Hardy Spaces on Homogeneous Groups. (MN-28), Volume 28 9780691222455, 069108310X

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HARDY SPACES ON HOMOGENEOUS GROUPS

by G. B.

Folland and E. M. Stein

Mathematical Notes 28

Princeton University Press and University of Tokyo Press

Princeton, New Jersey 1982

Copyright © 1982 by Princeton University Press All Rights Reserved Published in Japan exclusively by University of Tokyo Press; in other parts of the world by Princeton University Press.

Library of Congress Cataloging-in-Publication Data Folland, G. B. Hardy spaces on homogeneous groups. (Princeton mathematical notes ; 28) Bibliography: p. Includes indexes. 1. Hardy spaces. 2. Functions of real variables. groups. I. Stein, Elias M., 1931II. Title. III. Series. QA331.5.F64 515.7-3 82-47594 ISBN 0-691-08310-X (pbk.) AACR2

3. Lie

Printed in the United States of America The Princeton Mathematical Notes are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein

TABLE OF CONTENTS

INTRODUCTION

iii

Remarks on Notation

xii

CHAPTER 1: Background on Homogeneous Groups A. Homogeneous Groups

1 2

B. Convolutions C. Derivatives and Polynomials D. The Schwartz Class

15 20 35

E. F. G.

1+5 53 55 6l

Integral Representations of the 6 Function Covering Lemmas The Heat Kernel on Stratified Groups Notes and References

CHAPTER 2: Maximal Functions and Atoms Notes and References

62 79

CHAPTER 3: Decomposition and Interpolation Theorems A. The Calderon-Zygmund Decomposition

80 8l

B. The Atomic Decomposition C. Interpolation Theorems Notes and References CHAPTER k: Other Maximal Function Characterizations of H P A. Relationships Among Maximal Functions B. Construction of Commutative Approximate Identities Notes and References

97 107 111 113 113 128 139

Duals of

HP

A.

The Dual of

HP

B.

BMO

C.

Lipschitz Classes

156

Notes and References

182

CHAPTER 5:

CHAPTER 6:

spaces:

Campanato Spaces

1U6

Convolution Operators on

A.

Kernels of Type

B.

A Multiplier

HP

(a,r)

Theorem

Notes and References CHAPTER 7:

Characterization of H P by Square Functions: The Lusin and Littlewood-Paley Functions

Notes and References CHAPTER 8:

lUl lUl

Boundary Value Problems

l8U l8U 208 215

217 2k6 2^7

A.

Temperatures on Stratified Groups

2^8

B.

Poisson Integrals on Stratified Groups

253

C.

Poisson Integrals on Symmetric Spaces

26l

Notes and References

272

BIBLIOGRAPHY

273

Index of Terminology

28l

Index of Notation

283

INTRODUCTION

The object of this monograph is to give an exposition of the realvariable theory of Hardy spaces

(HP spaces).

This theory has attracted

considerable attention in recent years because it led to a better understanding in B

of such related topics as singular integrals, multiplier

operators, maximal functions, and real-variable methods generally.

Because

of its fruitful development it seems to us that a systematic exposition of some of the main parts of this theory is now desirable.

There are,

however, good reasons why in addition the theory should be recast in the more general setting where the underlying 3R group.

is replaced by a homogeneous

The justification for this wider scope, both in terms of the

structure of the theory and its applications, will be described in more detail below. Background:

Development of

The theory of

HP

H

theory

spaces is a multifaceted one with a rich history.

Here we can do no more than sketch its highlights . Originally functions

F

H

(0 < p < ») was defined to be the space of holomorphic

on the unit disc or upper half-plane such that

See also the overview by C. Fefferman [2].

su

Pn< 0, x G G ,

6 x

for

we shall denote them also by

6

reserving the notation

and call them

6

Sometimes we shall even write

is suggested by the case

G = Bn,

given by scalar multiplication.

x/r

for

with IR 0

6

—

6 , x.

6

as a

This notation

in which the natural dilations are

(Note, however, that we write the group

law multiplicatively, so the distributive law morphism property of

instead

for occasions

when it is required for clarity or when we wish to consider mapping.

rx

becomes

—

that is, the auto-

r(xy) = (rx)(ry).)

The analogy

is strong enough to persuade us to denote the group identity by

and to refer to it as the origin.

peculiar, but the equation

lim

The equation

rx = 0

xx" = 0

thus looks

comes out looking right.

Some examples may be in order. (1)

Abelian groups:

]R

is a homogeneous group with dilations given

by scalar multiplication. (2) group

H

Heisenberg groups:

If

n

is a positive integer, the Heisenberg

is the group whose underlying manifold is

(C x J?

and whose

multiplication is given by

H

is a homogeneous group with dilations 6r(z1,...,zn,t) = (rZ;L,...,rzn,r t ) . (3) Upper triangular groups:

matrices

(a. .)

such that

Let

G

be the group of all

nxn

real

a. . = 1 for 1 < i < n and a. . = 0 when i > j .

G

is a homogeneous group with dilations

These examples are all stratified groups.

It is also possible to

define other families of dilations on these groups.

For instance, on 3R

we can define d.

where

1 = d_ < d o < ••• < d , and on 1 - d n

H

min{a ,...a ,b ,.. . ,b } = 1

when we refer to 1

or

H

we can define

n b

a,

6r(x1+iy1,...,xn+iyn,t) = (r ^ i r where

d

and

±

a

b

y1,...,r \ + i r

a.+b. = c

X ^ t )

for all

j. However,

we shall assume that they are equipped with

the natural dilations defined in (l) and (2) above unless we state otherwise. Henceforth we shall be working on a fixed homogeneous group dimension

n

with dilations

the eigenvalues of

A,

{6

= exp(A log r)}. We denote by

of

d ,...,d

listed in increasing order and with each eigenvalue

listed as many times as its multiplicity, and we set 1 = d < d < ••• < d 1 - 2 - n We also fix a basis

G

Xl9...,X

and we define a Euclidean norm

of

g

Q

Thus

= d.

such that

II • II on

d = max{d.}. J

AX. = d.X.

for each j ,

by declaring the

orthonormal.

We may also regard this norm as a function on

obvious way:

llxll = llexp xll.

G

X.'s

to be

in the

The Euclidean norm is of limited utility for our purposes, since it does not interact in a simple fashion with dilations.

We therefore define

a homogeneous norm on

x -> |x| from

to

[0,~) which is

|rx| = r|x| x = 0.

G

to be a continuous function

C°° on

for all

xGG

G\{0} and

X ^ 0,

tends to

is

then,

0

or

II6 Xll

C

|X| = 0

and on

if and only if

^)\

is a strictly increasing function of r.

Hence there is a unique

and we may define a homogeneous norm on

|x| = l/r(exp~ x) G\{0}

116 Xtl = (Z^c2r

implies

°° along with

116 r vXll = 1,

101 = 0

(b)

|x""11 = |x| and

Observe that X = EVx.Gg

that

r > 0,

(a)

Homogeneous norms always exist; one example may be constructed as

follows.

If

and satisfies

for

x / 0.

r

which

r(X) > 0 G

00

C

Henceforth we assume that

(The fact that this function

manifold.) G

_is_ equipped with a_ fixed homogeneous

norm. x6G

and

r > 0

we define

B(r,x) = {yGG : \x~1y\ < r}, and we call B(r,x)

B(r,x)

the ball of radius

is the left translate by

image under (l.U)

6

of

LEMMA.

x

of

r

about

B(r,0),

x.

We observe that

which in turn is the

B(l,0). For all

xGG

and

r > 0,

such

by setting

follows from the implicit function theorem since the

Euclidean unit sphere is a

If

G

B(r,x)

is compact.

Proof:

Let us define l/d. p(x) = E°|c.| J

Then

p

and does not contain n

on it.

that

|x| > np(x)

Thus

B(n»0)

B(r,x)

0,

|rx| = r|x|

for all

x,

x •* |x| assumes a positive

and

p(rx) = rp(x),

and hence that

r > 0,

xGG.

There exist

(^ llxll < |x| < C2llxlll/d

If

y = exp(Zc.X.) J J

r llyll < llryll < rllyll a positive maximum x ^ 0

is compact

it follows

B(n,0) c {x : p(x) < 1}.

is compact, and it follows by dilation and translation that

is compact for all

Proof:

{x : p(x) = 1 }

C . Clearly

so the function

Since

(1.5) PROPOSITION.

Any

x = exp(z![c.X.)

satisfies all the properties of a homogeneous norm except that it

is merely continuous instead of

minimum

for

for C

r < 1.

C ,C

> 0

whenever

then

2 2d i ± llryll = (Ec.r J ) J

where

|xy|
|x|dHyll > C " d | x | d .

There is a constant

x,yeG,

2

By Lemma l.k, the Euclidean norm assumes

x = |x|y

| llyll < C ^ l x l ,

such that

|x| < 1.

and a positive minimum

can be written as

(1.6) PROPOSITION.

#

C > 0

such that for all

10

Proof:

By Lemma l.k9 the set

compact, so the function it.

Then, given any

{(x,y)e G x G : |x| + |y| = 1}

(x,y) ->• |xy|

x,yGG, set

assumes a finite maximum

is C

on

r = |x| + |y| . It follows that

|xy| = rlr^txy)! = r | (r^x) (r'V) I f Cr = C(|x| + |y|). #

By Proposition 1.2, Lebesgue measure on Haar measure on

G.

Q

We_ now fix the normalization of Haar measure on

by requiring that the measure of the usual norm, our measure is

B(l,0)

be_ 1.

r( (n+2)/2)/7r

denote the integral of a function or

f

G

G = B n with

(Thus, if

times Lebesgue measure.)

We shall denote the measure of any measurable

Jt

induces a bi-invariant

E c G

by

|E|, and we shall

with respect to this measure simply by

/f(x)dx.

The number Q = E^d. = trace(A) will be called the homogeneous dimension of |6p(E)| = rQ|E|, In particular,

|B(r,x)| = r^

A

( A G E ) if

f

on

G\{0}

f°5r = r f

Clearly we have

d(rx) = rQdx.

for all

r > 0, x 6 G .

always denote the homogeneous dimension of A function

G.

Henceforth

Q

will

and

g,

G.

will be called homogeneous of degree

for

r > 0.

We note that for any

f

jf(x)go xy

smooth, so by the mean value theorem and Proposition 1.5, |f(xy) - f(x)| < Cllyll < C'|y| = C |y | |x | X ' 1 . #

This proposition may be applied, in particular, to the homogeneous norm.

Combining this with Proposition 1.6, we see that there exists a

constant (1.8)

y

such that |xy| < y (|x| + |y|)

for all

x,yGG,

is

12

(1.9)

||xy| - |x|| < y|y|

Henceforth

y

and (1.9).

Clearly

for all x,yQG

such that

|y| < |x|/2.

will always denote the minimal constant satisfying (1.8) y > 1.

