Embedding and Multiplier Theorems for Hp (Memoirs of the American Mathematical Society)

Memoirs of the American Mathematical Society Number 318

A. Baernstei n II and E. T. Sawyer Embedding and multiplier theorems for HP(Rn)

Published by the

AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island, USA

January 1985 · Volume 53 · Number 318 (end of volume)

CONTENTS Introduction

1

1.

Embedding theorems.

4

2.

Fourier embedding.

11

3.

Multipliers.

18

4.

Proof of Theorem 1.

26

5.

Best possible nature of Theorems lb and lc.

36

6,

Proof of Theorem 3.

43

7.

Best possible nature of Theorems 3.

58

8.

Lower majorant theorem

70

9,

On a theorem of Pigno and Smith

74

Extension of a theorem of Oberlin

78

10.

References

81

iii

ABSTRACT The spaces

Hp (Rn) ,

0 O We prove two main theorems. The first gives sharp

conditions on the "size" of

f

which imply that

f

belongs to

Hp

The

conditions are phrased in terms of certain spaces K introduced by Herz. our theorem may be regarded as the limiting endpoint version of a theorem by Taibleson and Weiss involving "molecules".

We then use this embedding

theorem to prove a sharp Fourier embedding theorem of Bernstein-TaiblesonHerz type. Our other main theorem gives sharp sufficient conditions on for

m to be a Fourier multiplier of

Hp •

m E L~(Rn),

This theorem also involves the

K spaces and may be regarded as the limiting endpoint version of a multiplier theorem of Calderon and Torchinsky. We also prove three results about Fourier transforms of tions.

HP

The first establishes the "lower majorant property" for

second is an

HP(Rn)

distribuHp

and the

version of a recent theorem of Pigno and Smith about

The third result generalizes a theorem of Oberlin about growth of

1980 Mathematics Subject Classification Primary:

42B30, 42Bl5.

Library of Congress Cataloging in Publication Data

Baernstein, Albert, 1941Embedding and multiplier theorems for HP(Rn) (Memoirs of the American Mathematical Society, ISSN 0065-9266; 318 (Jan. 1985)) Bibliography: p. 1. Hardy spaces. 2. Embeddings (Mathematics) 3. Multipliers (Mathematical analysis) I. Sawyer, Eric T., 1951. II. Title. Ill. Series: Memoirs of the American Mathematical Society; no. 318. QA3.A57 no. 318 [QA331) 510s[515'.2433) 84-24294 ISBN 0-8218-2318-3

INTRODUCTION

The space f E

s'

HP(Rn), n~l, OO

t

,

is any function in

00

c0

with

We define

Many characterizations of it is proved there that If

f

Hp

HP(Rn)

are given in [FS] •

is independent of the choice of

is a function on Rn

t •

which defines a tempered distribution one

can ask what sorts of restrictions on the "size" of f E Hp •

In particular,

f

will imply that

Taibleson and Weiss [TW) have found one such set of conditions.

They call their functions ''molecules".

In §1 we present an embedding theorem

of this sort which includes the Taibleson-Weiss results and is "sharp" in several respects. J: X -+ HP , Hp

where

This enables us in §2 to prove sharp theorems of the form J

denotes Fourier transformation.

These results are

analogues of theorems of S. Bernstein, Taibleson and Herz for

Lp •

Received by the editors February 1, 1984. The first author was supported by a grant from the National Science Foundation and the second author by a grant from the National Science and Engineering Research Council of canada.

1

BAERNSTEIN AND SAWYER

2

In §3 we formulate a Fourier multiplier theorem for

Hp

which sharpens

up to their natural limits results of Calder6n-Torchinsky [CT] and Taibleson-Weiss. its sharpness.

In §4 we prove the embedding theorem and in §5 demonstrate

§6 contains the proof of the multiplier theorem and §7 shows

its sharpness. Finally, in §8-10 we prove three theorems about Fourier transforms of HP

distributions which follow easily from the "atomic decomposition".

first asserts that

HP

question of Weiss.

The second contains an

The

has the "lower majorant property" and answers a Hp

analogue of a recent theorem

of Pigno and Smith, while the last extends a theorem of D.M. Oberlin. In some respects this paper may be regarded as a successor to [TW], and we are grateful to Professors Taibleson and Weiss for their friendly interest and encouragement.

