Multipliers from H 1 Andreas Nilsson

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Ark. Mat., 36 (1998), 379 383

Multipliers from H 1

Andreas Nilsson

t o L p

1. I n t r o d u c t i o n

We shall prove the following t h e o r e m combining the multiplier t h e o r e m of H 5 r m a n d e r - M i k h l i n [5] for H a r d y spaces with a result of Johnson, Corollary 2.1 in [6]. T h e original version of H 5 r m a n d e r Mikhlin's multiplier t h e o r e m states t h a t under the assumptions of T h e o r e m 1 below with q = l the o p e r a t o r is of weak-type (1, 1). T h e H 1 case was proved in [4] a b o u t a decade later by Fefferman and Stein.

To simplify we will use the notation A j ={2J 1 ~ I~1 ~ 2j+1}"

T h e o r e m 1. Assume that m E C k ( R n \ { 0 } ) and

/ A ~ 12Jl~lD~m(~)[2d~<-2~J(2-q)/q' j c Z , J I,~l<k

where k is the least integer > n ( 2 - q ) / 2 q , and 1 <q<_2, then the convolution by K = ~ maps H 1 to L q.

As noted above the case q = l is the theorem of H 6 r m a n d e r Mikhlin, and the case q=2 gives us a weakened version of Johnson's result. The actual t h e o r e m of Johnson states t h a t if

Im( )l _< c,

then convolution with ~ m a p s H 1 to L 2. However, one can easily see t h a t our proof goes t h r o u g h in t h a t case even if we drop the derivatives in the assumptions. John- son also inverts the result, i.e. he proves t h a t every convolution operator m a p p i n g H 1 to L 2 is of this type. One should also observe t h a t if the kernel K has c o m p a c t support then the operator m a p s H a to itself. Finally, if we take m ( ~ ) = l ~ I ~ then we obtain the result of the H a r d y Littlewood-Sobolev theorem, i.e. the operator m a p s H 1 to L q if q = n / ( n - a ) .

Related results could be found in Bagby [1] and de Michele Inglis [3].

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380 A n d r e a s Nilsson

2. P r o o f

The proof interpolates the methods in Hhrmander [5] and Bj6rk [2].

Proof.

It suffices to prove t h a t the L q norm is bounded for atoms (with a bound that is independent of the atom). Let a be an atom with support in a ball with radius r. This means t h a t a also satisfies the following conditions

f a(x) dx O,

IlallL = <

r -n"

As the operator is translation invariant we can assume t h a t the atoms are centered at the origin. Since 2 / ( 2 - q ) is the dual index to

2/q

it is easy to see t h a t

(1)

[[aK[[Lq<--(/xl>2r* [Ka(x)[qdx) 1/q§*

We decompose m = ~

mj

by setting

mj(~)=m(~)O(2-J~)

where 0 C C ~ has support in the set 1 < [ ~ [ < 2 and for ~ht0

~ 0(2 J~)----1.

j - - OO

By Leibniz' formula we obtain

12Jl"lD mjl 2 c2nj(2-q)/q.

I~r<k

Using Parseval's formula and putting

Kj-~j

it follows that f ( 1 + 2 29lxl2) ~ IK~ 12

dx C2 nj<2-q)/q.

_<

(2)

Thus

(3) IIKulIL. dx

(1+22Jlx12)k(q/( 2 q))

In the same way (2) implies t h a t

(/xl>_t [[(j'q dx)l/q ~- v(2Jt) n(2-q)/2q-k,

(3)

w N c h s h o w s

M u l t i p l i e r s f r o m H 1 t o L p 381

(/ixl _:t < q)/2q-k

under the assumption t h a t

lYl<t.

Together with the following estimate, easily proved by S. Bernstein's inequality (which says in particular t h a t the Lq-norm of the first derivative of a C~ could be estimated by the radius of a ball, centered at the origin and containing the s u p p o r t of the Fourier t r a n s f o r m of the function, times the LU-norm of the function itself),

(/ ,Kj(x-y)-Kj(x),qdx)Uq<c2J+lt for ,y,<t,

this proves t h a t if

lyl

_<t

(f~ IK(x-y)-K(x)lqdx) ~/q<-C ~ min(2it,(2Jt)~(~-q)/2q-k)<~C.