We shall use (1.8) and (1.9) without comment

in the sequel. The following simple fact will be useful later: (1.10)

LEMMA.

Proof:

Since

For all

xjeG

and

s > 0,

|x| < y(|xy~ |+|y|) we have

1 + |x| < Yd+lxy^lKl+iyl), and we obtain the desired inequality by raising both sides to the

s-th

power. # We now establish our notation for some common function spaces on (with apologies for some slight inconsistencies). If

Q, c G, C(^),

(C (tt),C (Q)) denotes the space of continuous functions on compact support, vanishing at infinity). the class of

k

If

Q

G

is open,

C

G

(with

(k) (tt) denotes

times continuously differentiable functions on fi,

C*(a) = r\^ C^(ft),

and

C~(ft) = C°°(ft) n CQ(a).

usually omit mentioning it. Lebesgue space on

G.

For

If

0 < p < °°, L

0 < p < °° we write

llfllp= (j|f(x)| p dx) 1 / p ,

When

Q = G

we shall

will denote the usual

13

even though t h i s i s not a norm for however, a metric on

L

We recall that if function

for f

p < 1.

(f,g) -> Hf-gllP

(The map

is,

P

p < 1.)

is a measurable function on

G,

its distribution

X f : [0,°°) -> [0,°°] is defined by

(1.11)

Af(a) = |{x : |f(x)| > a}|,

and its nonincreasing rearrangement (1.12)

f* : [0, [0, Q a P A f (a) = sup t>o t l/p f*(t) < «,. It is easily checked that

[f]

< llfll

(Chebyshev's inequality).

not a norm, but it defines a topology on weak which is bounded from

L

to weak

L

[ ]

is

L . A subadditive operator

is said to be of weak type

(p,q).

We conclude this section with some results concerning integration in "polar coordinates". (1.13) o_f degree

PROPOSITION.

I£

f

is continuous on

-Q, there is ji constant

y

G\{0}

(the "mean value" of

i

for all g G l ^ U O ^ K r ^ d r ) , (1.1*0

and homogeneous

f f(x)g(|x|)dx = y I g(r)r"1dr.

f)

such that ————

— — —

Ill

Proof :

Define

L f : (0,«) + QJ by

Mr) = l sx

and using the homogeneity of

L (rs) = Lf(r)+Lf,(s)

is continuous, it follows that

for all

L (r) = L (e)log r,

Then equation (l.lU) is obvious when

g

r,s > 0.

and we set

Since

f, L

y f = L f (e).

is the characteristic function of

an interval, and it follows in general by taking linear combinations and limits of such functions. (1.15)

PROPOSITION.

Radon measure

a

on_ S

then

?

If

Let_ S = {x G G : |x| = 1}. There is a (unique) such that for all

u G L (G),

f u(x)dx = r fu(ry)r

da(y)dr.

J

Proof:

#

j

G

0 ji

let us define f on G\{0} by f(x) = |x|""Qf ( Ixl^x)

fGC(S),

satisfies the hypotheses of Proposition 1.13.

clearly a positive linear functional on against a Radon measure

a

on

S.

If

0 jS

f ->- y~

is

C(s), so it is given by integration g 6 C ((0,»)),

f f d x I ' ^ g d x D a x = f f(x)|x| Q g(|x|)dx =

j

The map

w?

f(y)g(r)rQ"1da(y)d

then, we have

Since linear combinations of functions of the form dense in

L (G), the theorem follows.

Remark:

The measure

o

f(|x| x)g(|x|)

are

#

can be shown to have a smooth density:

cf.

Fabes and Riviere [l] and Goodman [2], (1.16)

COROLLARY*

Let

C = a(S).

Then if

|x|a-Qdx = C c T ^ - a 0 1 )

J

0 < a < b < ~

if

a t 0,

if

a = 0.

and

aG

a 0, set

is the conjugate exponent to g-Jx) = 0

otherwise,

p. Define

and set

g 2 = g-g 1 -

Since

W

- f*Sl

x

f*g2(a/2)'

it suffices to show that each term on the right is bounded by C

depends only on

we have

1

p -q > 0

p

and

q.

On the one hand, since

q

Ca"

- (p1)

where = r

and hence

g1 fx)|P'dx = p'f a 5 '" 1 *g (a)da < p>f a 5 '"s1 * (a)da J 0 l " J o V o

= ^ - M P ' - I = ±U^'/r = (a/2)pl. P'-4

Thus for all xG G,

|f* g ; L (x) I < llfllpllglllpI < a / 2 ,

1

> 0,

18

which implies that

A

(a/2) = 0 .

On the other hand, since

,]V[

,00

|g (x)|dx =

A J0

S

(a)da = 2

q > 1,

.00

A (M)da + A (a)da •'0 s -"M S

and hence by Proposition 1.18,

p But then

g

(a/2) < [2llf*g 2 ll p /a] P

P _, pr(l-q)/q pr(l-q)/qp' P P n fl) < (£.) (_2_) tSL)

= C(p,q)a" ,

so we are done. §

We now summarize the basic facts about approximations to the identity. The following notation will be used throughout this monograph: a function on G

and t > 0,

we define

t = t"Qcj)o 6 1 / t ,

We observe that if G L

then

if $ is

< j > by

i.e., cf)t(x) = t~Q(j)(x/t).

.(x)dx

is independent of t.

19

(1.20)

PROPOSITipN.

(i)

If

feLP

(ii)

Lf

f

llf**.-afll T t

oo

->• 0

(iii) then

Suppose

(1 < p < « ) ,

i^GL1 then

and

U(x)dx = a.

llf* -afll

-> 0

as

Then: t + 0.

i s "bounded and r i g h t uniformly continuous, then

as —

If_ f

t -*• 0. ijs_ "bounded on G and continuous on an open set ft c G,

f*4>, - af •* 0 uniformly on compact subsets of ft as t -> 0. Proof:

If f

f^x) = f U y " 1 ) .

is any function on G

If f G L P ,

(1.21)

and y&G,

let us define

1 < p < «, it is easily seen that

llf^-fl! -*• 0 as y -> 0.

(Use the fact that only if f function.

C

is dense in LP.) If p = «,

(l.2l) holds if and

is (almost everywhere equal to) a right uniformly continuous We observe that f*cj)t(x) -af(x) = |f(xy""1)t""Q4)(y/t)dy-af(x)

=

f(x(tz)"1)4)(z)dz-af(x)

=J> Hence by Minkowski's inequality, llf*, < IIf Z - f l l |Uy( vz ) | d z . y - a f II t p - J p " Since

llftZ-fll

< 211 fII , under the hypothesis of (i) or (ii) it follows

from (1.21) and the dominated convergence theorem that

llf*.-af II •> 0. L p

20

The routine modification of this argument (with (iii) is left to the reader.

C.

p = »)

needed to establish

#

Derivatives and Polynomials

There are three common ways of viewing the elements of the Lie algebra g

of

G:

(l) as tangent vectors at the origin,

vector fields,

(2) as left-invariant

(3) as right-invariant vector fields. We shall have no use

for the first interpretation, but we shall need both of the others. Accordingly, let us denote by

Q

and

and right-invariant vector fields on of identifying

g

with

g

G.

the spaces of left-invariant We shall follow the usual custom

g , and in particular we shall think of the

exponential map as going from

gT

to

G.

(This point is not entirely

JJ

trivial, since the map which sends with

X

Xeg

to the unique

XGg-, which agrees

at the origin is an anti-isomorphism rather than an isomorphism:

that is,

[X,Y]~ = [?,X].

formula to

Hence if one applies the Campbell-Hausdorff

g , one obtains a different, although isomorphic, group law for

G.) We recall that in Section A we fixed a basis consisting of eigenvectors for the dilations d

r

l

d

,...,r

n

. The

X.'s o

differential operators on basis for

gR:

Thus, for

f G (T

6

X ,. .. ,X

for g

with eigenvalues

are now to be regarded as left-invariant G,

that is, Y.

and we denote by is the element of

Y , ...,Y gR

the corresponding

such that

Y.|Q = X.|Q.

,

X.f(y) = -£rf(yexp(tX.))|, n , j at j 't=0

Y.f (y) = -£-f (exp(tX. )-y) | . j at j 't-u

21

The differential operators

X.

and

Y.

are homogeneous of degree

d.,

for Xj(f°6 r )(y) =^f((ry)exp(r

J

tXj)|t=Q

d. „ d. = r d ^-f((ry)exp(tX.)) I n = r J(X.f ( dt j 't=0 j and similarly for

Y..

We adopt the folloving multiindex notation for higher order derivatives. If

I = (i ,...9in) GlT n ,

we set

2

••• X n,

X1 = X^X

Y1 = Y "4 2 ••• Y n.

By the Poincare*-Birkhoff-Witt theorem (cf. Bourbaki [l], I.2.T), the operators X (resp.

Y ) form a basis for the algebra of left-invariant (resp. right-

invariant) differential operators on

G.

Further, we set

Thus

|l| is the order of the differential operators

d(l)

is their degree of homogeneity, or, as we shall say, their homogeneous

degree. by I

We shall denote by

A

If .

A

is the set of all numbers

We observe that i c A

We pause to make two useful remarks. translations are isometries on skew-adjoint.

and

Y , vhile

the additive sub-semigroup of JR generated

0,d ,...,d . In other words, ranges over

X

2

since

d

as

=1.

First, since left and right

L , the operators

X.

Thus,

J(xJf)g = (-D^jftfg),

d(l)

Jf(Yxg) = (-D^

and

Y.

are formally

22

for all smooth infinity.

f

and

g

such that the integrands decay suitably at

Second, the operators

X

and

Y

interact with convolutions

in the following way: XJ(f*g) = f t t ^ g ) ,

YI(f*g) = ( Y ^ U g ,

( X ^ U g = f*(YTg).

The first two of these equations are established by differentiating under the integral sign, while the third is established by integration by parts: (XIf)*g(x) = |xIf(xy)g(y"1)dy =

(-1)I1'jf(xy)XI[g(y"1)]dy

= |f(xy)(YIg)(y"1)dy = f*(YIg)(x). We now investigate polynomials on called a polynomial if £,...,£ for

g,

Po exp

G.

A function

is a polynomial on

the basis for the linear forms on and we set

r\. = £. o exp J

J

. Then

which form a global coordinate system on polynomials on

G.

G

g

_ i (n = n 1

T

n , ...,n n

degree. If

N G I

—

will be

We denote by X ,.. . ,X

are polynomials on

G

G

can be written uniquely as

i ••• T^11, a^ffi) a

vanish.

Clearly r\

d(l), so the set of possible degrees of homogeneity

for polynomials is the set refer to its degree

G

and generate the algebra of

where all but finitely many of the coefficients is homogeneous of degree

on

dual to the basis

j.

Thus, every polynomial on

p = E^n

g.