We also thank John Fournier for suggesting that we look

in the direction of homogeneous Besov spaces in order to find sharp results. In [TW] the

HP

Lipschitz spaces.

distributions are defined as certain linear functionals on Latter's theorem about the atomic decomposition [L], see

[Wi] for another proof and §6 of this paper for a description of the result, shows that the TW spaces the

HP

spaces as defined by us.

where

1 N = [ n(- - 1)] ,

p

coincide with

EMBEDDING AND MULTIPL!Ea THEOREMS FOR HP(R 0 )

3

STANDING NOTATION (0.1)

N denotes the largest integer less than or equal to

(0.2)

~ denotes the shell

(0.3)

Hp = Hp(Rn)

(0.4)

Jf

(0.5)

lx E R0

:

2k

~ jxj ~

2k+ 1 } ,

1 p

n(- - 1) k E

z.

=J

f(x)dx , and dx denotes Lebesgue measure on Rn • Rn C denotes a constant depending possibly on n and p which can change from line to line.

(0.6)

For

(0.7)

We use the usual multi-index notation. each

x ERn,

~j

xi

denote the coordinates of

x .

For

~

=

(~ 1 ,

••. ,

~n)

with

a non-negative integer, ~n X

(0.8)

For

f E CN ,

basepoint

x

PNf

=0

•

n

denotes the N'th Taylor polynomial of

f

with

1.

If

f E Hp

I f(S) I

then

EMBEDDING THEOREMS

1

~ c Isl n(p- l)

[ TW, p. 105], so we expect that

will satisfy the vanishing moment condition

Jf(x)x 13 dx

( 1. 1)

0 ,

whenever the integrals make sense. 1

less than or equal to

f

N is the greatest integer

and integrals without limits are over all

n(- - 1) p

Suppose that

Recall that

does satisfy the necessary cancellation ( 1. 1) •

What kind of size condition on

f

will guarantee that

f E HP ?

One such condition has been found by Taibleson and Weiss [TW, Theorem 2. 9] , that

who followed up on earlier work by Coifman and Weiss [ CW] • 0 < p ~ 1 ~ q ~""

define a f E Lq(Rn)

where

(p,q,N,e)

q>1

and that

when

p = 1 •

Suppose

Taibleson and Weiss

molecule centered at the origin to be a function

which satisfies (1.1) and also

a and e are related by 1 p

and it is assumed that of TW asserts that

1

e > n(- - 1) •

(p,q,N,e)

p

+

n(- -

1)

Thus

a > .n(-p - -) q

molecules belong to

e

1

1

Theorem (2. 9)

Hp •

We are going to prove a stronger embedding theorem which can be regarded as a critical endpointcase of the Taibleson-Weiss theorem.

To

formulate this theorem, and some others in this paper, we must introduce 4

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) certain function spaces considered by Herz (H] •

Definition

and

K

5

K which, as far as we know, were first

Suppose that

consists of all functions

(a)

for which the norm or quasi-norm

<J

I }:;

(1. 2)

k:

-CD

lfla>b/a 2kab}l/b

~

is finite.

·a: b

a

(b)

n Ka '

L

with norm or quasi-norm

II fll Ka:, b

( 1. 3)

a

The usual modifications are made when

a

= • or b = ..,

The

.

K

spaces appear in (H] , where they are denoted

Flett [F]

gave a characterization of the Herz spaces which is easily seen to be equivalent to (1.2) •

They have been previously applied in Hp

theory by

Johnson (JO 2] • Elementary considerations show that the following inclusion relations are valid. (1.4)

b c t3,f)

= j(q>- P~)f

,

when

1

N < n(- 1) , p 1

( 1. 10)

where

n(- p

P~

denotes the N'th Taylor polynomial of

(1.8) shows that hence for

f

s'

x = 0 ,

defines a continuous linear functional on

0 < p < 1 we may regard

embedded in

q> at

1) ;::; 1

then

S' ,

and

1

K~(p - l)' P as being continuously

via the definitions (1.9) and (1.10) • 1

Note that if

f E K~(p- l),pc L1

moment condition (1.1) ,

and

f

satisfies the vanishing

then

q> E

s ,

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) so the distribution defined by

f

7

in the usual way coincides with the one

defined by (1.9) and (1.10) The Taibleson-Weiss for some

molecules are just the functions in

(p,q,N,e)

1

1

p

q

0:> n(- - -)

which satisfy ( l. 1) •

these molecules belongs to a space

By ( 1. 6) ,

each of

1

for some

O:>n(-- 1). p

Our sharp version of the Taibleson-Weiss theorem requires statement in 1 p

three separate cases, according as p

=

1•

NK~'q when O:>n(;-1),

By (1.4) we have

so this theorem generalizes

for

that by stating the theorem for

which

1

1

. K

0 p , 1

xn(p-

z

and the examples

1 z),q

2

X space, for instance f1

in §5 show that when

does not transform into

Hp ,

no

matter how

is.

