,2R2] (I I) , for all N > O. It thus follows that the in verse Fourier transform of f( the estimate

s) satisfies

69 From (10) we get

F.p(x,t)

l = 1 =

a

o

(Tts(-,S)

ds

* F",(·,ts»(x)-

s ds (Ts(·,slt) * F",(·,s»(x) - , o s at

and consequently, it follows that for every N

> 0, (11)

Since for 0

0,

From the discrete version of Lemma 6 we get that

L

(0 = 1 -

(¢U3 j O, r,(j3 i O)

j,(3;

belongs to V, and if a is large enough, then supp ( identity -if; = (f, ¢) holds. Whence,

L

-if;(O =

ca}, where the

¢(j3i O) +

¢(O)

i,(3;

where for i

= 1, ... , k,

'(t:.) = ('(1)(t:') '(k)(t:...,J.)) , T ... ,J , ... ,T

T ...,J

By estimating the Schwartz norms as before we get

IF,p(x,t)1

s c» L

j3iN

1

I F (y, j3 i t )l(l

n»

i,(3;

+ (Ix -

+ CN f

JRn IF(v, t)l(1 + (Ix -

YI/t))-N(j3 it)-ndy Yllt))-N en dy.

Now we define for N > 0

MN(X,t) and

=

sup 1F(y,j3 it)l(l s.e! ::;..

+ (Ix -

mN(x, t) = sup 1(j3it)\7 F(Y, j3 it)l(l

vllt))-Nj3i N,

+ (Ix -

YI/t))-N j3iN .

;,p; ::;..

Here, as usual, \7 F denotes the gradient in the space variables alone, and

Note that locally we can write

, = fi¢(i)

with fi E coo(R n). Thus, we

can use estimate (13) with tl\7 F(x, t)1 on the left-hand side, and with a > 0

71

arbitrarily smalL Whence, it follows that mN(x,t) cNMN(X,t), and with an argument similar to that in the proof of Theorem 1, we note that

IF1/J(x ,t)p1

cMN(x)P

L. + r» L.

1F.(y, Ii; t)I'( 1 + (Ix - YI/t))-Npli;Np

S v( Bet., t))

dy

1F.(y, t)IP (1+ Ix - yl)-Npw(y) dY).

Since the Schwartz norms of the 7](i) 's in Lemma 6 are uniformly bounded in {3 in the interval we can find a constant CN such that the above estimate holds for all {3 E [{3o, By integrating over (3 in this interval, the continuous version of Theorem 2(a) obtains. As for the proof of part (b), we need the following result. Lemma 7. Let and 'l/; be as in the hypothesis of Theorem 2 (b), and let N be a positive integer. Then there is a bounded vector-valued function h N with compact support on such that the function 'l/;N defined by

belongs to S(Rn), and

-

near the origin.

Let WO,WI, ••• ,WN denote the dual basis to 1,t, ... ,t N in the subspace of polynomials of degree less that or equal to N in L 2([1, 2]), i.e., for 0 i. k N we have

Proof.

r I, 2

W

·(t)t k dt 3

= {1

0

otherwise,

j

=k

denote the Taylor series at the origin of ¢ and Let Lj Pj(O and Lj respectively, where Pj = (Pj l, ... , Pjk) and similarly Qj are homogeneous polynomials of degree i- Then,

where the error term E(O satisfies for near the origin. According to our assumptions there are vector-valued polynomials qi = (qil,'''' qik),

72 i = 1,2, ... , N, such that the Taylor expansion of be written N

N

j=O

j=O

1/J of order N at the origin can

2:Qj(O = 2: 0 and f E ii; Here {3 E [{30, 1) is as in Lemma 6, and I is any number in the interval [1/2,1]. Taking then the inverse Fourier transform,

74 where

is the inverse Fourier transform of

l/q + 1/r/ =

=

Since

by the Hausdorff-Young inequality we get, with 1 ::; q ::; 2 and 1,

/1(1 + [xlr

::; C ::; C

=C

L

II Da (ij(i)'if;(t.») IlLq

L

(1

L

t 1al

(en

1

::; cmin(tA,t- B ) . The time has come to introduce some notations that will simplify the formulas that follow. Namely,

m(t,A,B)=min(tA,C B ) , We then have

* Ftf,{i)(.,S)(x)1 =

IL..

and

7J(y,N) = (l+lyl)-N.

- y)/S)F 0 and any

We then have

1F",(x,t)I P ::; c

f

m({3,j(Bq - c),j(Aq - s)

i

-

u, mq) dy.

75 Now, when p> q, from Holder's inequality it follows that

L 00

IF",(x,t)IP

c

m((3,j(Bq - c'),j(Aq - e)

i=-oo

x [ IF",(y,,(3it)IP7J,ait(x - y, mp -

JR "

On the other hand, when p N, and obtain

IF",(x, t)IP

c( L

(n

+ c)(l/q -

l/p»dy.

< q we invoke Theorem 2 instead, with some large

00

m({3,j(Bq - c:),j(Aq - c»7J,ait(x - z, mq)

i=-oo

Now we use Minkowski's inequality on Lq/p-norms for functions of the variables X Z, and observe that

(z,j) E H"

where

G(h,(3, t, x, y)

=

h+ho

L

m((3,j(Bq - c),j(Aq - c»g(x, y, (3, t,j)

i=-oo

and g(x, y, (3, t,j) equals

[ (1 + (Ix - zl/{3it»-m q((1 + (Iz - y!/(3it»-NP(3(h-i)NP)q/P({3itrn dz

JR"

c(3(h-j)Nq(l + (Ix - yl/(3it»-m q.

Here we used the elementary estimate

(1 + (Ix - zl/(3it»-m q(l + (Iz - y!/(3it»-Nq c(l + (Ix - yl/(3it»-m q(1 + (Iz - yl/(3it»-Nq-m q, and the fact that

[ 7J,ai(Z - y,Nq + mq)dz

JR"

c.

76

Consequen tly,

»-m

G(h,{3, t,x, y) :s; em({3, h(Bq - £),h(Aq - £»(1 + (Ix - yll{3h t

q

•

Thus, finally we get

1F",(x,t)IP

:s; e

f

m({3, h(Bq - £'), h(Aq - c'»

h=-oo

1 R»

IF,p(Y, S)IP 17,6h t ( X

-

y, mp) dy,

as we wanted to show. As for the weighted case, we use (14) and an estimate on some quotients involving t/, Indeed, if v E Dbo n RDbl' 0 < b1 :s; 17 :s; bo < 00, then

v(B(x, t»/t n/v(B(z, r»/r n

:s; e(l + (Ix - yl/ max(t, r»)bo-b 1 max ((r/t)n-b 1 , (r/t)-(bo-n)) . A similar argument to the one used in the unweighted case shows that

v(B(x, t»IF",(x, t)IP

L 00

:s;

Ce

m({3,h(Bp - (n -

bd - £),h(Ap -

(bo - n) - £»g(h,x,t) ,

h=-oo

where now

g(h,x,t)

= f

JRn

IF,p(Y,1'{3h t)I P( l X

+ (Ix -

yl/{3h t »

- m p + ( n + e)( ; - l )+

(1 + (Ix - yl/ max({3it, t»bo-b 1 w(y) dy .

To get the continuous version of Theorem 4 we only need to integrate the discrete version over l' on an interval, [1/2,1] say. This completes the proof. • Proof of Theorem 3. Assume first that f E S( Rn) is such that J has support contained in [0,00); we claim that for

(y,£)a(sy)'l/Jt(x - y) dy

converges to JR" F(y,£)'l/Jt(x-y) dy as convergent. Furthermore,

S""-+ 0, since this last integral is absolutely

which converges, since 1 * 'l/Jt E coo(Rn) is dominated by Cj,1J;,t(1 + Ixl)N for some N, to 1 * 'l/Jt(x) as E tends O. It then follows that, as asserted,

As for the weighted version, we use Holder's inequality and the fact, discussed in Chapter I, that {

JRn

W(y)-l/(P-l)(l

+ (Ix -

yl/t))-Nt-n dy

cv(B(X,t))-l/(P-l) ,

whenever w E A p and N is sufficiently large. • The first application of these results are estimates between some of the functions introduced in Chapter IV. Theorem 8. Let v be a doubling weighted measure with respect to the Lebesgue measure on H"; dv( x) = w( x )dx, and suppose (x,t) as t ---t O. We say that Ft/> converges nontangentially to L at Xo, if for every r > 0 lim

sup

IFt/>(Y, t) - LI =

O.

Analogously, we say that Ft/> converges radially to L at Xo, if lim 1Ft/>(xo,t) -

t_O

LI == o.

We then have the following results. Theorem 12. Let 0 < p < 00, .A > 0, and suppose cf> E S(Rn). If Xo in Rn is such that gt,p(Ft/> , xo) < 00, then F and 'l/J are Schwartz functions, 'l/J has nonvanishing integral, and .A > O. Suppose Xo is a point in Rn such that N>.. (Ftj>, xo) < 00 and so that Ft/>(x,t) converges nontangentially to AIR" (x, t) converges radially at Xo. Then, Ftj>(x, t) converges nontangentially at Xo. We remark that Theorems 8-14, with the exception of Theorem 10, hold with Ft/>(x, t) and F,p(x,t) replaced by Ft/>,r(x, t) and F,p.rCx, t) respectively, where

Ft/>,r(x,t)

= X[O,rJ(t)Ft/>(x,t).

Sketch of the proof of Theorems 8-14. Since the proofs follow rather easily from the estimates obtained above, we only sketch them; Theorem 11 requires an additional argument. Proof of Theorem 8. Since IF,p(Y, s)1 Mo(F,p, y) we can apply Theorem 1 (or 2) integrating this estimate first over s. Then, since v is doubling, the conclusion follows at once from (the weighted version)

82

Proof of Theorems 9 and 10. We use now the estimate in Theoreml, 2, resp. 4, in the definition of As in the proof of Theorem 4, the desired conclusion follows using Minkowski's inequality. • Proof of Theorem 11.

For e > 0 let

FE:(x, t)

=

inf IF",(y, t)l. ly-xl. > O. Then, there exist a number 0. > 0 and a constant ex > 0 such that if Xo is a point in R n for which then

xo)

xo),

0

..

The same conclusion holds with replaced by N>. and gp replaced by M o above. Assume for the moment that the lemma has been proved. Observe, then, that

Whence, from results in Chapter IV we get that

and consequently, we conclude that, except possibly in a subset of R" with v measure 0, we have that g;,p( FE:, x) < 00 for all c > O. By Lemma 7 it then follows that, except on a set of v measure 0, g;,p(F,x) c>.gl,v(Fe>.,x). Thus,

which is the desired conclusion. Since the case with gl,p and gp replaced by N>. and M o respectively, is proved in an analogous fashion, it only remains to prove the lemma. Proof of Lemma 15. 1. By assuming that

MN(X, t) =

We recall the proof of the weighted version of Theorem sup

yER" ,{3j ..(F..,p(F$,xo) < 00,0 < e < cN/2, provided N is large enough, the exact choice depending on the tempered distribution f. This means that MN(x, t) is finite for all (x, t) E Thus,

and ex does not depend on of the lemma. •

f nor does it depend on xo. This completes the proof

Proof of Theorem 12.

If

tends to 0 as r

-+

< 00, then

O. As in Theorem 6 we have

and consequently, N >..( F'1 > >., then N>"l(F..p,r,xo) is dominated by

CN>"l (Fht

sup

+ (Ixo -

YI/t))>"l

(1 + (Ixo - YI/t))->"l+>"

IF"l->")N>..(Fq"xo)+

+ (Iy -

xol/t))>"l

sup

IF,,,(y,t)l.