P

A

introduced above.

that is, max{|l| : a

If

^ 0}

P = £aTn , we shall —

as its isotropic

Further, we define its homogeneous degree to be max{d(l) : a^ f 0}. we denote the space of polynomials of isotropic degree

< N

by

23

P

, and if

degree

< a

aGA

by

we denote the space of polynomials of homogeneous

P . Since a

p c P* so c pN N dN

for

1 < d. < d - j -

for

j = l,...,n, we clearly have

NGI.

We can now give a more explicit description of the group law in terms of the coordinates

n . • Since the map 3

(x,y) -> n.(xy) J

is a polynomial on

GxG

which is jointly homogeneous of degree d. (that is, J d n.((rx)(ry)) = r n.(xy)), and since the Campbell-Hausdorff formula 2, ~

we must have (1.22)

ru(xy) = TK(X) + nj (y) + I ^ 0 > J ^ u

for some constants

C. . Since the multiindices J

I

and

J

in (1.22) must

satisfy

d(l) < d. and d(j) < d., it follows that the monomials 3 3 can only involve coordinates with degrees of homogeneity less than and in particular can only involve the coordinates

n 9. • . 9ri.

n ,n d., 3

. We note

two special cases: (1.23)

d. = l:

n.(xy) = n.(x) + n.(y).

(1.2U)

d. = 2:

n.(xy) = n. (x) + n .(y) + L

3 (1.25)

3 PROPOSITION.

3

3

For_ anv_ a G A ,

V V

_-, _n C.£n (x)n (y). 1

3

the space

k

l

P

is invariant

under left and right translations. Proof: function of Since the

From (1.22) it is clear that x

n.'s 3

for each

y,

n.(xy)

is in

and also as a function of

P y

as a for each

x.

generate all polynomials, the result follows immediately.

#

2k

Remark: G

P

is Abelian). (1.26)

is not invariant under translations (unless

PROPOSITION.

We have

' V

Proof:

P.. ,Q._ JK Jk

For

or

Consequently, it will not be of much use to us.

=X

k ^ j, and

N = 0

V

=

° ^

d

k < dj ^ ^

d =d

k J SSi

are homogeneous polynomials of degreee

xGG

define

different iable function

f

on

L x : G -> G G

by

Lx(y) = xy.

cL-d. K J

if

Then for any

and X G G ,

X.f(x) = (X.f)oL (0) = X,(foL )(0) = (3/3n.)(foL )(0) 3 J x j x j x since

X. agrees with 1,

#

—

will occur

3 will always denote this G = ]Rn

with equality when

and

|»|

is the Euclidean norm. (1.36)

LEMMA.

Proof:

Since

If

aGA

then

|l| < d(l)

On the other hand, since

d

max{ 111 : d(l) < a} = [a] .

for all

= 1,

for

I,

d(l) < a

implies

I = ([a],0,0,...,0)

|l| < [a].

we have

|l| = d(l) = [a]. # (1.3T) THEOREM (TAYLOR INEQUALITY). k = [a]. There is &_ constant class

C

cm

G

and all

C

> 0

degree

P a.

aGA

(a > 0),

such that for all functions

and f

x,yGG,

W where

Suppose

d(l)

is the right Taylor polynomial of

f

at_ x

of_ homogeneous

of

30

Proof: while

Let

YJg(0) = 0

g(y) = f(yx)-P (y). Then

Y g(y) = Y f(yx)

1 < m < k+1, that if

for

d(j) > a.

a-m < d(j) < a

for

d(j) < a,

We shall show by induction on

m,

then

|Y J g ( y ) | < C m I |

(1-38)

The desired result then follows by taking First suppose

a-1 < d(j) < a.

m = k+1

and

J = 0.

Then by the mean value theorem (1.33),

YJg(0) = 0,

since

|YJg(y)| < C I*=1 | y M s*P|z| a

then

Y g(z) = Y I f(zx),

so

S.

is dominated by the right

j

«J

hand side of (1.38) (since d(j)+d. > a-m+1 3 obtaining

sup

3 > l ) . If d(j)+d. < a we have ~* J ~* and we can apply the inductive hypothesis to

Y g(z),

z|°(Y) = exp Y, ^ X

where

. (Y) = [•••[[exp Y,exp X. ],exp X. ],...,exp X. ] X 1 l "" 1j ^l 2 j

[x,y] = xyx~ y~ .

By the Campbell-Hausdorff formula, for any

X,Yeg

we have [exp X,exp Y] = exp([X,Y] + higher order terms). Therefore, if we identify the tangent spaces of B Vn

and g

i s given by

and G

at the origin with

respectively, we see that the differential of ^. X l " # Xj

at 0

d°(O)(Y) = Y, X, ] , . . . 2

Now consider the map

from the (ZQ"" 1 ^ )-fold product of B

with itself into

G. Since

V

33

generates

g,

the preceding remarks (together with another application

of Campbell-Hausdorff) show that the differential onto

g.

Consequently, there exists

includes all of

G

xGG

with

is a product of

|x| < 6. 3 # 2 J -2

be written as the product of whose norms are at most as the product of |x|/ 0

d(j)(O)

such that the range of

Since a commutator of

elements, any J

xeG

J

N = Z^" v (3'2 -2)

exp(V )

j+1

with

$

elements

|x| < 6 can

elements of

By dilation, then, any

elements of

is surjective

xGG

exp(V )

can be written

whose norms are at most

#

G

is stratified, we have

A = If,

to be the space of continuous functions

derivatives

X f

and for f

are continuous functions on

on G

connection it is worthwhile to note that since

kGlf

G

whose

for

V

we define (distribution)

d(l) < k.

generates

g,

In this the set

of left-invariant differential operators which are homogeneous of degree j

—

that is, the linear span of

linear span of the operators

(l.Ul) There exist

{X : d(l) = j}

X. ••• X. ix ij

with

STRATIFIED MEAN VALUE THEOREM. C > 0

and

b > 0

is precisely the

1 < i < v - k -

Suppose

such that for all

|f(xy)-f(x)| f C | y | s u P | z | ? ) | y | j l ^ v

Proof;

—

G

fG C

for

k = l,...,j.

is stratified. and all

x,y6 G,

|x.f(xz)|.

The proof is identical to the proof of Theorem 1.33 except

that one makes the initial estimate only for Lemma 1.1+0 instead of Lemma 1.31.

#

yGexp(V )

and then uses

(1.1*2)

THEOREM (STRATIFIED TAYLOR INEQUALITY).

stratified.

For each positive integer

_____________-

______

G

there is a constant

is_ C

such

j£ _______

feCk

that for all

k

Suppose

and all_

x,yGG,

|f(xy)-P x (y)| < C k | y | k n ( x , b k | y | ) ,

where

P

is the left Taylor polynomial of

________

•£

degree

k,

_ _

______ _______

b

f

at

x

____

of homogeneous ___

_is_ as_ jLri Theorem l.Ul, and for

r > 0,

n(x,r) = s u P | z | < r j d ( I ) = k lx f(xz)-XIf(x)|.

Proof:

Let

g(x) = f(xy)-P x (y),

We shall show by induction on

|j|= k

then

X P

that if

= 0

for

d(l) < k.

d(j) = k-m

then

is a constant function, hence

X J P x ( y ) = X J P x ( x ) = X J f(x) m = 0,

0 < m < k,

X^O)

|X J g(y)| < C.|y| m n(x,b m |y|).

(1.1*3)

If

m,

so that

and so

X J g(y) = X J f(xy)-X J f(x).

(1.U3) is just the definition of

| j | = k-m+1,

and

suppose

| j | = k-m.

n.

Thus for

Suppose (1.U3) is true for

Then by Theorem l.Ul, since

X J g(0) = 0,

|x J g(y)| < C | y | s u p | z | < b | y | 5 l ; . f v

X.X J is a linear combination of J so by inductive hypotheses,

But

XIfs

|x.X J g(xz)|.

with

D(l) = d(j)+l = k-m+1,

|XJg(y)| j in If

GS

S

i f and o n l y

and

yGG,

y(x) =

(xy),

let y

us

Il.-IL x -> 0

for a l l

N.

define

c()(x) =

cf>(yx),

$(x) =

^(x"1).

(This notation will be used consistently in this section, but not afterwards . ) (1.U6)

PROPOSITION.

that for a l l

Moreover, Proof:

6 S

and

IIy- IL v -> 0

For each

N e l

there exists

>0

such

n

yGG,

and

C

Ily(|)-(J)II, , -> 0

?is_ y -> 0 .

F i r s t , b y Lemma 1 . 1 0 ,

Next we observe that

(YI$)(x) = (-1)'I'(X1^)(x"1),

and by Proposition 1.29

we have

" E|j|d(l) P IJ Y '

P

UGPd(j)-d(l).

37

Now

| j | < |l| < N

implies

d(j)-d(l) < d(j) < d | j | < QN,

(N+l)(Q+l)+d(j)-d(l) < (2N+1)(Q+1),

hence

and thus

-1 The estimate for

then follows immediately since

y

• 0 That

II 0.

#

Convolution is continuous from

N G IT

there exists

C

> 0

By Lemma 1.10,

ay. #

S^5

to_ S.

such that for all

More

C(>,IJJGS,

38

The dual space G.

If

fGS'

and

S'

of

(J> 6 S

S

is the space of tempered distributions

we shall denote the evaluation of

f

on

on

cj) by

(x)dx. Convergence in

S'

if

< f. ,cj> > -> < f , >

and o n l y i f If

feS

f

will always mean weak convergence:

and

(> G S

for

we d e f i n e

f# G S

In particular,

Proof:

and

If

S'

f#

by

f*cj> is continuous; in fact, I.

there exist

and for any

The continuity of M6l, |

feS'

in

N G 3N

and

C > 0

such

xGG,

f*(J)GS?,

to the existence of

for all

3

.

x)dy =

It follows easily from Proposition 1.1*6 that I

f. -* f

GS.

the convolution

f(y)*(y

thus

|

f

C > 0

i> Q S

we have

as a linear functional on such that )

for

S

is equivalent

39

Thus by Proposition 1.U6, if

N = (2M+l)(Q+l)

we have

= || < C(] The verification of the second statement is a simple exercise which we leave to the reader. (I.U9)

PROPOSITION.

feSf,

\\JGS and

# Suppose

ijj*cj>, -> aij; in

. 6 S

S

f*cf>, -> af

t

Proof:

and

L(x)dx = a. in

t

S1

•

as

Then for any

t -> 0.

~—~~

For the first assertion we merely repeat the proof of

Proposition 1.20, using the norms

II II / v instead of

necessary estimates are provided by Proposition 1.U6. follows from the first:

since

$ =

and

L

norms; the

The second assertion

($) = ( G S

we have , ,i|i> = -> a t t

as

t -»• 0.

#

The remainder of this section is devoted to some technical results which we shall need later.

The first one is a global version of the

Taylor inequality (1.37) for Schwartz class functions. (1.50) THEOREM. N = [a]+l. of_ cj> at_ x

If

(j)GS and

whenever

xGG,

let

of homogeneous degree

There is a constant

d(l) < a

a 6 A,

Suppose that

Px

a,

C,

independent of

and

|x| > 2y$ |y|.

b = min{b! e A : b f > a},

and

be the right Taylor polynomial and let

R (y) = (yx)-P (y).