Suppose next that

1

0:> n(p- -

1 z) '

Then (1.6) and (1.4) show that 1

n

- l},p Ko:,q c Ko:- 2• q c Kn(-p 1 2 1

(2. 5}

and we may define

THEOREM 2b.

0

in place of

n(~

-

t>

then :J

maps

X~'q

continuously 1E!2 Hp •

PROOF. If

When

1

z

n(P- 1) l

this follows from (2.2), (2.5) and Theorem 2a.

n(~ - 1) E z then K~'q c Y(p} for

p

1 2 dx)q/ 2 tqa.-l dt1 11q,

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) and this last expression is equivalent to

as required.

17

3.

f E HP

If p I~ /r<s>

then m f

c IIfl!

is a continuous function satisfying

1

1 P I~~"., I n [ TW, p. 105) • H

defines an element of

is well defined from Hp

f

then

HP

-t

if this mapping takes The function

where

s

MULTIPLIERS

s'

S' ,

and thus the mapping

We say that

Hp

-t

(m

'f)¥

m is a Fourier multiplier of

continuously into

m is said to satisfy a

f

Hp •

~ormander

condition of order

s ,

is an integer, if

O n(P -

z) ,

If

l!m511

a 2
1

~

::> :Ji(B '

2

q

)

if

Thus we have the

following corollary, which furnishes a practical sufficient condition for multipliers.

COROLLARY 1.

suppose

!!!.!!.

0

1

and

< ""

2.

£!.., equivalently

sup llm611Bo:,q < "" ' 6 2 then

m is !. Fourier multiplier of

Hp • 1

The corollary is sharp in that larger space

K t3,q

2

•

1

- -2), p Kn(2 P

cannot be replaced by any

EMBEDDING AND MULTIPLIER THOEREMS FOR Hp(Rn) When

p= 1

Theorem 3a becomes false.

0 1

The space

K1'

21 L

1

has to be

replaced by slightly smaller K-type spaces with appropriate weights Suppose that

w: IO, 1, 2, ••• } -+ [ 1,..,)

Define the space

f E L1(Rn)

K(w) to be the set of all

llf!IK(w) =

1 ~ w(k) ~ w(k + 1)

satisfies

...

J1 I < 1 1£1

+

E k= 0

X

<J

w(k) •

< ~ ,

1 2> = ~b ,

0

theorem for the endpoint case

1

but to get boundedness at the

it seems essential to have a multiplier 1

Kn(p- 2),p 2

m satisfying (3.6) will

4.

1 n(p--1)iZ,

Take Vx, t(y) = t

l' -n

fEY(p)

E t(

1

O2jxl

(hl'N

jf(y)j lxl)

I Yl < 2 1xl

Ig(x, t) I ~ C

sup t>O

(4.4)

(l.J)N- 1 lxl -n

J

'

lxl

(J 6 (x) + \If\\ 1 > , L

-n

dx,

p= 1 ,

where

J6(x) t

sup t-n N

we

,

= -co

I: k= -..,

mP 2 jNp 2k[n- p(N+ n)] j

deduce

(4. 6)

A completely analogous computation yields

when

(4.7)

To estimate maximal function.

J Jf dx

>

N •

we need some inequalities for the Hardy-Littlewood

FE L1(Rn)

For

1

n(p- 1)

(MF)(x) =

sup xEB

write

1

B

J

IFI dx ,

I I B

B denotes a ball.

where

Suppose

LEMMA.

1

n

F E L (R )

.!!!!! ~ F .!!. supported .!!! .! ball B • Then O 1

J:

dx

+ j' jxl

l

"'

E

J

k=O

~

J~ dx:=;c

"'

m~2jNp2k(n- p(N+n)]

110

r:

E

k=O j=k+l

j-1 . 110 E E mp 2JNp = C E j m~ 2jNp < C j=l k=O j j=l J a>

= C

J

I! fliP

•

Y(p)

Similarly,

(4.11)

J

Ixl > 1

J~ dx :=; C

= C(

Also, when

"' k+ 2 E E mp 2j(N+ l)p 2k(n- (n+N+ l)p] k = 0 j = -110 j

1 110 110 E E + E j = -= k = 0 j = 2 k

a>

m~ 2j(N+ l)p 2-kp)