Iy-xol 0, is essentially due to C. Fefferman and E. Stein [1972], where the support of the measure J.L is contained on the lines {(y, t) E R++l : t = 2- k , k = 0,1, ...}, and to A. Calderon and A. Torchinsky [1975] where the support of the measure J.L is contained in the upper half-space. The latter authors also introduced the condition (1) to replace the single function with nonvanishing integral. The estimates in Theorems 1-6 are extensions of the mean value properties of harmonic, subharmonic and temperature functions, where somewhat sharper results can be obtained.

CHAPTER

VI The Hardy Spaces

Previous chapters were preliminary in the sense that they covered some of the basic results leading to the theory of the weighted Hardy spaces. Another important property, namely, the fact that the "norm" of a distribution in these spaces can be computed in various equivalent ways, is the content of this chapter. Given a tempered distribution IE S'(Rn) and a Schwartz function E S(R n) with nonvanishing integral, let F4>(x,t) = 1* t(x). Assume that v is a doubling weighted measure with respect to the Lebesgue measure in H" with weight w, let a > 0, and for 0 < P < 00 set

= {I E S'(R n): M a(F4» E

1I/11m; = IIMa(F4»IIL::' .

We leave it up to the reader to show the basic topological properties of these spaces; for instance, with the distance function d(f,g) = III becomes a complete metric space when 0 < P S; 1, etc. '" Now, according to Theorem 10 in Chapter IV, is equivalent to 1I/11m; for all b > 0, and also for b = 0 under the assumptions of Theorem 12 in that chapter. Furthermore, Theorem 6 in Chapter V readily implies that the same provided that 'l/J is a Schwartz function with nonvanishing is true for integral. Indeed, let 0 < Po < P and note that

=

S;

Since v is doubling and pjpo > 1, by Theorem 3 in Chapter I, we see that the above expression is dominated by

.

= cI/Mo(F4»I/L:;',

86 and consequently, M 1(Ft/J) E By exchanging the roles of

1 and -1 < a - b, all conditions are met. The situation is altogether different when w E A p • Theorem 1. Suppose I is a tempered distribution in w E A p , for such that F",(x,t) = 9 * 0, and x in Rn, put

e=

E

).,MO,s(Sb ' (G))(z) < 8)'}.

Then we can choose 8, s independent of

E, T,).

and B(x,T) so that

1£1 :S "lIB(x,T)I· Since subtracting a constant from F leaves both M:,T (F) and unchanged, we may assume that F(x,T) = O. If E = 0 there is nothing to prove. Otherwise, since for z E E we have M:,T(F, z) > ),/2, to each z E E we can assign (Yz, t z ) E such that IF(yz, tz)1 > ),/2. Since t z :S T we can find a sequence {B(Yk, atk)} of pairwise disjoint balls corresponding to points Zk E E so that Proof.

e

UB 1(Yk, at k) '

(2)

k

Moreover, since for each k we have B(Yk,atk) B(x,(a + l)T) and tk > > 0, there can only be a finite number N, say, of such balls. Also, since

E

90

Jl} 2 {Mo,s(S;/2,2T(G» Jl} for each Jl > 0, for z E E we have S;/2,2T(G,z) 6)... Given 0 < t < T, let n(y, t) denote the cylinder {(y', s) E Rt-+l : Iy - y'l < (a + b)t/2,t/2 s t}. By the mean value inequality for the derivatives of solutions to the heat equation we have

(3) A moment's thought will convince the reader that if z E E and (z). Therefore from (3) it follows that

Iy - zl < at, then

n(y, t)

CbnJ [

G(y,t)

(z)

G(y', s)'l(bs)-ndy' ds

Cb nS;/2,2T(G,S)2

s

e(6)..)2,

(4)

where the constant e depends on b, n, but it is independent of f. Let now Ta.(Y, t) denote the tent, or inverted cone, {(y', s) E Rt-+l : IY' - yl < aCt - s)}, OTa.(y, t) denote its boundary, and (J denote the surface area in Rt-+ 1 • We claim we can choose 6 > 0 sufficiently small so that

Indeed, let y E B(z,aT), z E E. By the mean value inequality, with the left-hand side replaced by higher order derivatives of F such as s2llF(y, s), and (3), it follows that for t 1 / t2 rv 1,

(6) (z) and z E E, we have

Also, if (y, t), (y', t) are in

IF(y, t) - F(y', t)1 = Iy

y'l

111 (tV F(y' + s(y - y'), t), y

1 1

G(y'

+ s(y - y'), t) ds

y' )dsl

e(ly - y'l/t)( 6)..).

(7)

We combine these estimates. Since IF(Yk' tk)1 > ),,/2, from (6) it follows that IF(Yk,t)1 > ),,/2 - e6)" for tk/t rv 1 and, by (7), IF(y, tk)1 > ),,/2 - e'6).. whenever

91

(y, tk) E r:/

2,2T

(Zk)' Therefore,

IFI >

>./3 in a fixed proportion of aTa(Yk,tk), independently of k. Whence, since IB(Yk,atk)1 a(aTa(Yk,tk)), (5) holds. We are now ready to estimate 1£1. Let "-J

U=

2e/3,3T/2 (Zk)) \ (UN ( UN r(a+b)/2 k=1

(

T; Yk,tk

k=1

») .

The boundary au of U consists of two parts, namely

o.u = and

{(V,s) E aU:2 /3

aU2 = {(V,s) E au:s

s

3T/2} ,

= 3T/2}.

By (2) and (5) we conclude that

1£1

c>. -2

r

JehU

IF(Y, tW da(y, t) .

(8)

Next we estimate the right-hand side of (8). With no loss of generality we may assume that F is real valued. We begin by considering the integral JJu G(y, t)2 Observe that if (y, t) E there is a ball

B(Zk' cat), with c sufficiently small, so that (y, t) E (y') for every Y' E B(Zk, cat). Since (G))(Zk) 15>', it follows that

h>'}1

I{y' E B(zk,cat): S;/2,2T(G, V') Thus, if £1

(1 - s)IB(zk' cat)1

= (1- s)c(at)n.

= {y E B(x, T): S;/2,2T(G, y) < h>'}, we have

11

u

G(y,t)2dydt t

c

r S;/2,2T(G,y)2dy

J£1

c( h>.)2IB(y, T)I.

(9)

On the other hand, from (4) it follows that

G(y, t)

cb). ,

whenever (y, t) E U.

(10)

Moreover, since F(x,T) = 0, again from the mean value inequality, we conclude that

IF(y, t)

92

Let n(y, t) denote the normal unit vector to U at the point (y, t) of the boundary = (0, ... ,0, -1) on oUz, and n(y, t) = (nl(y,t), ... ,nn+l(y,t)) with nn+l(y,t) ca- l on oUI. Thus,

ofU; n(y, t) is defined er-a.e, Clearly, n(y, t)

du(y, t) :S cady,

a.e. (y, t) E oU .

(12)

Let Ut = Un {(v, t) E :t fixed}, and let dTt(y, t) denote the surface measure on the (n-1 )-dimensional boundary oUt of Ut. Then,

dTt(y, t)dt :S du(y, t) a.e. (y, t) E OIU. If U = {y E R'": (y, t) E U}, to each y E U there corresponds exactly one point (y,tl(Y)) E OIU and one point (y,tz(Y)) E ozU.

We are ready to estimate JaluF(y,t)Z du(y,t). For any y in

F(y,tz(Y))Z - F(y, tl(Y))Z = Whence integrating over

i

l2(Y)

tl(Y)

0 FF(y, t)z dt. t

71 with respect to dy by (12)

[ F(y, t)Z du(y, t) la2U

C;;l

[

la1u

71,

F(y, t)Z du(y, t)

we get

1: U

t

F(y, t)Z dydt .

Since F satisfies the heat equation we have

o F(y, t) Z = 2F(y, t)"!:) 0 1 F(y, t) = - F(y, t)tb.F(y, t). ut tn 1r

"!:)

Substituing this relation in the right-hand side above, by (11) it follows that

[ F(y, t)Z du(y, t) :S ca(c>.lu(ozU) i;

+c

Ji

tF(y, t)b.F(y, t) dydt

= I + J,

(13)

say. To estimate J we fix t and consider the n-dimensional set Ut. Ut is the union of finitely many balls minus the union of finitely many pairwise disjoint balls, and, as such, it can be approximated by smooth regions where we can apply Stokes' theorem to the identity

V· (FVF) Hence

Ii,

F(y, t)b.F(y, t) dyl :S

= IVFI 2 + Fb.F.

i, IV

Z F(y, t)I dy

+[

!F(y, t)IIV F(y, t)1 dTt(y, t).

lou,

(14)

93 8A2 + 8- 1 B 2 applied to the second integral By (10) and the inequality 2AB above, upon multiplying by t and integrating we get

Since u(oIU),u(lhU)

[

k1u

cIB(x,T)I, by combining (9), (13) and (14) we get

F(y, t)2 du(y, t)

c(8 + 82)IB(x, T)I

+8

[

k1u

F(y, T)2 du(y, t).

Since 01U is contained in a compact set where F is continuous, the last integral above is finite. So, if 8 < 1/2, we can move this integral to the left-hand side of the inequality and obtain

From (8) we now get that 1£1 is also dominated by the right-hand side above, and the desired conclusion follows upon choosing 8 sufficiently small. • To prove (1) we also make use of the following good-lambda inequality. o

Lemma 7. x > 0, let

Let

E = {z

E

°
0,

°< e < T
0, by (iii) and (iv) of Lemma 8 we find

which combined with (19) gives the desired estimate.

•

Lemma 10. Assume the assumptions of Lemma 9 hold, and that f is merely a tempered distribution. Let N(x) = Mza(max(jFI,G, IF1{JI),x), and for a fixed compact subset J( of Rt.+\ set Sa,K(G,X) = Sa(XKG,x). Let 0 < 17 < 1 be given, and for a ball B and a > 0, and A,E > 0, let U = {x E B: Sa,K(G,X) 2: 2A,N(x) EA}. Then, if B contains a point Xo such that Sa,K(G,XO) A, we can choose E independent of A and B so that lUI 17IBI.

97 Proof. A moment's thouhgt will convince the reader that if r denotes the radius of B, then, for x E B,

ra(xo) u {(y,t):O at - 2r U (r a(x) n {(y, t) : at

ra(x) Thus, if

T

Ix - yl 2r}) .

at}

inf{ t: (y, t) E K}, it follows that

ra(x) n K

(ra(xo) n K)

U

at - 2r

{(y, t):O

Ix - yl

at} (20)

Now, U is closed, and if x E U, we have G(y, t)2 £2.\2 for Ix - yl integrating over the sets described in (20) above we find that

Sa,K(G,X?

Sa,K(G,XO?