, such that

ko Proof: Y at

x

If

d(l) < a

then

Y P

of homogeneous degree

is the right Taylor polynomial of

a-d(l),

and

[a-d(D]

= N-l-111 .

Thus by the Taylor inequality

(1.37),

M Now J

Y Y

|z| < 3 |y| Z

implies

|z| < |x|/2y

is a linear combination of

|K| < |l|+|j| < N.

and hence, since

Y

Kf

s

d(J)

and hence with

|zx| > |x|/2,

d(K) = d(j)+d(l)

and

and

Therefore,

|l|+|j| < N

implies

d(l)+d(j) < QW,

The next sequence of lemmas deals with estimates for functions of the form 0

For every

such that

Nf = N+j(2d-l).

N,jG K

and every multiindex

I

there

Proof:

Since

Y (\p*9J 0

such

that su

where

W,s o ,..., £

Nf = N+Q+l+(Z^j.)(2d-l).

Proof:

The case

k = 1

is Lemma 1.52.

If

k = 2,

for

0 < sn < 1

h we set

\JJ! = ij;*8 S

U

S

2-

l

. S l

j

By Lemmas 1.52 and 1.53,

S

l

S

l

S

2

S

2

N+Q+l+j (2d-l)

)

I +j

+j

Since this holds for all

s 6 (0,1],

the assertion is valid for

The proof is now completed by an obvious induction on

k.

k = 2.

kk (1.55) PROPOSITION. there exists

Given

cj) G S and N,j

with

A . =

C > 0 such that for all J

f, . ^N, h

(1.56)

j 6B

k

sup__ ^n j(1+ 'x ) 3s y Lemma 1.36,

S We have proved the case

°

U»P = 0

N =1

-

The general case is now established by induction on and PGP

J

cf>P = 0

M4d]-

(,

for all

immediately.

= EX.4> 3 3

where

M

Pe

^ffdlN'

fr

°m

which the desired

N.

If

(j> 6 S

L P = 0 j3

for all

X1*^

where

result follows

#

(l.6l)

(a)

then

By inductive hypothesis, *. = l ^ ^ ) ^ ^

= 0

(where

for all P e P .M f °

for all

THEOREM.

For_ anv_ N G l

depends on, N)

L^P =

M P

there exist

cf)1,. . . 9^9ii)19.

. . ,i|;M 6 5

such that:

= 0 for_ all_ PeP , 1 < j < M, = 6.

Proof:

First pick

(|> 6 S

such that

U = 1

J polynomial

P

without constant term.

and

UP = 0

for every

J

(For example, let

cf) be the inverse

[Euclidean] Fourier transform of a Schwartz class function which is identically one near the origin.)

Set (j)' = d ,/dt | ,_-, • Since $. satisfies the same t t-~l t , T U ) = $(t£). o

If 4>GS, then to

f e S ' , we say that

f*c{>, ->• 0 t f

in

S»

as

f

vanishes weakly at infinity if, for any

t -> » .

f G LP

For example, if

vanishes weakly at infinity, since if

q

where

1 < p < «> , —

is the conjugate exponent

p,

l l f # 4 > . II

t»-

{1.6k) THEOREM. for any

f G 5f

< l l f l l I I * . II =

e

•tdt/t -> f

(j>cS,

I I f II llll

p q

t""Q/p.

U = 0, and J which vanishes weakly at infinity,

I

Suppose

p t q

I (J),dt/t = 6. 0 t

J

Then

51

Proof:

Let

we can write

a =

f1 Jo

4>,dt/t, t

$(£) = E £.$.(£)

3= with

f°° Ji

cj>,dt/t. Since t

°° .

£,

so

a

is

X

|(3/3^) $(^)| < C N (l+|c|)" $(t?)dt/t |^| > c,

for any 3

c > 0,

But

is also smooth near the origin.

and hence also

3 = 1- a = l-a(0) = 1. ,OO

3,

and are

Now observe that for

3 = 6—a = 1-a, 3,

so

s > 0,

,00

so that

rA fG5f,

then,

f*cj) dt/t = f*3£-f*3A-

Proposition 1.^9> and if A -> °° .

#

f

But

f*3

•> f

vanishes weakly at infinity,

as

e -> 0

f*3A "•" 0

3

is in

3 = dt/t = (|),dt/t, s j 1 st Jg t

If

N,

agrees with a Schwartz

class function except perhaps near the origin.

and

for any

and all of its derivatives

In other words,

Therefore

C . Also N

as

by

S,

52

Our final result shows that if

cj) G 5

and

U = 0

then

J always converges in

S',

Sf

THEOREM.

If_ 0, A -> °° i°_ iiL distribution which is

and homogeneous of degree Proof: with

(j) dt/t C

Then, if

converges in

away from the origin

-Q.

As in the proof of Theorem 1.6k we write

.GS. J

t

although usually not to the 6-function.

f (1.65)

cj) dt/t

JQ

I > Q+d

and

|^|

$(c) = 2^.$.(?)

denotes a homogeneous norm on g

r n dr1 |$(t^)|dt/t = |Z t ) •'O 100

J

^.$(t^)|dt J

d. -1

d.

t J kl J(] < CExn[

'JO 0

t

J

|c|

J

dt + I

J

t

k|

J

dt]

= C.

This shows that

Therefore

$(t£)dt/t •'0

I dt/t

J

t

converges pointwise and "boundedly, hence in

converges in

S'.

Moreover, for any

i|; 6 S

and

r > 0,

r

rA (• rA r rA/r (J), (x)i|>(rx)(dt/t)dx = 4>+(x)i|;(x)(dt/t)dx = +(x)i|;(x J t J J JQ e G J£ rt G Je/r * Letting

e -»» 0, A -»• °°,

distributions that

we see from the definition of homogeneity for

0.

(j) dt/t is homogeneous of degree

>

t

S'.

-Q. Finally, if

53

K c G

is a compact set which does not contain the origin and

multiindex, bounded as

X (x/t) t •*•«>,

vanishes to infinite order as

uniformly for

t -> 0

I

is a

and remains

X G K , S O the integrals

X 1 f . (x)dt/t = f t - Q - ^ 1 5 " 1 XX(|)(x/t)dt converge uniformly on

K.

Thus

. dt/t J

F.

0

is

C

away from the origin.

#

t

Covering Lemmas

In this section we present two useful covering lemmas, which are variants of classical results on IR

(1.66) WIENER LEMMA. positive function. open,

Suppose

(finite or infinite) sequence

Proof:

E c G

and

Assume that either (a)

| E | < °°, and B(r(x),x) c E

are disjoint, and

due to Wiener and Whitney.

E

for all

{x.}

in

E

r : E •> (0,») JLS_ an arbitrary ±s_ bounded, or_ (b)

xeE.

E

is

Then there exists a

such that the balls

B(r(x.),x )

E c W . B(Uyr(x.) ,x.) . 0 J J

We may assume that

sup

automatic, whereas in case (a), if

r(x) < °° .

sup

In case (b) this is

r(x) = °° there exists

xGE

XteHj

such that that

E c B(r(x),x),

so there is nothing to prove.

r(x ) > 1/2 sup x G £ r(x).

wise, we continue inductively: E. = E\V^ B(^yr(x. ),x. ). J 1 1 1 such that

If

If

E c B(Uyr(x1),x )

having picked E. = 0 j

Pick

we are done.

x ,...,x.,

we stop.

x GE

Other-

we set

If not, we pick

r(x. ) > 1/2 sup r(x). Observe that if J "t"-L — XGili . r(x.) < 2r(x.) (otherwise, we made the wrong choice of

such

i < j

x

GE. j +Jj

then

x . ) . Hence if

51*

B(r(x.),x.)

intersects

B(r(x.),x.),

we have

Uyr(xi) < |x~ x.| < y(r(xi)+r(x.)) < 3yr(x ) which is a contradiction.

Hence the balls

We claim t h a t the "balls

B(*+yr(x.) ,x .)

B(r(x.),x.)

3 3 cover E.

3 J is finite this follows from our construction.

{x.} 3 balls

B(r(x.),x.) °° .

Hence if there existed

xeE\\u/T B(l*yr(x.) ,x.) we would have r(x) > 2r(x ) for -i J J K. large, contradicting the choice of x . #

k

sufficiently

K.

(1.67) WHITNEY LEMMA. in

G,

and

r ,r ,... 1

C > 1.

Suppose

There exist

E

is an open set of finite measure

x ,x o ,...

in

E

and positive numbers

such that:

c.

(a)

E = \J. Btr^x.),

(b)

the balls

B(r.Ay,x.)

(c)

B(Cr.,x.) n E c = 0, «3

(d) where

M

are disjoint,

J

M [ R A C Y 2 ( 1 + 2 Y ) ] Q ,

M < [8CY3(1+2Y)]Q.

#

The Heat Kernel on Stratified Groups

In this section we assume that

G

is a stratified group.

On such

groups there is a natural analogue of the Gaussian kernel on IR , which plays an important role in analysis. As in Section C, we let d. = 1, 3

j = l,...,v

and we define the sub-Laplacian

L -- ^ .

be those indices for which L

of

G

by

The heat operator associated to

L

is the differential operator

3, + L

_____

on

Xi

Gx ]R,

where

3, = 3/3t

is the coordinate vector field on IR. By

a celebrated theorem of Hormander [l], L That is, if u (resp.

is a distribution on G

(3^ + L)u)

(1.68)

is C

and

3 + L t

(resp.

G*1R)

on some open set ft, then

PROPOSITION.

There is a unique

are both hypoelliptic. such that

u

must be

C°° function

h

Lu C

on ft.

on_ G*(0, 0, h(x,t) = h f x " 1 ^ ) ,

and

jh(y,t)dy = 1

for all

t > 0. (iii)

h(-,s)*h(«,t) = h(«,s+t)

(iv)

h(rx,r2t) = r"Si(x,t)

Proof:

for_ all s,t > 0.

for all. x G G , t > 0, r > 0.

By a theorem of G. Hunt [l], the operator

unique family

{y }..n of probability measures on G X>

for all s,t > 0

L

determines a

such that y * y

u^U

and such that for every

uGCL(G), U

Moreover, the fact that

L

h

u

"0

t

is formally self-adjoint implies that

be the distribution on Gx (0,°°) defined by

= \

\ u(x)v(t)dyt(x)dt

S >X>

3,(u*y, ) = - (Lu)*y, . y, is

symmetric (that is, dy (x~ ) = dy (x)). Let

= \i

Xi

S

y, :

( U G C Q ( G ) , vG c£( (0,»))).