I >~2t j=--oo

Since a is an a t o m it is easy to see t h a t this proves the required estimate for the first t e r m in (1). It remains to show t h a t

IIK*allL 2 ~ c r - n ( 2 - q ) / 2 q .

Since the m j ' s have b o u n d e d overlap it follows t h a t

IIKal125:* <C E

oo

f lmJ(~)a(~)l ~ d~

J

_<c ~ Ilmjl]~/ la(~)l 2~/(2 ~)d~

j - ~ J

oo ( J A ,(2--q)/q

<_ ~ c la(~)l ~/(2 q)d~)

,

j --oo J

using (3) for the last inequality. Thus we have reduced to showing t h a t

J := la(~) I ~"/(2-~)

d~ < c ~ -~(=-q)/".

j = - 5

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382 A n d r e a s N i l s s o n

We start with the case q = l . We get

J <_ Cllall2~ <_ Cr -~.

In the case q 2 we want to estimate

~ s u p la(~)l 2

j - c o Aj

by a constant. For this we divide the sum into two pieces

&:= ~ sup la({)l 2,

2J>r_l Aj

J 2 : = E sup l5({)12"

2J < r 1 Aj

To estimate J1 we introduce a function ~bEC~ which is assumed to be 1 on the support of a and

IIr <- Crn/2.

We m a y assume t h a t

~b(x)=r

where ~b0 E C ~ is independent of r. We observe t h a t

a({) =a,~({)

Thus we have to show t h a t

(4)

2J ~r--1

Let { E A j .

If r]~A~=Aj_IUAjUAj+I

it follows t h a t 1~-?71>2J-2. Hence

< Cr-2 f

~1--2n--2

drl < C~r-22 -j(n+2).

a i , f l > 2 j 2

By Schwarz' inequality the p a r t of (4) where r ] 6 A ; is bounded, because n + 2 > 0

and IlallL~ <_<~/2. We ~lso h~ve

sup f ~< ~ la({)t2~rr~llS_<cllallLIlr

2 j ~ r 1 A j g/~; 2j 2T__1 ;

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Multipliers from H 1 to L p 383

since IlallL2 < r - n / 2 and II~IIL 2 <_r n/2 this finishes the estimate for J1.

For J2 we use the fact t h a t

la(~)l<rl~l,

from which it follows t h a t J 2 = C ~ supl&(~)12-<Cr2 Z s u p l~12 -< c

2J<r 1 /kj 2J<r 1 Aj

T h e J1 and J2 estimates together provide the required estimate in the case q 2.

We shall now interpolate to get the general case. Let p = 2 / ( 1 - 0 ) where 0 < 0 < 1, let cj=I]SXjlIL~ and dj=ll~tX/llL2, where Xj is the characteristic function for Aj.

By interpolation we know t h a t

T h e q--1 estimate shows t h a t

Ildj IIg= <-c r-n/2

and the q=2 estimate t h a t

Ilcj IIz=

< c . This implies t h a t

0 1 0 C20tt2 20 1/2

0 1-0

<_llcjllz lldjll 1

So when we put p = 2 q / ( 2 - q ) the proof is complete. []

A c k n o w l e d g e m e n t . I would like to t h a n k my advisor J a n - E r i k BjSrk and also Per Sj61in for their assistance.

R e f e r e n c e s

1. BAGBY, R. J., Fourier multipliers from H I ( R n) to LP(R~), Rocky Mountain J. Math.

7 (1977), 323-332.

2. BJORK, J.-E., Some conditions which imply that a translation invariant operator is continuous on the Hardy space H I ( R ~ ) , Unpublished manuscript, 1974.

3. DE M1CHELE, L. and INGLIS, I. R., Sharp estimates for translation invariant operators mapping H 1 to L q, Boll. Un. Mat. Ital. A 17 (1980), 414 419.

4. FEFFERMAN, C. and STEIN, E. M., H p spaces of several variables, Acta Math. 129 (1972), 1 3 ~ 1 9 3 .

5. H O R M A N D E R , L.~ Estimates for translation invariant operators in LP-spaces, A c t a M a t h . 1 0 4 (1960), 93-139.

6. J O H N S O N , R., Multipliers of H p spaces, Ark. M a t . 16 (1977), 235 249.

Received November 25, 1996 Andreas Nilsson

Stockholm University Department of Mathematics S-106 91 Stockholm

Sweden

email: andreasn~matematik.su.se

Multipliers from H 1 Andreas Nilsson