E

=j

- 2

J

1 n(p - 1) = N ,

(4. 12)

(4.13)

Theorem 4b follows from (4.8)- (4.13) • Now assume

p= 1 •

In proving Theorem lc we may assume that

1\fll

1 = 1 . L

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn)

31

By (4.2) and (4.4) we have

(4.14)

computations like the ones above give

(4.15)

co

J

(4. 16)

Ixl > 1

J

J 2 dx~C

=1

Ixl > 1

lf(x)l(l+loglxl)dx~cl!fll*' y

0

Ixl > 1

J 3 dx

• CD m. 2J + E m.) j = -.., J j = 1 J

for

~ CJn

~ c ( E

Next, define again F(x) = 0

jmj~cj

E j

x f.~ ,

=

F(x) where

x E

f(x)

for

k~O

is fixed.

By part (b) of the lenuna, we have, recalling

~

R

=

I £1

~ C llfll *

•

Y

{2k-l~lxl ~2k+ 2 1

Then

J 1 ~ C MF

on

~

llfll 1 = 1 , L

s1

Let for

k E

s1

=

{k~O

~ e

-k

~

I , s2

= {k~O

:J_ 1£1 <e -k I ~

,

J ~

Since

:J_ 1£1

X

1

J 1 dx

log;~ X

1/2

~

C

J_ I fj ( 1 + log+ I fl + k) dx ~

for

0<xO •

from

p(N+ n) Hence, for

k

2

kn

-1

The last equation follows from (5.4) and the next-to-last one n • k ES ,

J k 2

0

and

Is I

Then

IIR*all 1 ~ C

We will show now how to

j E

P, ... , n}

is fixed.

By the atomic decomposition, there exist 1-atoms

H

and constants

A.j

with

EIA.jl

~c

and

""

R*a = E A.. a. 1 J J

with convergence in

Since

mE 1""

s'

.

Formally, we have

and Fourier transformation is an isomorphism on

s' ,

the

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) series on the right converges to

R*f

in

But

S' •

53

II (mij)"ll

1

!::

C ,

by

L

(6.21),

L1 •

so in fact the series converges in

... \IR*fll 1 !:: L

Hence

1\f\1 1 !:: C ,

c

E

1

Ift)

Moreover

!:: c

by the Riesz transform characterization of

H1

H

This completes the proof of Theorems Ja and Jb, modulo Lemmas 1 and 2 and (6.20).

PROOF OF ( 6. 20).

where

(6.22)

Also

where

Recall that

supp a c I J

IJ

a(y) R(y) dyl

I Yl

:S C 2j(N+l)(1+lxl)-r,

2J + 1

has rapid decrease.

Then

lead

56

BAERNS TE IN AND SAWYER

PROOF OF LEMMA 2 •

j ~0 •

Fix

( 6, 26)

We may assume

11811 K n(lp

1 •

1),p

1

k~j + 2

Suppose that

8*Q(x) =

k-2 E

J

x E

and

~

•

Write

Q(x- y) 8(Y) dy + (8

x.... ) * Q(x) A.c_

A.e,

.f.= _..,

CD

+

where

~ = ~ _ 1

U ~ U ~+ 1 •

IQ(x-y)l ~ 2-r(k- 1 ) , Write

J(.(..) =

J

If

181 dx •

x E ~ ,

.f,~k+2

while i f

E .(..=k+2

J

Q(x- y) 8(Y) dy

A.c,

.f, ~ k - 2 ,

y E A.(, ,

then

then

IQ(x-y)l ~ 2-.(..(k- 1) .

Then

A.f.

I8*Q(x)l::; 2-r(k-1)

J

k-2 "' E J(.(..) + 10 ,

Q E c""

1\Q\1 1 = 1 ,

and

supp

II xl ~ t1

> n(~ +

t>

IQ(x) I ~

satisfies

Q

C

Ixl-r

For fixed

for some

C ,

and so

c >0 •

Hence

satisfies the hypothesis of Leoma 1 for some

=e

f(x)

ix 1

1

(fo*Q)(x) E

unless

l .

m=

1

8 ~6 ~8

•

Then

supp me

n

Let

for every

Q

= cl],

h = ~ •

CX> 0 ,

~ 21

n(p - 2),q

K2

i t follows that

1

K2

The inverse Fourier transform of 1 1

These functions form a bounded set in Leoma 1 with

1} ~lsi

and

m6 (g)

m(!lg)

is

h E

n

by Corollary 1 of §2.