+(£.\?! roo {

11 1

+-

2r a /

(at)-n dy

J2r/a

V

at. Thus,

dt G(y,t)2(at)- ndy-. t

v T But, since the first integral on the right-hand side above is a constant that depends only on the dimension n, and since Sa,K(G,XO) .\, for x E U we have

(1 + CC)2.\2

Sa,K(G,x)2

11 1

+-

2r a /

dt G(y, t)2(at)-n dy-. t

v 7" Integrating now over U, and observing that

( x(lx - ylJat)(at)-n dx

1,

v Ju

and that the above integral vanishes if the distance of y to U exceeds at, and that therefore this integral is majorized by the function S1(y, at) associated with the complement of U as in Lemma 8, we obtain

1 u

Sa,K(G,X)2 dx

IU!(l

+ c£2).\2 +

t: 1 R"

7"

dt G(x,t)2S1(x,at) dx-. t

Moreover, since Sa,K(G,X)2 2.\ on U, the left-hand side of the preceding inequality is not less than 4.\2IU/. As for the right-hand side, it can be estimated by using Lemma 9. From the definition of N(x) and U we see that IFI,G and IF", I do not exceed £.\ on the support of S1 1 (x , at), so that by Lemma 9 we obtain

1 1 2r a /

7"

R"

dt G(x,t)2S1(x,at) dx-

3 _(£.\)2 2

1 R"

t

(S1 1(x, cr) +

2r a / -8 8 S1 1(x, at) dt)dx

7"

11

2T

+(4+c)(£.\)2

1

R"

7"

a /

t

8 -8 S1 1(x,at)dtdx. t

98 Furthermore, since Q l (x, at) vanishes if t 2r/ a and the distance between x and B is larger than 2r, this last expression does not exceed cl(eA)2jBj, where Cl is a constant. Combining these observations we get

from where the desired estimate follows by taking e sufficiently small.

•

A result of similar nature that is also useful in the sequel is Lemma 11. Suppose F is as in Lemma 9, and let B = B(xo, r) be a fixed ball. Then, given 0 < So < 1, there exist constants a > 0 and 0 < Sl < 1 such that for all A > 0,

> a;A}1 < sllBI,

I{y E

> A}I < solBI·

I{y E B:

implies

Proof. As the proof follows essentially that of the preceding material, we only sketch it here. Let r;(t) be a nonincreasing positive Co(R+) function equal to 1 for 0 < t 1, and equal to 0 for t 2: 2, and set 7](x) = r;(lx - xol/r). Let 0, as in Lemma 10, be the set where > a;A, where the constant a; is yet to be chosen. As above we see that, with Q associated to 0 as in Lemma 8,

Moreover, since

r

Joens

dy

=

rr

Jo r Joens

ll o

Rn

G(y, t)2Q(y,t) dydt t

G(y,t)2Q(y,t)7](y)dy-, &

from the above estimate, upon letting t2 -+ 0 and tl

t

-+

r, it follows that

(21) Consequently, I{y E B:

I{y E B nO:

> A}I is equal to

> A}I + I{y E B n Oc: 10 n BI + cla;2 IB I,

> A}I

99

where to estimate the measure ofthe second set we applied Chebychev's inequality to (21). Now fix a so that cta2 < 80/2, and then choose 81 above so that 81 < 80/2; since 10 n BI < 811Bj, this gives the desired estimate. Proof of Theorem 2. Since Ma(F, x) E from the results in Chapter V it follows that, with the notation of Lemmas 9 and 10, also Ma(N, x) E and IIMa(N)IILl:, cll M a(F )1I Ll:, . Moreover, since w E A oo, from Lemma 8 it follows that

v( {y E B: Sa,K(G, y) > 2)", N(y)

e)..})

crW(B),

all B.

(22)

For each x in {Sa,K( G) > )..}, let B(x) be the largest ball centered at x whose interior is contained in {Sa,K(G) > )..}. There exists, then, a pairwise disjoint countable family {B(Xi)}, say, of such balls such that {Sa,K(G) > )..} Ui B 1(Xi)' Since these balls are pairwise disjoint, their interiors are contained in {Sa,K( G) > )..} and v is doubling, it follows that

where B(Xi) denotes the closure of the ball B(xi). Each ball B(Xi) contains a point where Sa,K )... Therefore, by Lemma 10 and (22) above it follows that

Whence, since {Sa,I( get

> 2)..,N

e)..}

V({Sa,K(G) > 2)..,N

{Sa,K > )..} e)..})

Ui s.. summing over i

we

C27]V({Sa,K > )..}),

which in turn implies that

v( {Sa,K(G) > 2)"})

C27]V( {Sa,K(G) > )..}) + v( {N > e)..}).

Integrating this inequality with respect to ).. from 0 to measure d)"P, 0 < p < 00, we obtain

00

with respect to the

Now, according to the definition of Sa,K(G), since K is compact and contained in > 0, Sa,l«G) is continuous and has compact support, and so IISa,K(G)IILl:, < 00. Thus, from the last inequality, upon choosing C27] < 2- P we find that t

100

Whence, letting J( increase and exhaust the upper half-space, passing to the limit we obtain Proof of Theorem 4. The left- hand side inequality follows since the multiplier d. Chapter XI. As for the right-hand side operators Ti are bounded on inequality, we invoke Theorem e in Chapter V, as it applies to the fi'S. Let (x, t) E R++\ then for any q > 0, N > 0, we have

v(B(x, t))!(Ji * 1, M; is bounded on Lfjq(Rn).

•

Proof of Theorem 5. First assume that f E Since Vo is dense in d. Chapter VII, the operators T, can be extended continuously to as follows: Given f E let Um} C Vo be a sequence that conand set gi = lim m_oo Tdm. Then, if 9 E Vo, we have verges to f in (Td,g) = limm_oo(Tdm,g) = limm_oo(Jm, Tig) = (J, Tig); thus, fi = l:7=llim m_ oo Tdm = lim m_ oo Tdm = lim m_ oo gm = f. Moreover, II T d llm; = lim m _ oo IITdmIlH:; = clim m_oo !Ifmllm;, = cllfllm:.- It then follows that

liminf IITd EO>O

* E S(R n ) a Schwartz function with integral 1. Suppose that for some X > 0, N),(F""x) is finite a.e. in a set E. Then the nontangentiallimit lim

F",(y,t)

= g(x) ,

1­0

exists a.e. in E. Furthermore, g( x) is independent of 4> and a > O. Using estimates established in Chapter V we reduce the proof of Therem 2 to that of Lemma 3. Let H(x,t) be a function in R++ 1 which satisfies the heat equation, and assume that M za(H, x) is finite for almost all x in a set E. Then the

104 nontangentiallimit lim

._0

1"'-111 O. Thus, under our hypothesis, M 2a(H, x ) is finite a.e. on E and, by Lemma 3, we conclude that the nontangentiallimit lim '_0

1"'-111 0, and that

M>.CG, x, t) = sup IG(y, t)l(l yER"


'
0 so large that m - k > 1, and f so large that 2M + m + n - £ < -f'. Using the above estimate for and the above choice for the parameters we get

Thus, if £' is large enough,

JRn (1 + Ixl)-l'PW(X) dx

cs goes to 0 as s and that of Theorem 1. •

--t


p, we have

N (p, w) Finally, if

W

ifl 2k},

k:.= 0 ± 1,±2 ... ,

for each k, Ok is a bounded open set, possibly empty. We fix k for the time being, and consider a Whitney­like partition of unity {'iXB(z;,r;) is now infinite, then the partial

If the linear combination sums

>"X .

2- k

-

V1

form a Cauchy sequence in

From the above

estimates we get that is also a Cauchy sequence in and consequently, it converges to some distribution f E say, in the norm. By Lemma 2 the convergence is also in the sense of distributions, and we have

cll L.>'iXB(z;,r;)IILP J

..

.

This completes the proof of the second part of Theorem l. To complete the proof of the first half, assume first that f E Va and decompose it as in Lemma 3 with N > N(p, w). We thus get a sum L:i,k >'i,kf3j with >'j,k == 2k and f3j (00, N) atoms with support contained in balls Bj, where Uj Bj Uk == {/* > c2 k } , c an appropriate constant, and so that, uniformly in k, the collection {Bj} has the finite overlapping property. It then follows that for 0< S < 00,

L

(x)

e2ksXUk (x),

i

and

i,k

e L2kSXUk(X) k

cj*(xY·

This gives the estimate

II Lj,k >'j,kXB; IlL:">

Cs llj*IlL::'

0

< S < 00.

Now we show that L:i,k >'i,kf3j(X) converges to f in the Observe that by Lemma 3 the sum converges to f in the sense of distributions. Furthermore, by the second half of the theorem, which we just proved, the sum converges

121 in the and in the sense of distributions to 9 E since it also converges to I in the sense of distributions, it follows that 9 = I. Finally, for an arbitrary I in we choose a sequence Um} Vo such that it tends to I in the as m -+ 00, lilt 11m; :S 211 IIIH:;' , and 111m - 1m-111m; :S 2- mIl / Il H :;, for m 2: 2. Putting 91 = It, 9m = 1m - Im-1 for m 2: 2, it follows that I = E:=19m, with convergence in the Let 9m = Ek ).k,mbk,m denote the atomic decomposition given in Lemma 3 of gm E Vo. Observe that l::k,m ).k,mbk,m converges unconditionally to I and that if the balls Bk,m contain the supports of the atoms and P1 = min(l,p), then 1

II Lk,m ).k,mXBk,m 11:

:: / .

:S :S

This completes the proof.

r; "L

cL m

1

k ).k,mXBk,m

:S

11:

:: / .

c( L m 2- m p 1 ) II/I1H:;' .

•

Sources and Remarks. The atomic decomposition of distributions in HP(R) obtained by R. Coifman [1974] was extended by R. Latter [1977] to the spaces HP(Rn),O < p :S 1, and by J. Garda-Cuerva [1979] to some weighted spaces, again with 0 < p :S 1 and with w in an appropriate A q class. The proof of Lemma 3 is due to Latter.

CHAPTER

IX The Basic Inequality

An important role in the theory of weighted Hardy spaces is played by what we call here the basic inequality. This is roughly a principle which allows, among other applications, for the use of maximal functions to describe the dual to the Hardy spaces and to control the various operators that act naturally on these spaces. We present in Theorem 2 and 3 below two versions of the inequality. Theorem 2 holds for doubling weights and the proof uses the atomic decomposition. On the other hand, Theorem 3, which is a particular case of Theorem 2, uses only elementary properties of the heat kernel, but it is only valid for Ax> weights. We begin by discussing a third, and elementary, version of the inequality; this version is, however, sufficient for many applications.

Proposition 1. Let v be a doubling weighted measure with respect to the Lebesgue measure on Rn, v E D d , with weight w, and suppose that 1 E V o and 9 E Lfoc(R n). Then

Il..

I(X)9(x) dx l :s; c

l.

1*(x)Mf;:9(X)W(x)dx,

with c independent of 1,9 and P an appropriate class of polynomials of degree less that or equal to N = N(d).

Proof. Let 1 = L:k L:i 2 kf3 k,i be the atomic decomposition of 1 given in Lemma 3 of Chapter VIII; this decomposition is built from the grand maximal function 1* corresponding to a Schwartz function with nonvanishing integral and it has the property that there are balls Bk,i 2 supp f3k,i with the finite overlapping

123

property, such that Lk , i 2kXBk,l.(x) ef*(x). As in Chapter III, given a ball B, let PB(g) be a function in P such that

A[(g,B)= (1B) (lg(x)-PB(g)(x)ldx v

Then,

1=

L"

JB

I(x)g(x) dx

=

inf Mf':g(x).

xEB

'

kk'i fhAx)g(x) dx

kk" fh,i(X)(9(X) - PBk,i(g)(X)) dx.

