A Sharp (${H}^1 _ {L}^1$)continuity theorem for the Rubio de Francia maximal multiplier operator

A SHARP (HI,L I ) CONTINUITY

THEOREM FOR THE RUBIO DE FRANCIA MAXIMAL MULTIPLIER OPERATOR

YONG-KUM CHO

1. Introduction

Consider a function m defined in Rn and the associated convolution operator T given by the Fourier transform T= mj. The well-known Hormander-Marcinkiewicz multiplier theorem asserts that if m satisfies

for all multi-indices a, lal _~

[iJ

Let us turn our attention to the corresponding maximal theory. We shall be concerned about the operator described as

(1-3) T* f(x) =^{sup ITtf(x)l,} (TtJ)l~)=m(t~)j(o·

t>O

In this connection let us point out a theorem of Rubio de Francia [5] which states that if m E CHI (Rn) and for all a, lal ~ 1+1, (1-4) IDQm(x)1 ~ CQ (1 +Ixl)-a with some fixed a> 1/2,

Received August 12, 1994.

1991 AMS Subject Classification: Primary 42B20, 42B25.

Key words: Multipliers, Atoms, Fractional Integration Theorem, Dyadic Maxi- mal Operators, Square FUnctions.

then T* extends continuously to a mapping from IJ' (R.R) to IJ'(R.R) provided 2n/(n+^{2(1 -1)}< p <(2n -'- 2)/(n - 2a). .

The purpose of this note is to set up the (HI, L¹)-inequality for T*

under a mild assumption onm in' analogy with Honitander-Marcinkie wicz hypotheSis (1-1). Specifically Vie sha.I1 prove

THEOREM 1. Suppose that there exist positive numbers a, b, 8 with a+b> 1, 8> 3/2 suCh that

(1) Im(e)1 $ C(1+len-^{a, IVm(e)1 $ C(1} +len-^b,

(2) forallmulti-indices a of order lal = 1, 1+1, sup ( ID^am(e)1²de ^{$ Aa}R-^26•

I:::;R<oo JR:::;lel:::;2R

Then we have the apriori continuity inequality

tlT*! IIp(ll'') ~BII!HHt(ll")·

We remark here that the first hypothesis is required to ensure the L2 boundedness (see[5], pp. 398). Itis interesting to note that the sec-·

ondhypoth~isrequires decays on the derivatives ofm of only highest orders. Our method reveals that it's possible to have a slightly weaker assumption, namely,

THEOREM 2. The hypothesis(2)ofTheorem 1 may be replaced by (2)' sup ( { ID^am(e)12-Ede)^{H E $ Aa}R-²⁶

I:::;R<= JR:::;lel:::;2R

for an arbitrarily small €>0 and a, .Ial ^{= 1, 1+1.}

By the standard interpolation and well-known duality argument be- tween the Hardy space H¹(R.R) and BMO(R.R) ,the above theorems implicate instantly that

IIT*f ^{IILp(IlB) $ Cp} IlfIlLp(IlB), 1<^p< ⁰⁰

and moreover 'the pointwiseconv~rgenceproperty lim '{ f(x - ty) K(y) dy= f(x) a.e.,

t7*O JIl"

where K = m, f E V (R.n) , 1 < ^P < ^00. Throughout this paper, C will stand for a positive constant which ~ay be different in each occurrence.

2. Preliminary Settings

In preparation for the proof, we begin with collecting more or less known results and techniques ·which are relevant to our purposes (re- fer [5], [6] and [7]). Choose a radial bump function 1/J E coo(Rn) satisfying

{

1, 1~1:5 ¹

1/J(~) = ^{0, lel} > 2

Letusset 4>(e) = 'l/J(e)-'l/J(2e) so that supp(t/» C {1/2< lel <2}

and

1/J(e) +L ^{<,6(2-je) = 1 for all e=FO.}

'>1

,_

It follows that we have a Littlewood-Paley decomposition for m, m(e) = 1/J(e) m(e) + L ^{<,6(2-j e)m(e) = 'l/J(e)}m(~) + L ^mj(e)

j~1 j~1

and consequently we get the majorization T* f(x) :5 M* f(x) +2:j>1

Tl f(x) , where M*, Tj* denote the associated dyadic maximal opera- tors.

It turns out that M* behaves extremely nice (see [5], pp. 398) so we only concentrate on studying Tj, ^{j ;?} 1. A simple application of the fundamental theorem of calculus shows that

TJf(x):5 (2Sjf(x) Sj/(x)r/2

= 2i!2 (Sj f)(x) +2-i!2 (Sj f)Cx) , where

lOO ^{dt !}

Sj/(X) =(10 IT]fCxWT) ,

_ (1^{00 _} dt)l Sj/(x) = 10 ^{ITJfCxWT '}

(T]ftce) =mj(te)i(e), (TJftce) =miCte)i(e), miCe)=V'mjce) .e·

LEMMA 1. The condition (1) of Theorem 1 implies that T* is boundedin L²(lRⁿ).

Prool. The proof follows from the easy observation

Let us turn now to a brief description of the Hardy space HI (lRn) . Thisspaceisanextremely nice subspace ofL¹⁽Rn) ,elements of which are stable under various singular and maximal operators. Inparticualr,

where Ri .denotes the Riesz trailsf'OrIhs (Rif re^{e) = en}~, ^{1 <}

j =:;n. More importantly it admits the atomic decomposition for each element. A function a is called an HI (lRn)-atomifthere exists a cube Q such that

(1) (2)

supp(a) C Q

h

According to R. Latter [4], any I E HI(lRn) can be decomposed into f. = E AQ aQ where aQ is an atom and scalars AQ satisfy l:Q IAQI5 c^{III IIH1} ~ HI11£1(11")+L:j=I II^Ri I IIL1(1\") .Weshall need the following simple fact

LEMMA 2. Suppose that T is an L2-bounded sublinear operator andfor each HI -atom aQ supported in a cube Q,

1-

where cQ denotes the complement ofQ and Q the ·2-fold concentric dilateofQ. Then T ma.ps HI(lRn) continuously into LI(lRn). More precisely, we have liT f 11£1 5 CCIITII2,2 +A)IIIIIH1 .