57

Then we have

, Lu®v> =

r (

Lu(x)v(t)dy, (x)dt =

j

O jG 0

Jo

t

(l.u*y, ) ( O ) v ( t ) d t = t

J

0 JG

Lu(x)v(t)dy, (x t

i

)dt

- f 3, (u*y. ) ( O ) v ( t ) d t J t t

(u*y, ) ( 0 ) 3 , v ( t ) d t = t

r (

t

J

I f u ( x ) 3 , v ( t ) d y , (x)dt t t 0 jG

= < h , u ® 3,v> . x> But this says that

h

is a distribution solution of

by the hypo ell ipti city of dy,(x) = h(x,t)dx,

3 + L,

h

(3 +L)h = 0, so

is C°° on G*(0,«>).

Clearly

so properties (ii) and (iii) follow from the corresponding

properties of y . As for property (iv), we observe that since

x>

i(uo 0

by Corollary 1.70.

This again follows from the hypoellipticity of

-

~

—

G x B\{ (0,0)}. 8.+ L.

~

r(x,t) = (rx,r t ) ,

then

immediately that for any of degree

-Q-d(l)-2k,

h

kGl

X h(x,t)

as a homogeneous group with dilations

and any multiindex

I,

-Q.

It follows

k I 3 X h t

is homogeneous

that is,

(1.1k) PROPOSITION. Since

G x JR

is homogeneous of degreee

t)

Proof.

#

\j

We observe that if we regard

that

on

G

#

Proof: ~

c"

i

On the other hand

Corollary,

—

is

e •> 0.

(1.72)

—

h

as

+

r W h ( x , t ) .

h(-,t)eS

h(«,t) = 0

for

for each t < 0,

vanishes to infinite order as

t > 0.

it follows from Corollary 1.72 t •> 0

whenever

x ^ 0.

More

60

precisely, for any su

Nel

and any multiindex

P| y |=l 1X^(7,8)1 < C I N s N

But then for any

x f 0

which shows that

h(•,t)G S.

Remark.

G

and

t > 0,

0 < s < 1.

by (1.73) we have

In view of this result, the operators

can be extended to act on for all

in

for

I,

!

S ,

and we have

H.f ->• f

H, in

defined by (I.69) S'

as

t -*• 0

f 6 S'.

(1.75) PROPOSITION. is a constant

C > 0

For each

kel

such that for all

and each multiindex

I

there

t > 0,

J| 9 Jx I h(x,t)|ax 0 we observe that by (1.73), |8Vh(x,t)| < C(/t + | x |

where

C = sup{ | a V h ( y , s ) | : /s" + |y| = l}. Therefore f t -MQ + d(l))/ 2 d x + c f

| x | -Q-d(l)- 2 k d x

61

Notes and References Sections A through E:

Some of this material is folklore, and some of

it is derived from Knapp and Stein [l] and Folland [l]; see also Goodman [l]. The Taylor inequalities (1.37) and (1.42) and the results in Section E are new. Section F: [2].

For the original theorems of Wiener and Whitney, see Stein

The variants presented here, which are valid on arbitrary spaces of

homogeneous type, are in Coifman and Weiss [l], [2]; see also Koranyi and Vagi [1]. Section G:

These results are due to Folland [l].

62

CHAPTER 2

Maximal Functions and Atoms

HP

Here we begin our development of the basic ideas of After reviewing some facts concerning maximal functions on

theory. iP,

p > 1,

we turn to the "grand maximal function" in terms of which we define Atoms are also defined, and it is proved that "atomic HP q,a

is contained in

H ",

H .

namely

HP.

We shall be working on a fixed homogeneous group that if

P

(j) is a function on

G

and

t > 0,

we set

G,

and we recall

cj> = t

o 6 , ;

this notation will be used throughout. If M f

feS1

and

t(y)| : I x ^ y ] < t, 0 < t < «>},

(2.2)

M°f(x) = s u p 0 < t < a > |f#4>t(x)|.

The same definitions will apply if such that example, if M.f < M.f

-

(x,t) -»• f*. (x)

f

and

0,

6k

(ID)

IIM(A)fll

Proof: of

llfll

F i r s t we prove ( a ) .

M^f.

1

fcL ,

Given

ER = Then for each that

1

< C'p(p-l) p - A

xGE

|x~ y| < t

a > 0,

and

We f i x and

{x : Mf(x) > a

we can pick

fGLP,

for a l l

p

1 < p < .

A > Q and w r i t e R > 0,

and

Mf

instead

let

| x | < R}.

y = y ( x ) , t = t ( x ) , and cf> e A^

|f*.(y)| > a.

Thus

We write the last integral as a sum of integrals over the regions and

2kt < | z""1^-1 < 2 k+1 t -~

tQa < f

(k = 0,1,2...).

Since

(j> 6 A. A

—1

|f(z)|dz + ro 2' k A f

"

JB(t

'y)

|f(z)|dz

B(2 k+1 t,y)

< (1 • r 2- k(X - Q) ) SU p 2- ( k " l ) Q f |f(z)|dz. J B(2kt,y) ° Hence if we set A = A ( X ) = 2Q(1 (which is finite since

A > Q),

(2 k t)- Q f J

+

^2"k(X-Q))

for some

B(2Kt,y)

k = k(x) we have

|f(z)|dz > a/A.

|z y| < t

we obtain

|f(z)|dz + I" 2"k:

< f

such

65

But

|x y| < t,

so

B(2 t,y) c B(y2

t,x), whence

|f(z)|dz > f (2 k t) Q = — 2 — |B(Y2k+1t,x)|.

f B( Y 2

k+1

A(2

t,x)

^

•p

In other words, for each r(x) = y2 k(x)+1 t(x))

xGE

there is a ball

such that, with

f

B(r(x),x)

A' = (2y)QA,

|f(z)|dz > j,

J

B(r(x),x)

By the Wiener lemma (1.66), we can choose a sequence that the balls cover

B(r(x.),x.)

are disjoint and the balls

{x.}

in

E

so

B(Uyr(x.),x.)

E . Then a E K | 0,

be as above and set set

g(x) = f(x)

if

A

= (Uy)QAf.

|f(x)| > a/2A-L

Given and

fG L P

(l < p < »)

g(x) = 0 otherwise.

66

Then

|f| < |g| + a/2A , whence

Mf < Mg + a/2, whence

{x : Mf(x) > a} c {x : Mg(x) and thus by part (a), if

A(a)

> a/2}

A(a) = |{x:Mf(x) > a}|,

< -~r ngiL = —- f

|f(z)|d

Therefore

(Mf(x) P dx = p f aP""1X(a)da

J

J

< P

0

J

aP

[2A a

|f(z)|dz]da

u

0

J |f|>a/2A

2 Al |f(z)|

aP""2da|f(z)|dz

= 2A p I I G

0

2An(2A1)P"1p t

= -^-i

|f(z)|Pdz,

from which the desired r e s u l t i s immediate.

(2.5)

COROLLARY.

|(j)(x)| < A(l+|x|)~

If_

is_ a_ measurable function on

for some A > 0 and

(a)

|{x:M f(x) > a } | < AC^IIfll /a

(b)

HlVLfll t(y) = f*ipt(x) where

i|;(z) = 6 S

is

Then there is a ball B(r,0) we have

B(l,x) C B(rt,0),

on which so

+ ~"v9 - r(l+ Y M ~ ^ B(l,x) From this it is clear that p > 0

%

M x6L

only when

p > 1.

However, for any

we can exhibit large classes of distributions whose grand maximal

functions are in

L . The problem is to ensure that the maximal functions

vanish sufficiently rapidly at infinity, and this can be accomplished by assuming some vanishing moment conditions. We proceed to the formal definitions. If

0 < p < 1

and

aG A, we shall say that

a > max{a' G A : a1 < Q(p If in addition

-l)}. (if

a

only the case

p = q = 1.)

supported (i)

is p-admissible if

A = TT, this means that

1 < q < °°, we shall call the ordered triplet

admissible if

Suppose

a

is p-admissible and

(p,q,a)

L^

function

is admissible. f

there is a ball

p < q.

A

a > [Q(p~ -l)].) (p,q,a)

(The latter condition excludes

(p,q,a)-atom

is a compactly

such that B

whose closure contains

supp(f)

such that

llfll < |B|

Recall that homogeneity.

A

is a set of real numbers generated by the exponents of

See the definition on p. 21.

72

(ii)

If

B

j

f(x)P(x)dx = 0

for all

PeP . a

is any ball satisfying condition (i), we shall say that

associated to

f

is

B.

Condition (ii) in the definition of atom is the essential one. (i) merely imposes a normalization on later; it implies that

fG L

f

Condition

whose utility will become apparent

and that

lf| p < Finally, if

p > 0

we set

N = min{N G IN : N > min{b G A : b > P Thus if

p > 1

we have

N

=0,

while if

p < 1,

the smallest element of

A

which is p-admissible.

N

= [a]+l

(if

A = U

where and

a

is

p < 1,

The point of all these definitions appears in the following theorem.

(2.9) constant

THEOREM.

If

(p,q,a)

C = C(p,q,a,N) < °°

is admissible and

such that

N > N , there is a _ p < C for all (p,q,a)-atoms

IIM/__xfII ^

j

p _

_

f. Proof:

Lef

f

be a

(p,q,a)-atom

associated to

B = B(r,x ). Since

all the relevant definitions are invariant under left translations, we may assume without loss of generality that shall estimate

M/ sf on

B

and

B

x

= 0.

Let

separately.

B = B(2y$ r,0).

We

73

On

B

we use the maximal theorem (2.k).

If

q > 1

we have

j^ ( N ) f(x) P dx < (JM (N) f(x) a}.

Then f o r a l l

a > 0,

|Eo n B| < min{|Ej Ej , |fi|} ^/a,

|fi|}

The last two quantities in curly brackets are equal when Thus, since (2.11)

a = |B|"

.

p < q = 1,

I M,N)f(x)Pdx =

pa P

|Ea n S|da

|B|~ 1/P

= C". Next, suppose and

G S

and II II /N \ < 1- Let b = m i n { b ' G A : b ' > Q(p~ -l)}

c = max{c' G A : c' < b } , and for each

x G G let P

polynomial of f at x of homogeneous degree

be the right Taylor

c. Then by Theorem 1,50 there

is a constant

C > 0,

independent of

|(y-1x)-P (y" 1 )! < C|y|b

(2.12)

X

Therefore, if

Since to

, such that whenever

a

yeB

and

Q b

—

xGB°,

is p-admissible we have

a > c,

and hence if

q'

is the conjugate

q, = |t" Q |f(y)[*((y-1x)/t)-Px/t(y-1/t)]dy|

|f(y)||y|bdy

This being true for a l l

x e B°

and a l l j G S with

11*11 ( N \ < 1 5

Proposition 2.8 we have

J M ( N ) f(x) P dx < C

But

b > Q(p" -1),

(2.13)

(-Q-b)p < -Q, and hence

M (N) f(x) P dx < G|B| ( b p / Q ) " 1 + P |B| X - ( Q + b ) P / Q = C .

| B

BC so

|x| > 2y$ |y| ,

C

Combining (2.13) with (2.10) or (2.11), we are done.