=

(1)

At this point we should really say a word about the convergence of the integral e(l + Ixl)-N for any N > 0, and if defining I. For 1 E Do we have f*(x) tiP M 1 ,' 1I g(x) < 00 for some x, then we can control how fast the averages of 9 will grow on large balls about the origin. From this we see that JR" f* (x) Ig( x )! w(x) dx < 00, and the convergence of the integral defining I follows readily. Returning to the proof, then, observe that III is bounded by

e

L L 2kll(B k,i)A[ (g, Bk,i) s e L L 2kll(B k,i) k

i

i

k

se =e

x

iIJ3f . Mf:: g(x)

E ",I

k",.Mf::9(X)W(X)dX

i

R"

Lk,i 2kXB",i(X)M f:: g(x )w(x ) dx

[ j*(x)Mf::g(x)w(x)dx . •

JR"

Our next result improves on Proposition 1; to state it we need a notation. Given A 1, let 1)A

M:':Ag(x) = sup inf (B ,

B2{x}pEP II

(Ig(x) - p(x)1 dx.

JB

We then have, Theorem 2. Let II be a doubling weighted measure with respect to the Lebesgue measure on H"; II E Dd' dll(x) = w(x)dx, and suppose that 1 E Do, and 9 E Lloc(Rn). Then, given a Schwartz function 4> with nonvanishing integral and a constant A 1, there is a constant e independent of i.s such that

124

Proof. Let F(x,t) = f * 4>t(x), and pick), so that). > dA. Further recall that, properly normalized, rex) cN>..(J,x). Let Fj denote the collection of those balls B(y, t) such that 2 j < IF(y, t)1

2 j +I ,

j

= 0, ±1, ±2, ...

Since f E VO, jF(y,t)1 IIfIlLd- nll4>!ILoo. Hence, for each v. IF(y,t)1 - 0 as t - 00 and for each fixed i. all balls in Fj have bounded radius. We are, then, under the conditions of the covering lemma in Chapter I. If a denotes the constant in the covering lemma, and as usual aB(y,t) = B(y,at), let Fj == {B'} be a countable pairwise disjoint collection of balls in Fj with the property that to each ball B E Fj there corresponds B' E Fj such that B aB'. Now put i c(j) = a2 /.\ we claim that

{N>..(F) > 2k}

U U

2(h-k)/>"B

U U

c(h - k)B'

B'

BEF"

= u.,

say. Indeed, suppose that N>.(F,x) > 2k, and let ho = max{h:M2h/>.(F,x) > 2k +h } , d. the proof of Theorem 11 in Chapter IV. Then there is (y,t) E R++I such that Ix - yl < 2hol >'t and 2k+ ho < IF(y, t)1 2 k+ h o +I . Thus x E 2hol >' B k} with B E Fk+ho and, as asserted, {N>..(F) > 2 UB EF h 2h/>.. B u.. Next consider the atomic decomposition f = I:k I:i 2kf3k,i described in Proposition L; clearly Bk,i {N>..(F) > 2k} and U, Bk,i u; Having fixed k for the time being, we separate the Bk,i'S into two disjoint families as follows: 1. Those balls for which there are h 2': k and B' E 2c(h - k)B'j call them Bk,i 1 '

such that Bk,i

2. Bk,i 2c(h - k)B' for any B' E h 2': k; we call these balls Bk,i2' A word about the second family. Since each B k ,i 2 Uk,

n c(h -

Bk,i2

k)B' of; 0,

in this case we have B'

and

e.;

U

some h 2': k

I)k k

L1 h

(2)

5B k ,i2'

B'

U

U

B' .

III

by

/g(x) - PBk,il (g)(x)1 dx

Bk,il

+ c L2 k L k

B' E

(2)holds

Now, in the notation of Proposition 1, we estimate C

and

1

i2 Bk,i2

19(x) - PBk.i2(g)(x)1 dx

=L k

k 2 l 1(k) + c L 2kh(k),

k

125

say. We estimate Il(k) first. Fix, in addition to k, h consider those balls Bk,i 1 such that

=

Il,h,B,(k)

L iI, (3)holds

1

Bk.il

and

(3)

2c(h - k)B' .

Bk,il Then,

k and a ball B' E

Ig(x) - PBk.il (g)(x)1 dx

f Ig(x) - PZc(h-k)B,(g)(X)! dx JZc(h-k)B' v(2c(h - k)B')A inf MP)Ag(x) c2(h-k)dA/>'v(B,)A inf Md,PAg(x) :S c2(h-k)dA/>.

xEB'

(l,

I,v

2:h>k 2:B'EJ=:' Il,h,B,(k). Thus, the sum 2:k 2kII(k) is dominated

Clearly II (k) by

-

h

L2 k L2(h-k)dA/>. L k h>k B' E:P.h -

f

= c

L

h=-oo B' EJ=:'h

c

xEB'

l,v

};=

2

(f,

(1,B 2h L 2(k-h)(1-dA/>.) k 2 h } , the last integral above is dominated by

Now, since

c( 1ft ( L R

2h/A)

h,Zh 5,M(F,x)

:S

C

(1ft (Ml(F,

w(x) dX) ,

126

which is an estimate of the right order. The proof for the B k,i 2 's is similar. Indeed, for each fixed k we have I(k,i 2 )

=

1

B k ,i2

jg(X)-PBk,i2(g)(x)ldx

cV(2Bk,i2)A inf M:':Ag(X) xE B k , i 2

'

c ( " , ' " v( c(h - k)B') inf M1U,'PAg(x L...J L...J xEB' ,v

)l/A) A,

h?k

where the sum is extended over those B' E Bk,i 2 ' Whence, it follows that

and consequently, the sum

in the definition of the case 2 for

L:k 2 kh(k) does not exceed

Since this estimate is also of the right order, we are done.

•

We state now and prove Theorem 3. Theorem 3. Let v be a weighted measure with respect to the Lebesgue measure 2 on R" with weight w E A oo , and ¢(x) e-1l"l xI • If 1 E 'Do and 9 is locally integrable, and if F(x,t) = f * ¢t(x), then there is a constant c independent of I,g such that

=

Iln

I(X)g(X)dXI

c

In

M(F,x)Mf,vg(x)w(x)dx.

127

Proof. We achieve the proof through a number of steps of independent interest which we label lemmas.

Lemma 4. Let G(y,t) = tlV'F(y,t)1 and fix a ball B = B(x,h). Given 0 < < 1, there are constants a, So, such that for all ,X > 0,

So

y) > a'x}l < solBI

I{y E B:

I{y E B:

implies

y) > 'x}1 < solBI·

It follows from Lemma 11 in Chapter VI. By the mean value inequality, and the in that lemma is dominated by a multiple of conclusion follows from this at once. •

Proof.

°

Lemma 5. Let the functions F, G and the ball B be as in Lemma 4. Given wE A oo , let aVBw denote the average of w over B. Then, given < 81 < 1, there are constants aI, SI so that for all p > 0,

> alpavBw}1 < sIIBI implies I{y E B : G, y) > pw(y)}\ < sIIBI·

I{y E B

Proof. Let 1 < p < 00 be such that w E A p , and consider the set £ = {y E B: w(y) < b av BW}; here b is a constant to be chosen. Then there is a constant c such that 1£1 cp!(p-l) bI!(p-I) IBI, and consequently, we can choose b so that 1£1 < sIIBI/2. Also observe that with this choice of b,

Ib E

> bpavBw}1 < sIIBI/2 implies I{y E B: y) > pw(y)}1 < sIIBI·

Indeed, since

I{y E B:

y) > pw(y)}1 I{y E B n e-, I{YE

> pw(y)}1 + 1£1 > bpavBw}1 +sIlbl/2 < sIIBI·

Now we just apply Lemma 4 with So = $1 and ,X = P aVBw, and combine this with the above implication to obtain the desired conclusion. •

Lemma 6. In the setting of Lemma 5, suppose that v(2B) Av(B) for all balls B. Then, for any locally integrable function 9 and any ball B == B( z , h) we have f Ig(y) - PB(g)(y)Ih(x - y) dy cavBw inf ML,g(y) .

JRn

yEB

128

Proof. Let 'fI E supp 'fI {Iyl I}, be such that 'fit * p(y) = p(y) for all polynomials P of degree less than or equal to N = N( d), and all t > O. Then for y E 2 k - 2 B, and as a consequence for y E 2k +I B, there is a constant c such that

IP2kB(g)(y) - P2k-1B(g)(y)!

c(A/2 n)k Mf,vg(y) aVBw.

Indeed, we have

Ip2 kB(g)(y) - P2 k-1B(g)(y)1

L.

=1

7]((2

k- 2 h)-n(y - z)) (P2 kB(g)(Z) - P2 k-1B(g)(Z)) dzl

lk-IB Ip2 k- 1B(g)(Z) - g(y - z)1 dz lkB Ip2 kB(g)(Z) - g(y - z)1

+

h)n (v(2 k- 1 B) + v(2 k B)) Mf,vg(y)

c(A/2 n) k Mf,vg(y) aVBw.

Whence, if z E B, it follows that

L.

Ig(y) - PB(g)(y)Ih(x - y) dy hIn hlg(y) - PB(g)(y)I4>«x - y)/h) dy

+

00

1

r

hn 12 kB\2 k- 1B Ig(y) - PB(g)(y)I4>«x - y)/h) dy

k 00 Mf,vg(z)aVBW+ L4>(2 -

k=l 00

+ L 4>(2 k k=l

1

11

1)h

n

2

kB

Ig(Y)-P2 kB(g)(y)ldy

1

)

hn j2k BI sup Ip2; B(g)(y) - P2; B(g)(y)l· yE2 kB

It thus only remains to bound the two sums above. The first sum does not exceed

f

4>(2

k 1 - )

:n v(2 kB)Mf,vCg)(z)

k=l 00

cMf,v(g)(z) aVBw L 4>(2 k - 1 )(A/2 n)k2nk

cMf,v(g)(y) aVBw.

k=l

As for the other sum, it is bounded by 00

C

1

k

L 4>(2 k - 1 ) hn (2 kh)n L(A/2 n )j ML(g)(z) aVBw

k=l

j=l

129

The proof is completed by combining these estimates. Lemma 7. and 0 < 52 all A > 0,

1£11

Proof.

•

Let G(y, t) = g * 4>t(Y), and suppose W E Aooo Then, given a ball B

< 1, there are constants a2,82 independent of g and B such that for

= I{y E B:

> a2 A}1< 821BI implies 1£21 = I{y E B: Mf,2h(G - (P3B(g) * 4>d, y) > A}I < s2l bl· ,v(g)(y)

Let

£3 = {y E B:there is B'

3B, y E B' and

L,

IpB,(g)(y) - P3B(g)(y)1 dy > aA},

where a > 0 is a constant to be chosen shortly, and £4 = {y E B : there is B' so that y E B' and aVB'w > 6avBw}, and where 6 is also a constant to be chosen. First observe that 1£41 can be made arbitrarily small so long as 6 is large enough. Indeed, to each yin £4 assign B' 3B with aVB'w > 6 aVBw. Now, by a familiar covering argument, there is a countable pairwise disjoint subfamily {Bj} such that 1£41 c L: j IBjl c6- 1 L: j v(Bj)(avBw)-l c6- 1Ibl, and the is true. Whence, if £2 £1 U £4 we are done, and we can assume that this is not the case. Also, observe that if y E B \ (£1 U £3 U £4), then Mf,2h(G - (p3B(g) * 4>t),y) < A. To see this, for each such y, let (z, t) E and B 1 = B(y, t). In this case, there are constants C1, C2 such that if tP( u) = C1 e-c2IuI2, then 4>t( z - u) tPt(Y - u) for all u ERn. Consequently,

IG(z, t) - P3B(g) * 4>t(z)1 (Ig - PBl(g)l) * tPt(Y) = G1(y, t) + G2(y, t),

+ (IPBl(g) -

P3B(g)l) * tPt(Y)

say. An argument identical to that of the proof of Lemma 6 gives that G1 (y, t) cavBwMf,vg(y). Furthermore, since y ¢ £1 U £4, this expression in turn is dominated by c6 av BwMf,vg(y) c8a2 A < A/2, provided a2 is small enough (the choice depending on 6 of course.) Now, recall that polynomials satisfy the following property: There exists a constant K such that SUP2;B l lpl Kj l 1 fBllp(u)ldu. Thus, if we pick p(z) =

,J

130

PBI(g)(Z) - P3B(g)(Z), we have that G2 (y, t ) equals

L,

Ip(z)I,p,(y - z) dz +

«: r t lB I

to L+%\"

B,

Ip(z)/dz+cI:( j=O

sup zE2J+IBI

j j+1) Ip(z)I)(2 +1t)n:'l/J(2 t

(_1_ r Ip(z)1 dZ) 'l/J(2 j+1) IB

caA + c I:(J(2 n)j+1

111BI

j=O

caA + caA

Ip(z)I,p,(y - z) dz

A/2,

provided that a is small enough. Thus, y) < A whenever y belongs to B \ (£1 U £3 U £4). The assertion of the lemma follows once we estimate 1£3 \ £11, assuming there is a point Y E B \ (£1 U£4)' To this end, to each y E £3 \ £1 we assign a ball B' 3B such that

L, Ip'(z)1

dz > aA,

p'(z)

= PB,(g)(Z) -

P3B(g)(Z).