Proof. For f = E ^{Aq aq} belonging to the Hardy space, we have

liT^fllLl $ L ^IAql

f

= 2: ^IAqI

(10

!Q

Itfollows from the Cauchy-Schwartz inequality that

k

$ liT 112,21£1 11/211aq 112

$ liT 112,2IQII/2IQ 1-1/² $ CliT 112,2,

Therefore,

liTf IILl $ C( IIT1I2,2 +A) L ^{IAqI :5 C(1ITI12,2+A)11f IIHI. 0}

3. Proof of The Main Theorem Notice first that

and we may view our square functions Sj, Sj as Hilbert space-valued linear operators

Sj f(x) = {T}!(x»)P«O,oo)jdt/t) , Sj f(x) = ('1')!(x»)L2«O,oo)jdt/t)

so that we may appeal to Lemma 2 for their mapping properties. Take an HI-atom a supported in a cube Q.

LEMMA 3. For j ~ 1 and an arbitrarily small € >0,

(3-2) j __CQ ^{ISj a(x)Idx ~ C2j(1+E- 6) •}

Proof. Using the translation invariance of our operator, we may as- sume the center of Q lies at the origin. By the Cauchy-Schwartz inequality

Denoting by l(Q) the side-length of Q, we observe that

1-

1-

CQ lo ^{CQ t}

(l ,

(Q)

100) 1

= + _Ix121-2EIT}a(x)12dx-

o ^{I(Q) CQ} t

= (1)+(ll).

With Kj = mj, KJ(x) = t-ⁿ Kj(x/t) ,we note that

( )^l~

IT}a(x)1 =^IKj*a(x)1 =/Iall2

h

so

q~ d

(1) ~ IQI-^{1 {} l_lxI21-2E { IKj(x - y)12_dydx--!.

la ^CQ lQ ^t

Since Ix - yl ~ Ixl wheneverx E cQ, y E Q,

Applying the basic properties of the Fourier transform and the frac- tional integration theorem of Hardy-Littlewood-Sobolev ([7], pp. 354 ), we get

f ^{Ix121- 2E IKj(x)12dx =} f IIxl-^E( lxi'Kj(X))1^{2 dx}

J.... Jll"

2

= c f _1~I-n+E *L ^{DPmj(e) de}

J." ^IPI=I

= e L f Ilel-^n+E *DPm,(O1 2 d~

IPI='J."

~ ^c L

(f

IPI=' J...

H we make use of Holder's inequalty, then

and therefore by hypothesis (2), we are led to

For the part (H) we use the vanishing property of atoms to write down

Tl^{a(x) =}

k

=

k

= I:

11

where the last equality comes from Taylor's theorem. Hence

1 ) 1/2

ITl^a(x)1 _{~ CIQI~-!}

(L

(Il) ~ CIQI~-l ^{{OO {_ IzI21-2£ { {I} ID] [(} (z _ sy)12 dsdydz dt

l'(Q) lcQ l Q1^{0 t}

~c _IQI~ {CO t21 - 2 £-n-2 ( IzI21- 2£IV"{ Kj(z)12 dz dt

l'(Q) l ...ⁿ t

~ CIQ\~-~-l

In

~^C IQI~-~-l 2: (In ^{IvP(Cmj({» II'de}

riP,

(PI=1 ...

~CIQI~-~-122j£ 2:

1

JPI=1 ;!j-l~lel~2i+l ⁾

as before. Here recall hi == 1 and the hypothesis (2) shows then (11) $ CIQI-U- - ;:^-122;(1+£-6) .

Combining both estimates, we finish the proof. 0 Now it's a relatively easy matter to state

LEMMA 4. For j ;::: 1 and an arbitrarily small € > 0, (3-3)

Proof of Theorem 1. Since 11 Si112,2 < 2-ia , II5'i112,2 < 2ⁱ (1-b),

Lemma 2 gives us

IISifllLl < C(2^{i(l+t-6)+2-ia ) Ilf11H1,}

l15'i/llL1 :::; ^C (2^{i(2+t- 6)+2i(1-b») IlfllRl}

and consequently (3-1) shows

Upon summing this geometric series, the proof is now completed. 0 References

1. Y. K. Cho, Multiparameter Maximal Operators and Square F'unctions on Prod- uct Spaces, Indiana U. Math. J. 43, No. 2 (1994), 459-491.

2. R. Fefferman andK.C. Lin, A Sharp Marcinkiewicz Multiplier Theorem, Prepr into

3. L. Hormander, Estimates for Translation Invariant Operator! in LP Spaces, Acta Math.104 (1960), 93-140.

4. R. Latter, A Characterization ofHPCIl~n) in terms of Atoms, Studia Math. 62 (1978), 93-101.

5. J. L. Rubio de Francia, Maximal Functions and Fourier Transforms, Duke Math. J. 53, No. 2 (1986), 395-404.

6. E. M. Stein, Singular Integrals and DifJerentiability Properties of F'unctions, Pinceton University Press, 1970.

7. _ _ , Harmonic Analysis, Princeton University Press, 1993.

Department of Mathematics College of Natural Sciences Chung-Ang University Dongjak-Ku, Seoul 156-756 E-mail address:[email protected]