We have

.,

= m = f •

6

8 ~6 ~

m satisfies (7.4).

p>O

= m(6E;) -n

1

when

Then

so

cQ

n(p - 2),q

q>p

Define

Q2:,0 ,

.

Qc

L

r

satisfying

8

T)(S)

f(6

-1

=0

x).

From

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) Since

f E L

Take

x E

(f0 *Q)(x)

1

~

to show

,

,

it

even.

k

k 2:. 2

J

=

f ~ HP

Then

~

.!!

Since

the middle integral is

it follows that for all sufficiently large

p

(f 0 *Q)(x) 2:. c 2

fo*Q ~ Lp ,

Hence

and

-nk/p

f ~ LP ,

I:

e (k) < .., •

k

-1/p

2:.

-nk/p -1/p c 2 k •

k

,

as required.

Next, we construct the function for which

>2

the first and third integrals are

Q(O) > 0

Since

f ~ LP

suffices to show

k _ 1 Q(x- y) f 0 (y) dy + J

IYI

61

Take a subsequence

M •

S c 2 z+

Define

k ES

2-nk/p

X

Q be as in the construction of

e(k)

above show that If does.

e (k)

s ,

m and define

M=g.

g(x)

If

k E

E ~

otherwise.

0 ,

Let

'

satisfies (7.2), then Lemma 2 and an analysis like the one M satisfies (7.5) but that

(Mh)¥

~ Hp

does not satisfy ( 7. 2) we replace it with a sequence

We may suppose

e ( 1)

=1

•

Define inductively integers

.t j

,

whi.:l: :

z: ~

62

by

BAERNSTEIN AND SAWYER

t0 = 0 ,

and

tj + 1

1 ~(2tj) ~(2tj+l)~-2 ~ ~

Then Construct

B(k) ~

De f"1ne

and

M using

is the smallest positive integer for which

B(k)

k~1 ,

8 (k) ,

bY

satisfies ( 7. 2) •

8(k) •

Also,

8(k)

~

e (k) •

Then

sup I!M6 11

6>0

~ sup IIM6 11 6>0

K(e,p)

0

\1616\1

0 .

EMBEDDING AND MULTIPLIER .THEORE~5 FOR Hp(~n) As we pointed out in §3,

2: w(k)

with m =0 •

-2

= "'

63

there exists a nondecreasing sequence

such that the only

m

satisfying

Thus, some supplementary hypothesis such as

11m :1

sup

6 >0 " 6' K(w)

S C w(k)

w(2k)

w(k)

'"'

is

is needed

for the theorem to hold.

PROOF.

Choose a set

S = \k 1 , k 2 ,

="',

lim (ki+l- ki)

~ 3 , .•.

!c

z+

with

ki < ki+l,

and

i-tm

2:

w(k)

-2

= .., •

kES

k 1 ?. 10

We also assume that

~ E c~

~

,

'!

0 ,

with

supp

Yc

I: k ES

Then k E S

m

and

Define

.

WrLte

li

(S) = 0

unless

2k- 3 soS2k+ 3

f 0 = ..

k

6 = 2 e ,

k k m(2 (sli- 2 el))

is

F(x)

!lsi< 1\ .

for

i ?_1

Take

Define

1 w(k)

2k- 3 S li S 2k+ 3

for some

k ES •

For fixed

we have

and fix

where

ki + 1 - ki ?. 10

and

1

r

>n •

B S e S8

Then

f0

satisfies

The inverse Fourier transform of

BAERNSTEIN AND SAWYER

64 which satisfies

IF(x)l ~ C 2

(7.7)

C which depends on

for some

-2kn

r

all x

,

but not

E •

An argument like the one used to prove Lemma 2 of §6 shows that the

F*~

functions

also satisfy (7.7)

1 ,.. (F*T]) w(k)

¥

m~

Since

= --

v

it follows

that

< 1 6\~(x)l v -

(7.8)

C _1_ 2-2kn w(k) '

I616 (X) I 0 •

w(.f,) S cw(k)

t\M w(.f,) S C ( k) w(k) ,

for

l~tS2k

t>k ,

Thus, the right hand side of (7.9) is majorized by

C + C 2 2k(r- n)

~

(i)M

.f, = 2k \k;

•

w(k)

2 (n- r)t

and also

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(R 0 ) 0(2(n- r) 2 k).