By a familiar covering argument it follows that

1£3 \ £11

IBil

c

s :A

L,. Ip'(z)1 dz

:A 2;L, IPBj(9)(Z) - g(z)1 dz + :A L, Ig(z) - P3B(g)(z)ldz J

J

J

J

s

J

J

c, Lv(Bj)Mf vg(Yj)

aA

.

J

'

+

c,

aA

r Ig(z) - P3B(g)(z)1 dz,

l3B

where Yj f/. £I, and the Bj's are pairwise disjoint. Thus, this last expression does not exceed

which can be made as small as we want provided az is small enough. Corollary 8. Let g, G and w be as in Lemma 7. Given 0 < constants a3, 83 such that for all A > 0,

I{y E B: Mf ,vg(y) > a3A}1 < 831BI I{y E B:

83

•

< 1, there are

implies

G - (P3B * 4Jt), y)

> Aw(y)}1 < 831BI·

131

Proof. We combine Lemma 7, with X there replaced by aVBw>", and the observation in Lemma 5 to the effect that I{y E B: w(y) < bavBw}1 < cb'7IBI for some 'TJ> 0 when w E A oo , to obtain the desired conclusion at once. • Now, back to the proof of Theorem 3. Set F(y,t) = f * (!>t(y),F1(y,t) = tl'VF(y,t)I,G(y,t) = 9 * ¢tCy) and G1(y,t) = tl'VG(y,t)l. To each ball B = B(x,h) and 0 < s < 1 we assign a number A(B,s) = AB as follows: Let AJ,Ag

be such that and

y) > Agw(y)}1
E S(Rn) defines a continuous linear functional Z,, say, on by means of

On the other hand, it is not hard to see that every continuous linear functional Z on can be represented on a suitable dense class by a tempered distribution ge E S'(Rn), i.e.,

To see this, observe that there exists an integer N > 0 and a finite number of Schwartz norms II . Ilk depending only on p and w such that if f E S(Rn), then f belongs to Hiu(Rn) if and only if xC{ f(x)dx = 0 for /0'1 ::; N, and ::; c 2:k IIfllk; we do not prove this statement at this time for it is a particular case of results proved later in the chapter. So, having framed S(Rn) (Hiu(Rn))* S'(Rn), we seek a closer identification of the space of functionals. In particular, we want to replace the Schwartz

JRn

135

class in the arguments above by a space, to be denoted by A, which is as large on n A with an element as possible, and then to identify £ E ge E A *, the dual space to A. Moreover, elements in A * should be easily described as either functions or measures. If A contains the finite linear combinations of atoms, or if at least the representation of £ E by 9 E A* is valid for finite sums of atoms, we hope ge. In fact, the estimates to derive estimates for the sharp maximal function . we obtain are 1. If 11£11 denotes the norm of £ as a functional and 0 < p < 1, for an appropriate A we have

Mf·:

2. In the notation of 1. above, if 1 P < 00 and lip + lip' = 1, and if P denotes an appropiate class of polynomials, the degree depends on p, t/, we have The appropriate A in condition 1 above is determined from the basic inequality. Also, if the left­hand side in 1 above is finite for a locally integrable function g, it follows from the basic inequality that such g's define bounded linear functionals on HI;;(R n). Similarly, when 1 P < 00 and f E 150 , by the basic inequality and Holder's inequality we have

IL..

f(X)9(X)dX\

c

L..

M1(Fet>,x)Mf;:g(x)w(x)dx

II M Thus, any locally integrable function

f:: gilL:: .

9 such that II Mf:: gilL::


1, then we may pick q = pr'. When q = 00, it is not always possible to satisfy simultaneously conditions 4 and 5 above, because the dual of a space given by an Loo-norm is not easily described. This difficulty is overcome by requiring that A contain all finite linear combinations of continuous atoms. In addition to the choice of q described above, we associate with the weight w the set I w = {t>O: k,,(l+ XI)- tPW(X) dX < j I

OO}

o. A will then be described in terms of q and to. Let N' be the largest integer such that n + N (j. I w , i.e., n + N + 1 is the smallest integer that belongs to I w • We consider two cases, to wit, I w is an interval of the form (to, 00) or [to, 00) for some to >

137

Case 1. to f. n + N + 1. Case 2. to = n + N + 1. We begin by discussing Case 1. Pick t1 so that to < t1 < n + N + 1. If 1 < q < 00, we let A consist of those locally integrable functions 1 which belong to where u(x) = (1 + Ixl)t1q-n, and A" of those locally integrable g's that belong to (R n), where v(x) = (1 + Ixl)-t 1Q'+(n/(q-1». Specifically, let

A

(Ln U(x)IQ(1 + Ixl)t1Q-n dxY/Q =

= {I E L}oc(Rn):


1 and w rt. A p • Nevertheless, the results spaces. proved in this chapter include Hormander's multiplier theorem for The main method of proof used in this chapter is the atomic decomposition of the weighted Hardy spaces. In addition, we present other methods, including pointwise estimates involving Lusin and Littlewood-Paley integrals, as well as pointwise estimates in terms of sharp maximal functions. In fact, we find that each method is more appropriate for a different range of values of p, depending on the weight w. We begin by introducing a notation. By Ixl '" t we denote the fact that the values of x lie in the annulus {x E R'": at < Ixl < btl, where 0 < a ::; 1
0 and multi-indices a with in addition,

IDCXm(O { ( J{I{I"-'R}

lal

f when f is a positive integer, and,

- zW

s c(lzl/ RpR(n/q)-!a!

for all lzl < R/2 and all multi-indices a with lal = j = integer part of E, and f = j + 'Y when f is not an integer. We reserve the notation k(x) for the kernel that corresponds to the inverse Fourier transform of m in the sense of distributions. It is a natural question to consider how the behaviour of k reflects the fact that m E M(q,.e); first a definition. 0 and 1 ij 00, we say that k verifies the condition For a real number i M(ij,i), and we often write k E M(ij, i), if I/ii

(

(

ID13k(x)l ii dX)

for all multi-indices 1131 < strictly less than i and i =

i,

i

cR(n/ii)-n- I131,

all R

> 0,

and, in addition, if j denotes the largest integer

+ 1',

I/ii (

(

J{lxl"-'R}

ID13k(x) - D 13k(x - Z)lii)

c(lzllR)'Y R(nfij)-n-i if 0 0, and all multi-indices 13 with 1131 = j. In case q = 00, the integral expression above should be replaced by a supremum norm in the usual way. We open with some remarks. At a first glance, the condition M(ij, £) may appear a bit unusual when £ is an integer. It may seem more natural to use all multi-indices /3 with 1/31 i in the first expression above rather than considering the differences. However, such condition is too restrictive as our next result shows.

152 Lemma 1. Suppose m E M(q,f), 1 q 2. Given 1 ij such that lip = ma:x(1/q, 1 - l/ij). Then k E M(ij'£), where f

00,

let p

= l - nip.

1 be

Observation. We have assumed that the multiplier m is a bounded function. However, the condition M(q,l) on m already implies that m is bounded provided that f > »t« Moreover, the functions R) have Lipschitz f-n/q norms which are uniformly bounded in R in the annulus '" 1} if 0 < f - n/q < 1; to prove this one may use Sobolev's inequality on expressions of BMD-type, the details are left to the reader. Now, let the "M(q,f) norm of m" denote the infimum of all possible conin the definition of the condition M( q, f); similarly we may define the "M(ij, f) norm of the kernel k". Also, if 1] is an arbitrary Schwartz function, then multiplication of either m or k by 1] only increases the M(q,f) or the M(ij,f) norms by at most a constant factor cTj, depending solely on 1]. This is readily seen by using Leibniz's differentiation rule, and it is especially simple when 1] is supported in an annulus. We also note that the condition M(q,f) is invariant under the dilation t > 0, and similarly, the condition M(ij,f) is invariant under the dilation k(x) - t-nk(x/t), t > O. Therefore, if mE M(q,f), then so does 1](te)m(e), uniformly in t > o. Similarly, if k E M(ij,f), also does 1](tx)k(x), uniformly in t > o. Finally we observe that the conditions M(q,£) and M(ij,f) are monotonic in q,f and ij, i, respectively. In other words, if_q q1 e f 1, then A:!(q,fl implies M(qllf1), and similarly, if ij ij1 and e f 1, then M(ij, f) implies M(ij1,ft}. This observation follows from Holder's inequality and the mean value theorem together with Minkowski's inequality. Proof of Lemma 1. Since q p and ij p', by the monotonicity in the conditions, M(%,l.) implies M(p,f) and M(p',f) implies ¥(ij,f). Hence, it suffices to show that if f = f - nip and mE M(p,f), then k E M(p',l). There are several cases depending on whether f is an integer and on whether we are estimating expressions involving n fJ k(x) or the differences n fJ k(x) nfJk(x - z). First we split the multiplier m into several parts, and treat each one of them separately; the proof is then based on the Haussdorff-Young inequality. We do not do all cases, but we hope that the indications we give in the cases we discuss in detail are sufficient to make it clear how the remaining cases are handled. First, since k( x) _ en k( x It) corresponds on the Fourier transform side to - m(tO, t > 0, and the conditions M(ij,f) and M(q,l) are invariant under these dilations, we may assume that R = 1. Let ¢ be a nonnegative Schwartz ¢(2- iy) = function with support contained in {1/2 < Iyl < 2}, such that i 1 for y =I- o. Set 1](0 = 1 ¢(2- O and

mo(O = 1](e)m(e), = and let ki(x) denote the inverse Fouries transform of

i = 1,2, ... i

= 0,1,2, ...