#

75

After these preliminaries we now make the following definitions. If

HP

0 < p < °°, we define the Hardy space

to be

HP = {fcSf : M(N )f6LP}. P If H

(p>q.5a)

P

is an admissible triplet, we define the atomic Hardy space 00

to be the set of all tempered distributions of the form *

q,a

(the sum converging in the topology of (p,q,a)-atom,

X. > 0,

and

Sf)

where each

f.

E-A.f. 111 is a

z!°X? < °°.

Several remarks are in order concerning these definitions. (1)

If

p > 1

then

(2)

The condition

HP = LP,

and

H1 c L1,

by Theorems 2.U and

2.7. II4>II / v < 1 in the definition of M/ sf is P P essentially the weakest one which allows the arguments in the proof of Theorem 2.9 to be carried out. 1,

However, the )< 1

W/w

the condition

are

P

II^IJ,

s < 1 means that |(x)| < (l+|x|)~ , and one can P replace the number Q+l by any X > Q as in Theorem 2,k. We shall not pursue here the point of precise decay conditions on that in some problems the optimal conditions on

0 = N . Moreover, it will follow p

H? * = H P

for all

(p 5 q 5 a)

is admissible.

that one needs some control over the functions grand maximal function, and the smaller

(k)

If

p

called an atomic decomposition of We now define topologies, on p > 1 and

we use the P

p q,a

L

H

= H

N < N . The moral is

€ S

which enter into the

is, the more control one needs.

f G H P , the representation q, a

combination of atoms is far from unique.

p

even when

Bearing this in mind, one sees from the

preceding remark that the theory breaks down when

P

N > N

On the other hand, we shall also show in Chapter 3 that

whenever

if

it follows from Theorems

f = E~X.f. of _L i i

f

as a linear

Any such representation will be

f. H

norm on

and

H

for p < 1. (Of course, q,a H .) Namely, we define the quasi-norms

by

p P (f) = inf{EA P : ZX.f. q, a l ii into

is an atomic decomposition of

(p,q,a)-atoms}.

f

77

(f,g) -*• p P (f-g) q,a respectively, -which make them into topological

It is easily verified that the maps HP

are metrics on

and

H

(f,g) -> pP(f-g)

q,a vector spaces (not locally convex, unless P

nontrivial point is that

p (f) = 0 q_,a

p = l).

implies

and

The only slightly

f = 0,

but this follows

from the next proposition. (2.15)

PROPOSITION.

If

(p,q,a)

HP c HP c Sf, q,a

is admissible then

the inclusions being continuous. Proof:

Let

we have (since

C = C(p,q,a,N ) be as in Theorem 2.9. p p < l)

pP(f) = |[M (N ) (EA.f.)] P < ZXP |[M (N

Hence -> f | for a l l #

Jn

J

and

Remark:

The same proof shows that if

given the obvious topology, the inclusion all

if

H?__v H^\

c

is defined, by (2,lU) and 5'

is continuous, for

N6I. HP

is complete.

(2.16)

PROPOSITION.

Proof:

We need only consider

{f.}

is a sequence in

converges in

HP

and it suffices to show that EpP(f.) < «>, the series

such that

Ef.

H . However, the partial sums of this series are Cauchy in

H , hence in

S1

by Proposition 2.15, so the series

to a distribution

f.

p

P

p (f) < Ep (f.) < «

so the series converges in

If

f

converges in

and

P P

fGH .

Similarly,

H . #

We conclude with two more remarks about atomic (2.17)

Ef. 3

We have

p P

hence

p < 1,

is a

(p,q,a)-atom

then

p

HP

spaces.

(f) may be strictly less

than one, since there may be more "efficient" atomic decompositions of than

f

such that write

itself.

p (f) > 1-e. q,a

f = EA.f.

would obtain for all

However, for any

P

pP

with

e > 0

there exist

If not, then for any

EA? < l-(e/2).

(f) < (l-(e/2))2,

j, which would imply

(p,q,a)-atom

Decomposing each

and by induction,

f = 0.

(p,q,a)-atoms

f. pP

f

f f

we could

similarly, we (f) < (l-

79

(2.18)

If

N f = E X.f.

is a finite linear combination of (p,q,a)-atoms,

then

j|fp < SJ jxPlfJ* < zJxP,

hence j| j|f |* hence




___________

independent of

#

f, i, and a,

—

_____

such that

Proof:

~~™~~~""~

Let

u , ...,TT J_

m

(m = dim P ) be an orthonormal basis for

with respect to the norm (3.5).

Then by the properties of

X

J

J

1

X

"

x

J

a

£.,

1 = ( L ) " 1 [ h . ( y ) | 2 C . ( y ) d y > |B(2r x ) f 1 f J

P

a

B(r/1+Yx)

U.(y)|2dy J

B(IAY,O) where

TT.(Z) = 7T.(x(r.z)). j

j

i

Since

dim P

are equivalent), there exists C > 0 1 sup,T|

w

< °° (so that all norms on

such that for all

TT ., J

we obtain

-

1 J. h/ »

PGP, a

|p(z)|2dz)1/2.

, , - l Y ^ z ) ! < C_([

|1| (x) = (r /t) Q f*$ J-

where

O

1

( x ) - (P.C.)*, (x) X .

1

$(z) = 4>( (r. /t)z)c(x(i\z~ )). If

|x. x(r.z

1

U

$(z) # 0 we must have

)| < 2r., which implies that |z|
N.

——_

87

Proof:

Suppose

(T^ri,xi) Case I :

eS,

llIL

v < 1,

and

t

> 0.

n Q C. t

< r.. — i

We w r i t e

b.*cj), ( x ) = it

f * $ , (w) - ( P . C. )*. ( x ) t lit

-1

If

zG supp $

and since

then

|x7 w(tz~ )| < 2r.,

whence

|x^ x| > ^r^^ > . -1

-1 X. X

-1 X. X

i X. X

W

x. x

i X.

2yr.

-1 - 21 Therefore, if

Pick

|l| < N,

t '•

by Lemma 3-7 we have

-1 I

K. X

_ - 1

where

since

t/r. < 1.

Also, if

y e supp b.

and

x^B.

we have

|y """x| > cjx^xl, so by Lemma 3.8,

< C,(r./|xT 1 x|) (N+1)(tl+l) Mf(w) +V t-« 1 2 ~ 3 1

since

t < r., Case II:

b < N,

and

r. < |xT x|.

t > r..

Let

a' = a

if

a < N

.,x.)

and

a' = min{a"GA : a" < H}

1

if

a > IT; thus

we write at

z

a' = max{a" G A : a" < b}

(yz) = P (y)+R (y) where z z f

of homogeneous degree

a .

P

z

and

N > [a']+l. For any

zGG

is the right Taylor polynomial of

By Theorem 1.50,

R

satisfies z

(3.3A)

|YXR (y)| < C 7 |y| b - d ( l ) |z|- Q - b Z

Also, if (3.15)

d(l) > b

—

then

|Y\(y)|
2Yj3M|y|).

I

—

YIRz(y) = Y ^ y z ) , (

(

C fl | Z |- Q - d(l)

)

(

so

)

(|l| < N,

d(l) > b,

| Z | > 2y|y|).

Now, by the construction of b.*cj>, (x) = t 1

Q

t

b.

we have ((yxT1)/t)dy

b.(y) R n (x.xj/t

J 1

X

JPi(y)ci(y)R

((yxi1)/t)dy,

_±

where (r.z)w

x

= R

If

z G supp $ 9

(3.16)

we have, as "before,

|z| < C r./t,

which now implies that

and 2

Y

3

(r.z)w"" x. |-^ ±

N

"bounded by a constant. |j| < N

, x i lx ,

T.

By Lemma 3-7, the derivatives of

(3.16), for

~)?.(w(r z" 1 )).

£.(w(r.z

)) with respect to

we have

-' -""^

(r

iz)w

X

i,b-d(l) ,Xi X -Q-b

x. x 10

- c i r tj I t I

C

/jtxQ .

10

are

Therefore, by the estimates (3.lU), (3.15), and

r

C

z

I

J

\

r

i

-i

xQ+d(l) /

t

90

since

|x. x| > r..

Therefore, since

supp $ c B(C ,0),

we have

and hence

K^/t^f*^ (w)| < ^^(r./lx On the other hand, by (3.lJ+) and Lemma 3.8,

X

-1 -1 i ,b ,Xi X

Combining these last two estimates, we are done.

and

(3.17)

THEOREM.

f 6H

There is a constant

.

Suppose

0 < p < 1, A^,

N = N , p

a

independent of

is p-admissible f, i, and

a,

such

that

Mb.(x) P dx < A q f Mf(x) P dx. g Moreover, the series

Eb.

converges in

i H , and _if_ L

is as in (3.U),

P fM(Eb.)(x) dx < LA f Mf(x)Pdx. 1 ^ in 'a

J Proof:

By Lemmas 3.12 and 3.13, we have

< A P f Mf(x)Pdx + A^aPf (r /|x^-x|) p((iH)) dx "

3

\

BC. '

91

where

b > Q(p~ - l ) .

Hence

p(Q+b) > Q, so

BC 1

Hence, since

B. c ft9 fMb.(x)Pdx < C j [

< 2C

Mf(x)Pdx + a P | B . | ] B.

f Mf(x) P dx. i

This proves the first assertion, and since

H

is complete, the second

follows from the estimate

Z. |Mb.(x)Pdx < 2C Z. I Mf(x) P dx < 2LC I Mf(x) P dx.

Remark: Mf(=M/ Eb.

vf)eL

If we replace the assumptions

N = N , p

f6HP

#

by N > N , - p

, we obtain the same conclusions, except that the series

converges in the topology defined by the maximal operator (3.18)

THEOREM.

Suppose

M• >. •

N > 0, a G A , and f e L 1 . Then the series

Eb.

converges in L , and there is a constant

and

a,

A^, independent of

such that

E|bi(x)|dx < A 6 j|f(x)|dx.

Proof: By Lemma 3.8,

i

+A a|B |

f, i,

92

Hence by (3.M and the maximal theorem (2.U),

S ( M1

N , 0 < p < I)

"good part" (3.19) A7,

and for

fGS1

such that

f S L . We now investigate the

g = f-Zb.. LEMMA.

independent of

Suppose f

and

Zb. a,

converges in

S1.

such that for all

There is a constant

x G G,

Mg(x) < A7aZ.( r-i ) ^ D + Mf(x) X Ax), " T X Ix xl+r n where

b

is as in Lemma 3.13.

Proof:

If

x^Q,, by Lemma 3.13 we have Mg(x) < Mf(x) + Z,Mb,(x)

Mf(x) + Z A a(r /Ix^xD^^

since that

|x. xl > r.. 1

1

'

x6B. , and let K

On the other hand, if

xGft

let us choose

k

such

1

J = {i:B. n B. f 0}. Then IK

card(j) < L, —

and as

93

above we have

^

2

Q+b

Hence it suffices to estimate the maximal function of As in the proof of Lemma 3.13, we fix Suppose

(f)GS,

IIC>II /N\ < 1 5

and

g+E. , b. = f-E

b..

wGB(T~r ,x,) n Q,C. t > 0.

