Embedding and Multiplier Theorems for Hp (R^n)

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)

1980 Mathematics Subject Classification. Primary 42830. 42815. Library 0' Cong..... Cltaloging-in·Publication Da" 8aernslein. Alben. 1941. Embedding and multiplier theorems for II/'(R II )

(Memoirs 01 the American Mathematical Society. ISSN 0065·9266: 318 (Jan. 1985)J Bibliographv: p. 1. Hardy spaces. 2. Embeddlngs (Mathematics) 3. Multipliers (Mathematical analysis)

I. Sawyer. Eric T.. 1951Society; no. 318.

II. Title. III. series: Memoirs 01 the American Mathematical

OA3.A57 no. 318 (OA331) 51051515'.2(33) 84·24294 (ISBN ().8218·2318·3 Subacriptionl end orders for publications of the AmeriCan Mathemattcal So1

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 •

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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