153

Case 1. We start, when i = 0, by estimating the expressions D{3 ko(x) and D f3k o(x) - D{3ko(x - z), which are essentially the inverse Fourier transforms of and (1 respectively. Since S 1 and 1] has support contained in < 2}, we get

Sc

2: 2- (l{3I+ 00

i

n / p)

S c.

i=O

Now, when estimating D{3ko(x ) - D{3ko(z - x), we have in addition the factor 11- eizo 1 S when S2 we get 1- eizo on the Fourier transform side, and since

Case 2. Next we consider D f3k i (x), i > O. Since L:1O'I=i IxO'I2: c> 0 for [z] when '- is an integer we get

("V

1,

By the Hausdorff-Young inequality this expression does not exceed c

2:. (in

IO'I=J

R

is a Schwartz function we see that = satisfies the condition M(p,'-) uniformly in i, and furtheri more, its support lies in rv 2 } . Hence, for each a in the above sum, the terms Now, since

154

are bounded by c2i(n!p+/.Bj-lal) and, since [o] conclude that

= j = £ when £ is

an integer, we

lip'

(

(

1D.Bki(X)IP'dX)

If £ is not an integer, i.e., £ = j

2: Ix

a

:::; c2i(n!P+I.BI-£) .

(2)

+, with 0 < "t < 1, we use the inequality

sin(x· z",)!

c> 0

for Ixi

rv

1,

l"'l=i

for some suitably chosen Z"'; in fact we may choose z'" = a/lal. Since multiplication by the sine factors corresponds to taking differences in the Fourier transform side, as before we see that

{

(

1D.Bki(X)IP'dX)1!P'

:::; c

2:

lal=i :::; c

(1

2:. (1

lal=J

Ix'" sin(x· z",)D.Bki(x)IP' dX)l!P'

+

n

+ z",)) ­

­

­ zo,))IP

R

Since 2-il.Blemi(O E M(p,f), £ = j +" its support is contained in the annulus i i rv 2 } and I ± z",1 :::; 1 < 2 /2, each term in the last sum above is bounded i)"2i(n!p+I.BI-i) by c(lz",I/2 :::; c2(n!p+I.BI-£). Thus, we also get (2) when £ is not an integer. Finally we estimate the terms involving the difference D.Bki( x)D.Bki(X ­ z) when 1,81 = J, where J is the largest integer less than l. When £ is an integer we first invoke the inequality LI"'I=£ Ixal c > 0, and obtain Case 3.

(

(

ID.Bki(X) ­ D.Bki(X ­ z)IP' dX)l!P'

:::; c

2:

lal=£

(1

lip'

Ix"'(D.Bki(x) ­ D.Bki(x ­ z))IP' dX)

The Fourier transform of the function x"(D.B k i( x) ­ D{3k i( x ­ z)) is essentially equal to D.B«1 ­ which, by Leibnitz's differentiation formula, can be expressed as a sum of prod ucts of terms of the form c a1D"'1(1 ­

155 with all multi-indices a1 with 0 a1 S a. We now use the estimates 11 - ei,;,zl and IDi(1 - eiZOX)1 S clzpa d for a1 =I- 0 and Ixl '" 1, together with the condition M(p,e) which is satisfied uniformly by the functions Thus, by the Hausdorff-Young inequality we get

(

f

Ix a(D f3 k

l / P' i(x) - D f3k i(x - z))IP' dx )

S c2i(n/p+If3Hal)lzI2i

L

+c

2i(n/p+If3Ha-all)lzlla-11

S c2i(n/p+If3I+l-lal)lzl ,

when

Izi

s 1. Thus, when e is an integer,

( f

ID f3 k i(x) - D f3 k i(x - z)IP' dX)l/

P'

S c2i(n/p+ 1.BI-t)lz!2 i.

Note that when 2i > l/lzl we get a better estimate using the triangle inequality and the estimate in case 2. When e is not an integer we may use sine factors instead, as we did in case 2, and note that this corresponds to taking differences in the Fourier transform side. After using Leibniz's formula we get, in the Fourier transform side, a sum involving expressions of the form calDal(ei(';-zo 1. Further, let 1 ::; q ::; 2, s ::; p ::; 00, and suppose that m E M(q,f), where f > n/q and f 2: nmin(s/p,l/p' + l/(rp)). Then the multiplier operator associated with m is into itself. a bounded mapping from By a duality argument and general properties of the weights, the proof of Theorem 5 is reduced to Theorem 5, reduced version. Let l/ be a doubling weighted measure with respect to the Lebesgue measure on Rn, dl/(x) = w( x )dx, and assume that the weight w E A p n RHr . Let 1 ::; q::; 2, and suppose mE M(q,f), where f is such that f > nmax(l/q, l/p' + l/(rp)). Then, the multiplier operator associated to into itself. m is a bounded mapping from Assuming the reduced version of Theorem 5 for the moment, we will see how the general version follows. First, we may assume that the inequality f > n min (sip, lip' + l/(rp)) in Theorem 5 is strict since w E As n RHr implies that w E A s - e n RHr +e for some E > O. Next, we observe that under the hypothesis and its dual of Theorem 5, the reduced version applies to the space L:_1f(p_l)(Rn). To see this we only need to check that w E As implies that w-1/(p-l) E Api n RHr 1 , with "i = (p - 1)/(s - 1) and l/p + l/rlP' = p/ s. In this case, by a duality argument, it follows that a multiplier that is bounded on is also bounded on We may now proceed with I

Proof of Theorem 5, reduced version. We show that liT ::; ellfllL::' for f E Vo; the operator T then extends to a bounded mapping on The proof is very similar to that of Theorem 4, and we indicate the differences with that proof here. Let f E Vo. We then decompose f into a sum of (00,0) atoms such that the corresponding sum of the characteristic functions 'L: j AjXBj is bounded in the Next we choose Po

by

and limN--+oo

II

Ajaj -

= O.

> prIer - 1) and f > n(l - 1/po); the last choice is possible

162

since £ > n(l- (r - l)/pr). Since the (00,0) atoms aj are (Po, 0) atoms as well, and since by Theorem 4 the operator T is bounded on LPO(R n ) , we may apply Lemmas 2 and 3 on each atom aj. In this way T('Ef=l Ajaj) can be written as a double sum 'Ef=1 'E:'o(Aj/Ci)b ij, where the bi/s are (Po,O) atoms with support contained in 2i H B j • By Lemma 4 in Chapter VIII we get )-1

(Ai/Ci)bijll L

P

w

:s

c

)-1

(Aj/Ci)X2 i HB·11 . J L" w

It is at this point that we invoke the fact that w E RH r and Po > prier - 1). Now, by Lemma 3 in Chapter VIII, since w E A p , this last norm is dominated by in

c(2 rei)

)-1

AjXB·11 J

.

As pointed out in Lemma 2, we have Ci c2i (n + e) for some e > 0 when £ > nmax(l/q, 1- lip). Summing over i we obtain

IIT(I::1

cllI::1

As for the limiting argument needed to obtain the estimate for T i, it is the same as that in the proof of Theorem 4 and is therefore omitted. This completes the proof of Theorem 5 in the general and reduced versions. • We now state the multiplier result for the weighted Hardy spaces. First we recall that if w E As for some s land p s, then is identical to and in this case Theorem 5 applies. When < p < s we have the following result.

°

Theorem 6. Let v be a doubling weighted measure with respect to the Lebesgue measure on H"; dv(x) = w(x)dx, and suppose that v E DnB with 1 0 s. Further, let 1 q 2 and 0 -

l

n(B-l){s-p) p(s-l)

{>n

p

+ max

_ 1...) and q'

(1

1... + l) rp

q' p'

if 1

0, and e > O. The order of the moment condition needed is directly related to this number 0, also when applying Lemma 3 in Chapter VIII. More precisely, we need to decompose f into (oo,No) atoms, with No > 0 - n - 1, and apply Lemmas 2 and 3 on (Po,N o) atoms with Po > prier - 1). We recall that we have 0 = nO/p when 0 < p 1, and 0 = n + n(O - l)(s - p)/p(s - 1) when 1 < p < s. We can choose Po such that £ > 0 - nmin(l/r/,l/po). With the notation of Lemma 2 it follows that l> 0 - n, and we conclude from this lemma that Ci e2 i( H £) for some E > 0, and a constant e > O. This is precisely what we need. The limiting argument is quite similar to that given in the proof of Theorem 4. This time the finite sum of atoms fN is a Cauchy sequence in By the a-priori estimate of T acting on finite sums of atoms, T fN is also a Cauchy sequence converging, in the to some distribution g. As before we can show that TfN also converges to Tf in the L 2(R n)-norm. This means that T I» converges to both 9 and T f in the distribution sense. Thus, 9 = T i. and liTfilm; ellfllm;, and the proof is complete. • Alternate Proof of Theorem 6, 0 < P 1. Instead of using the full strength of Theorem 4, second half, and Lemma 3 in Chapter VIII, we only need to invoke the following estimate on a single atom, namely,

II allm.

eV(B)l/ p

,

for any (Po, No) atom a with support contained in the ball B, provided that Po > prier - 1), No > (nO/p) - n - 1, and v E DnB and w E RHr . Under these

164 Let, then, f E Vo(Rn). As before we decompose f into a sum Lj Ajaj where the aj's are (00, N) atoms with support contained in the balls B i» say, and such that the partial sums L,%l Ajaj converge to f in the As noted in Chapter VIII, this can be done in a such a way that Lj Arv(Bj) As before we also get that T(L,%l Ajaj) = Lf=l(Aj/Ci)b ij, where the bij's i+4Bj. are (Po, No) atoms with support contained in 2 We then have

provided Po follows that

I 2::

1

> prier -

1) and No

2:: 0(Aj/Ci)b ij

s

l:::

2::

> (nO/p) - n -

1. By the triangle inequality it

2:: 1

inB o(2 /cn 2:: 1 Arv(B j)

e

2::

inB o(2

As before we can pick Po so that '- > (nO/p) - nmin(1/r/,1/po) and obtain, by e2i«nB/p)+c) for some E > 0, e > O. We conclude then, Lemmas 2 and 3, s, that ellfllH:;" A limiting argument gives now liTfl\H:;' ellJllm" and this concludes the proof. • An interesting result in the unweighted case is the following corollary to Theorem 6. Corollary. Let 1 q 2, 0 < p 1, and suppose that m E M(q,'-), where '- > n(1/p - 1/r/). Then the multiplier operator associated to m is a bounded

mapping from HP(Rn) into itself.

We pass now to discuss a different approach to deal with multipliers, namely, that of pointwise comparison of sharp maximal functions. For reasons of clarity and emphasis we depart from the notation introduced in Chapter III and, given a weighted measure v with respect to the Lebesgue measure on H", dv(x) = w(x)dx, we put

M!:: f(x)

=

1

1 sup inf ( (B) If(y) - p(yW w(y)dy v B

)l/r

We prove our results only for singular integral operators. This is done for simplicity and the reader should have no difficulty in extending them for general multipliers.

165

Proposition 7. Let v be a doubling weighted measure with respect to the Lebesgue measure on R n , dv( x) = w(x )dx, and suppose v E D n e and w E RHr, where r,O > 1. Further, let the kernel k E M(ij,l), 1 ::; ij ::; 00, and assume the associated operator T satisfies

LITl(xWO dx ::; clBI,

0

< qo < 00, or ITI(x)] ::; c if qo = 00,

for all balls B and all bounded functions Finally, let 0::; d1 < 00, min (l,dt) ::; dz < Then, we have

supported in B with 11/1100 ::; 1. 00, and also d1 ::; min (ij/r',qo/r').