If

t < r , we write

where -1

We observe that

r\ = 0

on

B ,

so if

z G supp $

K.

where

C

> 0,

and hence T 33rrkk//t) | < Y(|y| + Iw^xl/t) < Y(|y|+T

Therefore, by Lemma 3.T 9 for

111 < N

we have

and

y = z((w

x)/t),

K

C

o

< C.a
r. ,

let

x)| =

Also, by Lemma 3.13»

K.

*(z) = ^(zCCw^xJ/t)).

by Proposition 1.U6 we have

xQ+b

t < r . ~

If

k

II*II, ^ < C ,

Since

I w ^ x j / t < T.r / t
T r.

i

since

w^^.

Hence

< (C6+l)r.
N , a is p-admissible, - P Mg e L , and there is a_ constant A n , independent of f

_______

___

such that

_______________

Q

JMg(x)dx < Agcx1"1? JMf(x) P dx. (ii)

Suppose

N > 0,

aeA,

__ constant A ,. independent of Proof:

(3.21)

f

and

f e L . Then

and

a,

such that

gGL°°,

II g I! < A a.

(i) By Lemma 3.19,

JMg(x)dx < A1al± f ( — ^ 'Xi x ' + r i

Let C =

) Q+b dx+ f Mf(x)dx. nc

and there is

Then the first term on the right of (3.21) is bounded by A^CctE.r; = A^CaE |B. | < LAvCa M = f

i

i

(

i

i

-

C'aM.

f

Hence

|Mg(x)dx < C f a | n | + [ Mf(x)dx < C'a-cT P [ MfU^dx + a 1 ^ f Mf(x) P dx < C ' a 1 ^ JMf(x)Pdx. (ii)

If

feL

then

g

and t h e

1

Thus by Lemma 3.8, for for almost every

1

xeft

x6fl°

11

we have

we have

b.'s

are functions,

and

1 1

QC

|g(x)| < LA^a, while by Theorem 2.6,

|g(x)| = |f(x)| < CMf(x) < Ca.

#

This completes our discussion of the Calderdn-Zygmund decomposition. As an immediate corollary, we obtain the following important result. (3.22)

COROLLARY.

Proof:

If

decomposition of

If

0 < p < 1,

fGH

and

a > 0,

f

of degree

a

let

HP n L1

is dense in

f = g a + Eb?

and height

a

HP.

be a Calder6n-Zygmund

associated to

Mf = M/

xf, P

were

a

is p-admissible.

By Theorem 3.1T5

p P (E.b a ) < C f 1 X

so

pP(E.ba) •> 0

as

"

a -> ~.

by Theorems 3.20 and 2.7.

#

Mf(x) P dx,

/

• {x:Mf(x)>a} Hence

g a -> f

in

HP

as

a -»- °°, but

g^L1

97

Remark: "good parts" to

M, vf

N > N and JVLT>>feLP - p \N)

The same proof shows that if g

of the Calderon-Zygmund decompositions of

are in

L

f

then the

associated

and [M(N)(f-ga)(x)]Pdx + 0

B.

as

a -• co.

The Atomic Decomposition

We now aim to prove that Hardy spaces coincide with atomic Hardy spaces. Suppose

0 < p < 1,

N > N , a

is p-admissible, and

such that

f

k

~D

M/ sfeL . For each

Zygmund decomposition of

f

kGE,

of degree

let a

is a distribution k

f = g + E.b.

and height

2

be a Calderonassociated to

M/-,xf. We shall label all the ingredients in this construction as in Section A, but with superscript

k's:

for example,

fik = {x : M(fj)f(x) > 2 k } ( We now need two more definitions. of on

k P., P a

we define

-bk = (f-Pk)?k,

B k = B(r k ,x k ).

First, by analogy with the definition

k+1 P.. to be the orthogonal projection of

k+1 k (f-P. )c

with respect to the norm

IIPH2= (J ? k+1 )- 1 ||F(x)| 2 C k+1 (x)dx.

(3.23) Tc+1

That is, P..

is the unique element of

P

Second, we define Bk

= B(2r k ,x k ).

such that for all

Qe?a>

(3.2*0 B.

LEMMA.

c B(T2r.,x.).

for which

B.

+

Proof:

If

I"]

B.

("b)

For each where

L

j

r^ +1 < Uyr^ and

there are at most

L

is_ as_ in. (3.M.

AV

n B. + 0 we have

|(xJrV; +1 | dist(x. ,{Q ) ) > T r. .

Thus _ k+l _. ,f k+l / o 3ON k+l l8y 3 r. = T 2 r. < dist(x. ,(

so that

From this and (3.25) it follows that

values of

i

99

so that if

~k+l y 6 B. , J

This proves (a), and (b) follows from (a) and (3.^). (3.26) k,

LEMMA.

There is ji constant

A ~,

#

independent of

i, j , and

such that 1, v 1 . o k+l (y)| < A2 Proof:

The argument is essentially the same as the proof of Lemma 3.8,

and we indicate only the necessary modifications. orthonormal "basis for show that for some

P a

C > 0

If

IT , ...,TT

is an

with respect to the norm (3.23), it suffices to independent of

But by Lemma 3.8 and its proof (with

IP^ty)! < C2 k+1 ,

i, j , k, and £,

P.,C.

l^y)! < C

replaced by

for

Hence :+lx-l fpk+1

r k r k+li

k+1

so we need to show that

r1 f f t ^ 1 ! < c2k+1.

T,

P.

11

k+1

, C.

)>

100

Now

Mr where

WG B(T r k + 1 , x k + 1 ) n (^ k + 1 ) C -j

J

./ x

/ k+lxQ/f k+lx-lr

so it suffices to show that i

and

j

k k+ln, / k+1 -lvv

) l7^-^

$(z) = (r. P ( k .

those values of

and

«]

li$ll/ % < C.

J(w(r. z )),

However, we need only consider

A ^k+1 k B. n B. ^ 0,

such that

k k+1 C.C. vanishes identically, and for these values of 3.7 and 3.2U yield the desired result. (3.27)

LEMMA.

For every

Proof: C

k+1

For each

(x) $ 0.

x

x tthe series

and

j, Lemmas

) = 0, where the

S'.

j, P

and by Lemma 3.2U there are at most Thus for each

k+1 k+1 E. (E .P . \ . i J IJ J

there are at most

Moreover, for each

i

#

k6E,

series converges pointwise and in

since otherwise

k+1

L

L

values of

is zero unless

values of

k+1 k+1 E.E.P.. (x)c. (x) i 3 ij

J

i

j B

for which k

n B k + 1 f 0,

for which this happens.

is actually a finite sum,

and by Lemma 3.26, i k+1 k+1 L . L . \r . , \X. ) C, .

i V - f

-> 0

f in

HP

as

k -> + °° , while by Theorem

k ->--«>. Therefore

(convergence in

Now, using Lemma 3.27 together with the equation

„

Lk+1=

E . C .D.

k+1

X i D . =

, k+1 b .

we have

JJ

=

(3.29)

1J1J0

E.h.

where all the series converge in

5'

be the

as in the preceding lemmas, where

uniformly as

-g )

f = g +E.b.

and

h^ = (f-P^)c^- Z.[(f-P^+1

S1).

102

From this formula it is evident that ihk(x)P(x)dx = 0 Moreover, since

for all

k+1 E.£. = x

P k r k + rks

But

|f(x)| < C J /

\f(x) < C 2 P 2.6, so "by Lemmas 3.8 and 3.26,

P

k+1

r k + 1 +Z

for almost every

llh B

i cx, 1 (2C 1 + A 2 + 2 L A 2 + 2 L A 1 Q ) 2 k

k+1 P.. = 0 k

Lastly, since

Lemma 3.2U that

h.

unless

a.

is a

*

k+1

rk+1

x^^

by Theorem

C^.

~k ^k+1 B. n B. f 0, it follows from (3.29) and k k

is supported in

x!f = C 2 2 k T^ / p |Bj| 1 / p we see that

P

J = C, IIMfll p .

n

Therefore, if we set

and a^ = h^/xj,

(p,°°,a)-atom,

oo

Bf^r^xJ.

and that

p a p " 1 | { x : Mf(x) > a } | d a

103

k k E E.X.a. K. i 1 1

Thus the series decomposition of

converges in

D Ir and defines an atomic °° 5 a

f.

It remains to remove the restriction that arbitrary element of in and

HP n L1

fG L . If

f

is an

H , by Corollary 3.22 we can find a sequence pP(f_ ) < (3/2)p P (f), p P (f ) < 2~ m p P (f) 1 ~ m — f = Z.A1f1 be the atomic decomposition of m i m m

such that

f = z!°f . Let lm

constructed above.

Then

f = E E.X f m I m m

{f } m for m > 1, f m

is an atomic decomposition of

f,

and

(3.30)

THEOREM.

(i)

If

(p,q,a)

(ii)

If_

fGS1

N > N ,

Suppose

0 < p < 1.

i s admissible then then

f G HP

HP = HP

i f and o n l y i f

pP ^ pP

and M, x f G l ?

.

f o r some

p P ( f ) * IIM ( N ) fll P .

and

Proof:

(i)

(p,r,a)-atom

I t i s e a s i l y checked t h a t i f

is also a

(p,q,a)-atom,j

l < q < r < ° °

emu. hence ncii^c and

F H

r 5a

r~ H M c

then every . Therefore

Q_ 5 a

by Proposition 2.15 and Theorem 3.28, c HP c HP5 HP c HP °°,a q,a where the inclusions are continuous. (ii)

feHP

If

then

M, s f e L P

for all

Conversely, the proof of Theorem 3.28, with

N

N > N

since

replaced by

M/ vf < M/ N

(cf. the

remarks following Theorem 3.IT and Corollary 3.22), shows that if then

(i),

f G HP °°,a

fGH

P

for any p-admissible

and

P

p (f) * "M (N) fll£.

a,

#

and

x

pP (f) < CllM/-Txf IIP. °°,a vNj p

M,NvfGLP Thus by

Theorem 3.30 is the rock on which our subsequent investigations wi be founded, and we shall frequently use it without referring to it explicitly. dense in

As a first application, we prove that smooth functions are

H .

(3.31) and

C > 0

LEMMA.

C > 0

$GS.

such that for all

Proof: and

Suppose

For_ anv_ N 6 l

there exist

fG S',

We first observe that for any

N6 JT

there exist

feS'

sup

Nf > N

such that for all cj),^G5,

(This follows from Proposition 1.1*9 and its proof.) if

N1 > N

From this we see that

and ,t(ieS,

£ ,t>o

! f ** e **J *

sup

£ 0

M ( K 1 ) ( ( f - g ) * + e ) + lim

Hence by the maximal theeorem (2.U), for any

a > 0,

|{x : lim sup £ _ > ( ) M (Nt) (f*cj) e -f)(x) > a } | < C'llf-gl^/a < C'6/a.