The last sum is easily shown to be side of (7.9) is

0(1) ,

uniformly in

5 ,

11£511

sup 5 >0

65

Hence, the left hand

and we have shown that


0 ,

q

w(k) ;::: 1 It remains to construct the function

h •

Choose

k 1 < k 2 1 , b1 = 1

ON A THEOREM OF PIGNO AND SMITH

I b j I~

and let

be a sequence of positive numbers

and

b.

1

--l±...!. > b. - q '

(9. 1)

J

Pigno and Smith [PS 1], see also [PS2], assumed that

q2;:2

and used

the method of Cohen-Davenport to prove the following theorem about Given an analytic function

f E H1 (n)

there are measures

~·

J

H1 (n)

E M(n)

satisfying

f(.t)

j = 1, 2, •.• '

.(, E Z ,

and

We are going to use the atomic decomposition to prove an analogous result for

Hp(Rn),

O=C(p,q,n)

where

depends.Q!l q

butnotQ_!!

!h.! • J

where

Examples of the form supp cp c ! IS I < 1} '

nHP

which belong to

by Corollary 1

Ol

we have

We may assume

~

clsl ,

q2:2 •

By (8.3)

so that 0

( 10. 3)

~

k

=-co

To analyze the case

k2: 1

we introduce the unit cube

Q

and the

numbers

a(t)

as in §8. length

~

Let C(n) ,

.s:k

=

= It E zn so

sup sEQ

:

n

If <s + t) I ,

Q+ t

meets

tEZ ,

sk

l .

Each intersection has arc

80

BAERNSTEIN AND SAWYER

By

~older's

I: ( k= 1

inequality, with

1 1 q+ qt =

1,

J

sk

~ C(

!:

2 -kq I /q) 1/q I

k=l

Now each q;:::2 •

.{,

belongs to at most two of the

(

. !:

I:

k= 1

.t,Etk

tk ,

a(.(,) q) 1/q

and we are assuming

Hence

where we have used (8.2).

This inequality, with (10.3), proves Theorem 8 •

REFERENCES [A]

A.B. Aiexandrov, A majorization property for the several variable Hardy-Stein-Weiss classes (in Russian) Vestnik Leningrad. ~· No. 13 (1982), 97-98.

[C)

R.R. Coifman, A real variable characterization of Hp,

Studia Math.

51 (1974), 269-274. [CW]

R.R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. ~· Math. §££. 83 (1977), 569-645.

[CT]

A.P. Calder6n and A. Torchinsky, Parabolic maximal functions associated with a distribution, II, Advances i£ Math. 24 (1977),

101-171. [F)

T.M. Flett, Some elementary inequalities for integrals with applications to Fourier transforms, Proc. ~Math. 22£. (3) 29 (1974),

538-556. [FS]

c. Fefferman and E.M. Stein, Hp spaces of several variables, ~ Math. 129 (1972), 137-193.

[H)

C. Herz, Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms, l· ~· ~· 18 (1968), 283-324.

[Ja]

s.

Janson, Generalizations of Lipschi.tz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math l· 47 (1980),

959-982. [Jo 1] R. Johnson, Temperatures, Riesz potentials, and the Lipschitz spaces of Herz, f!££. ~ Math. ~· (3) 27 (1973) 290-316. [Jo2]

R. Johnson, Multipliers of Hp spaces, ~·

[L]

R.H. Latter, A characterization of HP(Rn) in terms of atoms, Studia Math. 62 (1978), 93-101.

[LS]

E.T.Y. Lee and G·I. Sunouchi, On the majorant properties in Lp(G), T8hoku Math. l· 31 (1979), 41-48.

[MS]

S. Minakshisundaram and 0. Szasz, On absolute convergence of Fourier series, ~· Amer. Math. 22£. 61 (1947), 36-53.

[M]

A. Miyachi, On some Fourier multipliers for HP(Rn) Tokyo 27 (1980), 157-179.

[0]

D.M. Oberlin, A multiplier theorem for H (R ) ,

1

M!!· 16 (1977), 235-249.

n

l· E!£. 2£1.

.f!2£.. Amer. tl!£!l .

.[££.

73 (1979), 83-88. [ P]

J. Peetre, ~Thoughts Dept., Durham, 1976.

~~Spaces,

81

Duke University Mathematics

82 [PT]

BAERNSTEIN AND SAWYER J. Peral and A. Torchinsky, Multipliers in Hp(Rn), O