I

::; CMd2,1I(J*)(x),

IE Vo ,

provided l > nO/min (ij/ r' , d 2 , 1) - n / ij' and P is the class of polynomials on degree less than or equal to No, where No is greater or equal to the integer part of i. A remark before we proceed, namely, the assumption on T I holds if T is bounded on LqO(Rn) or more generally ifT is of weak-type (q,q) for some q > qo. Proof. By using Holder's inequality in the definition of the various maximal functions involved we may assume that d1 > 0 and d z ::; min (ij/r', 1). Thus, we have two cases to consider, namely, 0 < d1 ::; dz ::; 1, and 1 = d2 < d1 < 00. Set d3 = max (dt, dz). From the hypothesis we may assume that ij < 00 and that l is not an integer. This will facilitate the writing of the proof although still the simplest case is ij = 00. We may also assume that qo < 00. By a translation and dilation argument it suffices to estimate

LITI(X) - p(X)ld1w(x)dx where B = B(O, 1) denotes the unit ball in R" and pEP is a suitable polynomial of degree less than or equal to No = integer part of l. According to the atomic decomposition we write 1= Lj >"jaj as the sum of (oo,No) atoms aj with support contained in balls B j and so that the corresponding sum of characteristic functions of these balls satisfies

L>..jXB;CX)::; CdJ*(x)d,

all 0 < d::; 1.

i

When d1

= d3 = d we proceed to

estimate l / m a X( d ,l )

(

LITaj(x) - pj(x)ldw(x)dx ) . ::; c sup z ?"

( 1. ) v 21B

1

ZiB

XB·(x)w(x)dx, J

166

for some E > 0, and we then get the desired estimate by summation; here Pj is a suitable polynomial of degree less than or equal to NoMore generally, when d1 ::s; d3 , the idea is to split Taj( x) into two functions, Taj(x) = j(x) + Wj(x), say, and to estimate

and

I

( v(B)

f

1B IWj(x) -

Pj(X)l

d

)1/maX(d3.1) 3 W(x)dx

by the right-hand side above. In this case Tf(x) and with p(x) = L:j AjPj(X) we get

,

= L:jAjj(X) + L:jAjWj(X),

(4) Here we used Holder's inequality, the triangle inequality and the notation

Assume for the moment that the integrals on the right-hand side above can be estimated as indicated. Then the first sum on the right-hand side of (4) is bounded by

167

In a similar fashion, the second sum on the right-hand side of (4) is bounded by c sup 1. [f*( x )min (d3,l) w( x) dx . c-o v(2 t B) B

12;

Combining these estimates it then follows that CMd3,v(J*)(x). The main part of the proof still remains to be carried out, namely, we must define the function iC x) and Wj( x) and show they satisfy the desired properties. Let rj denote the radius of B j . We then let

and

We want to show the estimates

(5) and

(6) for some E > 0 and for some polynomial Pj of degree less than or equal to No depending on Band aj' Let j be fixed, and let i o be the smallest integer i such that 2iB n Bj f:. 0. First we estimate the right-hand sides of (5) and (6). When rj 2 io , since there is a ball Bj in Bj n 2 io +l B with 2 io+l B coB j for some dimensional constant Co, these quantities are at least greater than or equal to

c2- i ov(B j n 2 io +1 B)jv(2 i o+l )

c2- ioe

.

Also, when rj < 2i o , Bj is contained in 2io +1 B, so that the right-hand sides of (5) and (6) are larger than

168

Next we estimate the left-hand sides of (5) and (6); we do (5) first. We consider several cases. When i o > 1 it is easy to check that j( x) vanishes identically on B. When i o = 1 and rj 1, we use the reverse Holder's condition on the ba1l2Bj and the fact that I2B.ITaj(x)lqO dx cl2Bj i to get that IB J cv(2Bj) and thus the left-hand side of (5) is less than or equal to

which in turn is dominated by the right-hand side of (5). Next, the case io = 1 and rj > 1 is very similar. We use the reverse Holder's condition on the ball B and the fact that I2B IT(X2Baj)(x)lqOdx clBI, to get that

dx

cv(B).

Thus, the left-hand side of (5) is bounded above by a constant, while the righthand side of (5) is bounded below by a positive constant. In order to estimate (6) we consider the Taylor expansion of order No - 1 of k(x - z). When r i 1 we expand k(x - z) as a function of z around x i» the center of B]. We then have k(x - z) = Px(z) + Rx(z), where Px(z) is a polynomial in z of degree less than or equal to No, and

IRx(z)1

c

L

l1ID,8k((X - Xj) - s(z - Xj)) - D,8k(x - xj)llx - XjlNo ds.

1,8I=No The assumption k E M(q,l) implies that 1/ii

[ ( J{lx-xj

IRx(z)lq dX)

s c2- injq' (2- irj)l

whenever z E B j and 2i > rj. So, using the moment condition on the atom a we get that k*aj(x) = IRn Rx(z)aj(z) dz, and with Pj(x) = 0, by Holder's inequality and the reverse Holder's condition, we get

Since dist (x j, B \ 2B j) RxC z) above we have

c2io, by Minkowki's inequality and the estimates on

169

Thus, observing that d3 / max (d 3 , 1) (6) is bounded by

= d2, we conclude that the left-hand side of which can in turn be dominated by

the right-hand side of (6) if d2 (l + (n/ij'» nO + E. From our assumptions, this inequality holds provided E > 0 is small enough. When ri > 1 we expand k( x - z) as a function of x about the origin and write k(x - z) = pz(x) + Rz(x), where pz(x) is a polynomial in x of degree less than or equal to No, and

when i

> 0 and x E B. We define the polynomials Pj( x) then by

and get Wj(x) = fR" R z(x)(l- X2B(z»aj(z)dz. By a simple computation using Holder's inequality on the sets {izi 2i } and summation we get fV

We thus conclude that the left-hand side of (6) is dominated by c2-iod2l, and that (6) holds when rj 2io provided we choose E so that 0 < E < l. In case io 1 < rj < 2 , by Holder's inrequality it follows that

Thus, the left-hand side of (6) is bounded by the above expression raised to the power d2 , and we conclude that (6) holds in this case provided that E is chosen so that d2 (l + (nfil» > nO + E. The proof is thus complete. • Proposition 7 leads to weighted norm inequalities for the singular integral operator T. Let v be a doubling weighted measure with respect to the Lebesgue measure on H"; dv(x) = w(x)dx, and assume that v E D n8 and wE AsnRHr . We will apply Proposition 7 to the weighted measure Vb defined by dVb(X) = w(x)bdx, where 0 b 1. Observe that Vb E Dn8b and w b E RHr p where Ob = b«(}-l)+ 1 and rb ::= rib. Also w E ASb(Vb), where l/sb = l/s + bf s', Now, from Theorem 2

170

in Chapter III it follows that there is a polynomial P of degree less than or equal to No, depending on f E Vo, such that

liT f

-

:S

cIlMJ::'s

l

(T

As we have seen in Chapter III, there is

O,II,S g(x) < -

for 82

81

> 0 small enough.

> 0 such that

1/ d 1 0,lIb,S2 g(x) < - C8 2

d1,lIb

g(x)''

in the first two inequalities above we used that w E A oo • This observation together with Proposition 7, give the pointwise estimate

f E u«,

MJ::'s(Tf)(x):S CMd2,IIJf*)(x) ,

provided k is as in Proposition 7 and i > nfh/ min (ij/ rb, d 2, 1) - n] if. Moreover, since the maximal operator Md2 , lIb is bounded on when p/d 2 = Sb, we get IITf :S f E Vo, provided i> nOb/min(ij/rb,p/sb, l ) - nfij'. It is not difficult to check that the polynomial p above is zero when Indeed, by direct estimates we get

r ITf(x)IPodx=O(R J{lxl-R}

n

as

) ,

f

E Vo.

R-HXl,

and by Holder's inequality it follows that

r ITf(x)-p(x)IPodx=O(R J{lxl-R} when

liTf - pilL:' < 00, if Po

n

as

) ,

R-too,

is chosen small enough. Whence

r Ip(x)IPO dx = o(R J{lxl-R}

n

) ,

as

R

- t 00,

and consequently, p( x) = O. We have thus shown the estimate

provided i> nOb/ min (ij/rb,sb, 1) - nfij'. Next we minimize this last expression by choosing appropriately b, 0 :S b :S 1; we only consider some cases in the theorem below. Note that we may replace the on the left-hand side above by the HI;;(Rn)-norm of Tf. A way to see this is to express the HI;;(Rn)norm of T f by mean of its R" )-norm and that of its Riesz transforms of high enough order, assume that we know the fact that such transforms are bounded on and then note that they commute with T. In fact, choosing a suitable b this method gives

171 Theorem 8. Let v be a doubling weighted measure with respect to the Lebesgue measure on H"; dv(x) = w(x)dx, and assume that v E D n 9 where {} 2: 1 and wE As n RHr where 1 < or,s < 00. Further, let k E M(ij,l), 1 :s; ij:S; 00, denote the kernel of a singular integral operator T so that there is qo, 0 < qo :s; 00, with the property that

hITf(xWOdx:s;cIBI,

O n/ij when p 2:

S

:s; rfij'.

> 1.

Proof. We need only pick b appropriately. For the cases (i), (ii) and (iii), we choose b = 1, (l/p)(s - p)/(s - 1), and 0, respectively. Also observe that coin.cides with when p 2: s > 1. • For inequalities with w E As, 1 < s :s; p, it is in some cases preferable to use the following estimate involving unweighted maximal functions; the estimate does not involve the atomic decomposition.

Proposition 9. Let k E M(ij,l), 1 < ij :s; 00, I > 0, and suppose that the associated singular integral operator T preserves Lqo(R n), 1 :s; qo < 00. If o :s; d1 :s; qo and d2 2: max (qO, ij'), we have

where P

= constants.

Sketch of the Proof.

where B supp fi

We may assume that d1

= B(O,I) is the unit ball. Write f = 2i+2B \ 2 i B for i = 1,2, ... For T fo

> O. We want to estimate

Ii, where supp fo 2B and we use the assumption that T

172

preserves Lqo(Rn), and for T f; we observe that with c, we have

Tfi(X)-Ci=

f

JRn

(k(y-z)-k(-z))f;(z)dz,

=-

JR" k( -Z)fi(Z) dz,

i=1,2, ...

Using Holder's inequality and the condition M(ij,i), it follows there exists 0
1. Further, suppose the kernel k E M(ij,l) , 1 < ij ::; 00, is such that its associated singular integral operator T preserves L s» (R n ) , 1 ::; qo < 00 Then, for I E Vo, 0

::; eIl / IlB:; , 1::; P < 00 ,

liT111m;

provided that

i> n(() -

1) and pr' ::; min (ij,qb).

Sketch of Proof. Whenever w E A oo, from the basic inequality and Proposition 11 it follows that

Ii..

(k

* J)(x)g(x) dxl s

c

i.

dx,

9 E

where M 1 (F, x) denotes the nontangential maximal function corresponding to an extension F of I. The condition pr' ::; min (ij, qb) is used to make sure that is bounded in the by the norm of 9 in when d 2 = min (ij,qb) The conclusion of the theorem follows readily from this remark; we leave the details to the reader. • 0

Also, rather than using Proposition 11 it is possible to invoke an implication between the sharp maximal function and the truncated Lusin function which essentially leads to the same duality argument as in the proof of Theorem 12; the assumptions here are that dv(x) = w(x)dx, v E DnB, () 1, and W E Aooo Indeed, let I E Vo , and if.,p(x) = V'e-1rlxI2, put H(x,t) = !.,pt *(k*J)(x)l. Given a small positive number s, there exist positive numbers Sl, a1 and 11, depending only on s, the kernel k and w, such that for any ball B with radius hand 11 > 0,

I{x E

- P'

> a1a}1 < sllBI implies I{x E B: S't(H,x) > aw(x)}1 < slBI·

The condition on the kernel k which is required for this implication depends only on () and d, and the proof is left for the reader. The methods described above have further applications. For instance, if T is a sublinear operator which maps each (q,N) atom into a sum 2-ieai, say, where the ai's are (q,O) atoms and E > 0, and such that the support of the ai'S is contained in balls B, 2 i B, where supp a B, then T extends to a bounded mapping from into 0 < p < q, provided wE RHr , r = (q/p)', and E > c5( w, p); here c5 is an in Lemma 3 in Chapter VIII. These considerations apply, for instance, to pseudo-differential operators. We do not consider them in detail here since the main ideas are already apparent in the singular integral operator case. Finally, we describe Stein's method using the Lusin and Littlewood-Paley functions to obtain pointwise estimates.