Since

6

is arbitrary, we are done.

#

in

and the dominated convergence

M/ ,>>(f*(j) -f)

This is certainly true if

has compact

106

oo

Remark: and

C

fP = 0

oo

p

is not a subset of

for all

PGP

constant multiple of a

where

H a

(p,°°,a)-atom,

for

p < 1.

However, if

is p-admissible, hence

f

fG C

will be a

fGHP.

We conclude this section with some comments about vector-valued functions. X-valued

If

X

is a Banach space, we can consider the space

tempered distributions on

linear maps from

S

into

X.

G,

Sj, of

i.e., the space of continuous

The whole of

HP

theory up to this point

can be developed in this context, merely by replacing absolute values by X-norms

in appropriate places.* For example, if M f(x) = sup y

|x y| X, a

we set

PT(X) = AUJTTTT. Then for any

Q G P , a

L(Q) =

QP dy. I -I i

Thus the construction

of the

polynomials

However the reader should take care because not all results in H^ theory are extendable mutatis mutandis to the case of Banach space-valued functions. Some theorems require the L 2 boundedness of appropriate operators, and those may be essentially restricted to the case when the Banach space is a Hilbert space. See also the remarks preceeding theorem 6.20 below.

107

P.,

etc.,

in the Calderon-Zygmund and atomic decompositions can be carried

out in the Banach space setting.

C.

Interpolation Theorems

In this section we prove two results on interpolation of the real method.

We recall that if

X

and

V

H

spaces by

are quasi-normed linear spaces

embedded in some topological vector space, the Peetre K-functional on

K+V

is defined by

K(t,f) = inf{llgllx + tllhlly : g e X , h e V, f = g+h}

and the interpolation space

[X^L

(t > 0 ) ,

( O < 0 < 1 , 0 < q < « )

is defined by

( t " 9 K ( t , f ) ) q d t / t ] 1 / q < 00}.

We refer the reader to Bergh and Lofstrom [l] for a detailed exposition of these matters.

(We remark that to fit

setting, we use the quasi-norm

(3.3*0

THEOREM.

q" 1 = (l_0)p~ 1 + e r " 1 ,

Proof: [ ,] 8,q that

If

(p )

r = °°.

[H p ,H r ] f i

into this

p .)

0 < 0 < 1,

and

= Hq.

By the reiteration theorem for the interpolation functor

We may also assume that Thus we suppose that

r

boundedly, so it maps

[LP,L°°]Q = L q ,

and

p < 1,

0 < p < 1,

First, the subadditive operator oo

b,q

(p < l)

rather than

0 < p < r < °°,

then

spaces

(cf. Bergh and Lofstrom [l], Theorem 3.11-5), it suffices to assume

well known.

L

H

M,

p o o

P ~ q

O < 0 < 1 ,

maps

H

to

and L

q = p/(l-0). and

L

to

-Q oo

LH ,L J

N >N ,

N

since otherwise the result is

f,q

to

[L ,L ]

8,q

whence i t follows that

boundedly.

[HP,L°°L

o ,q_

But c

Hq.

108

To prove the reverse inclusion, by Corollary 3.22 it suffices to showthat

H q n L 1 c [HP,L°°]

. If f G H q n L 1 , let F be the nonincreasing

rearrangement of Mf = M,

\f on (0,«>), and for t > 0 let f = g + £b. p be a Calderdn-Zygmund decompositon of f of degree a and height a = F(t P ) associated to Mf, where

a

is p-admissible.

Then by Theorem 3.17»

IIM(rb*)llp < c f Mf(x) p dx = c f F(s) p ds. 1 P J ~ Mf(x)>a Jo Therefore, making the change of variable

t P -> t

and then using Hardy's

inequality (cf. Stein [2], p. 272), ,00

(3.35)

,00

f

t

P

(t~9IIM(£b.)ll ) q d t / t < C t " 9 q ( x p Jo " Jo Jo

= Cp"1 r J

F(s)Pds)q/pdt/t

J

o

o

= Cp-V q/P [V 6 q / p (tF(t) P ) q / p dt/t Jo

= C

[ F(t) q dt

= c'IIMfllq. q.

Moreover, by Theorem 3.20, (3.36)

["(t1-9^*!! )4dt/t < C rt ( l - 6 ) < 1 F(t p ) < 1 dt/t 0

J

0

"1 r

= Cp

I t^ 0

= C'llMfil'1.

^/H/^F(t)Vddt/t =

Cp

_1 J

0

109

But c l e a r l y +tllg t !l oo ,

K ( t , f ) < IIM(Zb^)ll so "by adding (3.35) and (3.36) i f to them i f

q > 1,

or applying Minkowski's i n e q u a l i t y

we have [

so that

q 0, let f = g+Zb.

a is p-admissible.

a and height

Then each

in the notation of Section A ) ,

X. = lib. II |B. p 1 / P ) - 1 5

T

to_ the space of measurable functions such that

Zygmund decomposition of f of degree M/p.\f9

and suppose

llTfllP < CpP(f)

T is weak type Proof:

L

0 0,

let us define the tangential

by

if and only if

AeL

P

,

provided

A > Q/p.

In defining the nontangential maximal function

replace the cone for any

eS9

{(y,t) : |x~ y| < t < °°} by

a > 0 without changing the p > 1,

L

M,f

we could

{(y,t) : |x~ y| < at < °°}

properties.

it follows from Theorems 2.k and 2.7 that if

G S

and

U = 1 then (U.3)

llM^fll ^ llM^fll ~ IIM, T vfll * P 4> P (N ) p

for all

fGHP.

If

G = ]Rn,

this result remains true for all

p > 0.

The crux of the matter is the following result, which shows how any tyeS can be expressed in terms of (k.k)

PROPOSITION.

such that any tyG 5

B*^I(N)
0

can be written as

Moreover, for any m , N 6 l such that

If

and its dilates:

there exists

0 ^ 2 - ^ 1 * 1 1 (H).

C

> 0,

independent of

\\>9

116

Proof:

Since

Taking Fourier transforms, we see that (U.5) is equivalent to

$(0) =

|C| < e ,

U =

1,

there exists

e > 0

k

and hence

| $ ( e 2 ~ £ ) | > 1/2

—

for

such t h a t k

\$\ < 2 .

—

partition of unity on 3R supp C. J

c

then (i»-.6) obviously holds. (k) II\|J "fivrV

reader.

and

Let

supp C

for

{r }" be a K (J

~

such that

B(2 J ,0)\B(2 J ~ 2 ,0),

|U)| > 1 / 2

c B(l,0),

II (3/3£;)Ic .11 < C2"' 5 ' 1 '. J °° -

The estimates for

11$

If we define

and nence

"(ro}>

for

follow easily from this construction; details are left to the

#

We do not know if this proposition remains true for an arbitrary homogeneous group non-Abelian.

G;

certainly the above argument is worthless if

is

However, we shall now show that a variant of this proposition

is true for general

G

when an additional restriction is imposed on

and from this we shall deduce the relations (k.3) for such

Definition: such that

G

= 1

's.

a commutative approximate identity is a function and

I

*, = ,*cj) S t T> S

for all

,

GS

s,t > 0.

We postpone until Section B a discussion of the problem of finding commutative approximate identities.

For the present, we assume that we have

such a function in hand and proceed to work with it.

If

denote by

with itself.

the N-fold convolution product of

N

H

observe that (^)* = U * \ and J/ = ( f^

N

$

eS,

we shall We

117

(k.7)

LEMMA.

Suppose

0 < J < N < «» and and

5

a)

then

3^*'

N+1

N S

' = 0

SS

Proof:

and

rh. = j.

1—

Since

J

3

the commutativity of

XI

is a limit of difference quotients of (j) *(3^ ) = (9g(f)s)*(J)

cj> implies that

!

for an

s,

y J-

In particular, if N > 1,

(..8)

a s ,f^ - ^ f * O A ) < ^ = (.+i)*«.O8*B, T

which proves the lemma for

s

T

sTs

s

j = 1.

We now proceed by induction on

8 U * * ) = zhbA * 3 S s Ys

Expanding

3 0

is_ a_ commutative approximate identity.

such that;

118

|(l+|y|)N|e(s)(y)|dy < CsNil^ll(3N+3), where

(ID) N

C

depends only on

and . Proof:

Fix

ceC°°([O,l])

0 < c(s) < sN/N! Also, let

w

(s)

for

= a/

N+1

1/2 < s < 1, '

and we claim that these (a)

such that

N+1

'

s)

0's

and

?(s) = sN/N! 3^(l) = 0

for

for

0 < s < 1/2,

0 < j < N+l.

he as in Lemma k.l. We set

have the required properties,

Consider the integral

'0 We integrate by parts the fact that

N+l times.

Because of the properties of

1 *(N+2) 3d *i|> remains bounded as

s -> 0

C

(by Lemma k.'J and

Proposition 1.58), there are no boundary terms in the first

N

integrations

by parts, and we obtain I = -(

S

But

jcj>*(N+2) =

S-U

(Uf+2 = 1,

JQ

S

S

so by Proposition 1.U9,

* * ( N + 2 ) » * I S =Q

T h u s b y Lemma ^ . 7 ,

f1

[\*e(s)ds. S

and

119

(b)

Observe t h a t

| 8 N + 1 c ( s ) | < CsN

and that

8 N + 1 ^(s) = 0

since

for

s < 1/2.

Hence we

need only show that

But this follows from Proposition 1.55(U.10) For any M,

THEOREM.

NGl

Suppose

there exists

xf(x) < CT f(x), Proof:

Suppose

as in Theorem U.9-

#

is a commutative approximate identity. C > 0

where

T

such that for all

f e S'

II^IL

, < 1.

UlM+Jj -

Then for any

Write

\|> =

x,yGG,

*^st* t

st+

(yz"1)||0(s)(z/t)|t~Qdzds

< f1 [ T " f ( x )([ x "Vst 1

- JJoo JJGG * N ,

x

f 0

J

x€ G,

is defined by (k.2).

^ G 5 and

f*

and

- N , i x v G

%

- l i

NNI

f 1 * 0 ( ) d s

JQ s

120

But if

|x~ y| < t,

then

-1 ^

| + 1 < y(l+ |w|) + l < 2 Y (1+ | W | ) .

Therefore, by Theorem k.99

M.f(x) < (2 Y )Vf(x) f f s-N(l+ |v|)N|e(s)(v)|dvds

from which the desired result is immediate. # (k. 11) feS', C,

and

COROLLARY.

Suppose

0 < p < °°.

If

independent

Proof:

of

_

f,

is_ a_ commutative approximate identity,

M.f 6 L P

—

^

pP(f)

such t h a t

f G HP,

then

< CllM.fll P .

-