175

Theorem 14. Let 1/J be a Schwartz function with vanishing integral and let ¢Y = (¢Y(l), ¢Y(2), ... , ¢Y(d») be a vector-valued Schwartz function satisfying condition (1) of Chaptert V. Let k E M(2,f), and for I E Vo put Fc/>(x, t) = (¢Yt * J)(x) and G",(x,t) = 1/Jt * (k * J)(x). Then,

IEVo, provided A :s; f when f is an integer, and A < f when f is not an integer. We need only to observe that the functions {1J(r)}r>O

Outline of the Proof. defined by

1J(r)(f.) = k(f.lr)1/J(f.) ,

r > 0,

verify the conditions of Theorem 4 in Chapter V with A = 1, B any positive number, and the integer m :s; £. By the remarks following Theorem 8 in Chapter V we get the desired estimate when f is an integer. Since the case when f is not an integer was not considered in Chapter V, we give a direct proof here. We may assume that d = 1, that the support of 1/J is contained in {1f.1 "" I}, and that ¢Y(O = Ion the support of 1/J. Then,

1/J(tOk(O}(O = k(01/J(tO}(f.)¢y(tO, and consequently,

G",(y, t)

= (1/Jt * k) * F(·, t)(y).

Since 1/Jt * k satisfies the condition M(2,f) uniformly in t > 0 and is supported in the estimates in the proof of Lemma 1 it follows that

{1f.1 "" lit}, from

f

1(1/Jt

* k)(y Wdy :s; c2- i n(2i Itt- U

,

2i

t,

and, by Plancherel's formula, that

f Thus,

l1/Jt*k(yWdy:S;cC n

,

t>O.

JR" l1/Jt * k(y)/2(l + (IYllt)) 2Atn dy :s; c, and we get G",(z, t) :s; c

f

JR"

1Fc/>(y, tW{1

+ (Iz -

yl))- 2At- n dy.

Substituing this estimate in the definition of Sl,2( G"" x) we obtain the desired estimate after changing the order of integration. This completes the proof. • If the Schwartz function 1/J satisfies in addition 11/J(01 c> 0 on the annulus {1f.1 "" I} and the vector-valued function ¢Y in Theorem 13 in addition satisfies ¢Y(O) = 0, then we have

Ilk * 111m: ""

and II/l1m; "" IISl,2(Fc/»IILl: , when w E A oo • Using Theorem 13 together with Theorems 4 and 5 in Chapter V we get the following result.

II Sl,2(G",)IILl:'

176

Theorem 14. Let 1/ be a doubling weighted measure with respect to the Lebesgue measure on E"; 1/ E D n(}, 0 1, and let dl/(x) = w(x)dx, with w E As, 8 > 1. Further, let m E M(2,f). Then, the associated multiplier operator T satisfies the estimate

provided

0 0, by definition there is f E :F with f(s) 1I111F II xIlA. + E. But,

IIfllF and the inequality

IIf(it)II Ao also holds as

y-s

E

let

+ it)IIAl = e-.\(I-s)lIf(l + t)IIA

= SUPt IIf(it)IIAo /

IIgllF =

oX

IIf(1 + it)IIAl

> 0 is arbitrary.

=

x and

Y,

•

This observation concerning the norm in As motivates our next result; first some notations. If 0 is a real number, 0 0 1, and wand v are A,>o weights, we let (1) O(x) = w(x)(v(x)/w(x))8, dO(x) = 8(x)dx. Thus, informallly, the "0" measure corresponds to w, while the "I" measure corresponds to v. Also, if 0 < Po PI < 00, and zEn, let p(z) be defined by 1 1- z z -=--+-. p(z) Po PI

When 0 quantity

< z =s
t(x) is uniformly continuous and bounded for z in nand (x,t) in any compact subset of {(x, t): x E H"; t > O} and analytic for z in the interior of n; here 4> is a Schwartz function with nonvanishing integral. If f(j + it) E H? (R n ) and SUPt IIfU + it)II H,Pj < 00 for j = 0,1, then fez) belongs to when z = sand

IIf(s)lIw

"

rv

IIMo(F("s»IILP

"

s

y-s

rv

i

IlMo(F(·,l IIf(1 + it) II Hrl

+ it» II Lr

Y

1

Y

Here M« (F(., z» denotes the radial maximal function of F(·, z) defined with 4>. On the other hand, if f E Do, there is a function f( z) of the variable zEn such that F( x, t, z) has the properties stated in the first part of the Theorem, f(s) = l, and

IIf( u

+ it)IIHP(u)

,,(u)

rv

IIMo(F(·, u

+ it»II LP(u)

cIlMo(F(·,s»IIL:

,,(u)

rv

IIfIlH;,

u

+ it

En.

Combining Theorems 2, 3 and well-known properties of intermediate spaces we obtain the desired interpolation theorem for analytic families of operators. Theorem 4. Let 0 < PO,Pl < 00, and suppose (Ao,At} is an interpolation pair of Banach spaces. Further, let 1), 1)s, 0 s 1, be defined by (i) 1) = A o n AI, and Ti, = As, 0 s 1; or (ii) 1) = {f: f = 'E j Ajaj where the ai's are (oo,N) atoms with N = N( w, v) appropriately chosen}, and 1)s = Hf(R n ) ; or

180

(iii) V = {f: f = Lj AjXEj is a simple function}, and V s = Also, let 0 < qo, ql < 00, and suppose (B o, Bd is an interpolation pair of

Banach spaces. Suppose Rand R s , 0 :s s :s 1, are defined by either (iv) R = B o + B l , and Tc, = B s , 0 :s s :s 1; or (v) R = {f: f E S'(R n )} , and R s = HZ(R n ) , where l/q = (1 - s)/qo + S/ql, P, = sq/ql, 0 :s s :s 1; or (vi) R = f: f is Lebesgue measurable}, and R s = Finally, assume that {Tz } is a family of linear operators defined on V with values in R for z E Q . Also, suppose that, in case (iv), for each continuous linear functional e on Eo + Bl, l(Tz!) is continuous and bounded for z E Q, analytic for z in the interior of Q and all f E V; in case (v), that Tzf * 'k (V(Bk)/IBkl)l/P XEk I LP '" IILk>'kXEk wl/pll LP= IILk>'kXEk '" I L k >'kXBk . This completes the proof.

•

Proof of the second half of Theorem 3. Let f = L: k >'kak, supp ak Bk, denote the atomic decomposition of f into (00, N) atoms, with an appropriately large N; we can also use (q,N) atoms for q and N sufficiently large. This decomposition may be assumed to have the further property that

all r>O.

(4)

To define fez) we put

x ( )= kz

>.p/p(z)

k

(V(Bk»)(Z-S)P/POPl ii(Bk) ,

where dii(x) = v(x)dx, and set

fez)

= L>'k(z)ak. k

We begin by checking that fez) has the desired properties. Clearly f(8) = f and for u + it E fl, by Lemma 8 we have

IIf( u + which in turn is dominated by

ell L

k I>'k( u + it)lxBk

186

Whence,

IIf('ll +

1

)I/P(U) p(u) 1 XBk IIL"(u) ( IBk I Bk w(x)I-ILV(XY'dx

ell L k

,,(u)

p/p(u)

e L..Jk Ak

Thus, by (4) with r

XBk

IIP(U)

L"(U)·

"

= plp('ll) there, we get

IIf('ll +

..)

eIlU*)P/p(u)

,,( .. )

,,(u)

ellJ*lIi:

.

In the case 0 < Po PI 1, the above estimates can be obtained in a somewhat simpler fashion, without making use of Lemmas 8 and 9, which are fairly complicated when 0 < P < 1. Indeed, we have

IIf( u +

L IAk( u + it)!p(u)v(Bk) = L At(v(Bk)lv(Bk))Uv(Bk) , k e

,,(u)

k

which, by Lemma 7, is less than or equal to

eLk At (lk W(x)I-ILV(X)1L dX) Next we show that fez) E observe that

IAk('ll + it)1

ellfllfI" .

+ HEl(R n) for z in Q. Fix 'll +it E Q and = Ak(O)I-U Ak(1)U .

Thus, for any k, at least one of the inequalities holds. Setting fo = Lk Ak('ll + it)ak, where the sum is taken over those k's for which IAk( 'll + it)1 Ak(O), and II = f - fo, we get that h E (R n ) , j = 0,1, and IIhIl H? c k Ak(j)XBk ilL"; cllfll;It, j = 0,1.

IIL

Hfj

J

187

It remains to show that F(z, x, t) is uniformly continuous and bounded when (x,t) lies in any compact subset J( of {(x,t):t > O}, and analytic for z in the interior of Q. Given E > 0, let IN = Li"=1 Akak be a finite sum of atoms such

that

111- INllw,. ::; C 112: k=N+1 AkXBkII 00

< E.

Now, it is readily checked that N

FN(Z,x,t)

= IN(Z) * . condition, 12 Calderon's complex interpolation method, 177 Covering lemma, 2

Vo, 60

Decomposition of weights, 18 Distributions, 60 "continuous" representation formula, 63 "discrete" representation formula, 63 extension, 60 Doubling Db condition, 2 critical index, 12 Grand maximal function, 86 Hardy-Littlewood maximal function, 2 restricted weak-type (p,p), 4 type (p,p), 3 weak-type (1,1),3 weighted version, 5 Hardy spaces, 85 atomic decomposition, 111 complex interpolation, 179 dense class, 103 dual space, 134

multipliers, 150, 161, 162, 164 Heat equation, 86 basic inequality, 126 mean value inequality, 86 nontangentiallimits, 103 Hormander's multiplier theorem, 159 John-Nirenberg inequality, 33 Kernels, 151 M(g,!.) condition, 151 Littlewood-Paley function, 48, 49 Local maximal function Mo,J.L,s, 31 Local sharp maximal function 30 ' , Lusin or area function, 49 Multipliers, 150, 161, 162, 164 M(q,l) condition, 151 Nonisotropic metric, 59 Nontangentiallimits, 103 Nontangential maximal function, 48, 56 Normalized distance function, 13 Oscillation, 32 A"fU,B),32 Pseudo-differential operators, 174 Poisson kernel, 63 Radial maximal function, 56 Reverse doubling RD d condition, 11

Index Reverse Holder RHr(JL) condition, 6 critical index, 11 Sharp maximal function 30 Singular integral operators, '150, 172 on atoms, 157 Tangential maximal function, 56 Telescoping chain of balls, 15 Telescoping sequence of collection of balls, 15 Weighted measures, 1 doubling Db condition, 2 reverse doubling condition, 11 relative doubling condition, 13 relative reverse doubling condition,13 Weights, 1 Al (JL) condition, 5 Ap(JL) condition, 4 Aoo(JL) condition, 6 RHr(JL) condition, 6 decomposition of, 18

193

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