Weak type $L^1 (R^n)$-estimate for certain maximal operators

J. Korean Math. Soc. 34 (1997), No. 4, pp. 1029-1036

WEAK TYPE £l(JRn)-ESTIMATE FOR CERTAIN MAXIMAL OPERATORS

YONG-CHEOL KIM

ABSTRACT. Let{Ath>obe a dilation group given byAt= exp(-P logt), whereP is a realn xnmatrix whose eigenvalues has strictly positive real part. Let v be the trace of P and P* denote the adjoint of P. Suppose that lC is a function defined on Rn such that IlC(x)1 ~.ft(lxIQ) for a bounded and decreasing function .ft(t) on R+ satisfying .ft0 I . IQ E Ue:>oL¹«1+ Ixl)e:dx) where Q =

1₀⁰⁰exp(-tP*)exp(-tP)dt and the norm I . IQ stands for IxlQ =

J(Qx,x),x E Rn. For f E Ll(Rn), define !JJlf(x) =SUPt>o IlCt *

f(x)1 where JCt(x) = t-vJC(Ai/tx). Then we show that !JJl is a bounded operator ofLl(Rn)intoLl,oo(Rn).

1. Introduction

Let P be a real n x n matrix with eigenvalues Ai, Re(Ai) > 0, and let v be the trace ofP and P* denote the adjoint of P. Then we consider the associated dilation group defined byAt = exp(Plogt), t >

O. ThenA; =exp(P*logt). We see that ingenerallA;xlisnot strictly increasing in terms of t. However, one has the following fact ( see [6]

); there exists a positive definite and symmetric real matrix Qso that IA;xIQ = J{QA;x,A;x) is strictly increasing in t, where the norm

I ·IQ is defined by IxlQ = J{Qx,x) for x E JRn; in fact, it turns out that

Q=

1

Received August 13, 1997.

1991 Mathematics Subject Classification: 42B25, 42B15.

Key words and phrases: maximal operator, weak type Ll(Rn)-estimate, quasir- adial majorant, maximal weighted Ll-space, maximal Bochner-Riesz operator.

The author was supported in part by Korea Research Foundation and GARC.

1030 Yong-Cheol Rim

Observe that, since Q is positive definite and symmetric, this norm satisfies the triangle inequality and we have that for x E Rn

(1.1)

where IIQII denotes the operator norm ofQand /-l0 > 0is the smallest eigenvalue of the orthogonal matrix which makes Q diagonalized.

For a notation, we write I E Ll,oo(Rn) ( which is called weak-Ll space) if

11/11£1,00 =;supsl{x E Rnll/(x)1 >s}1 <00.

8>0

For simplicity, we denote the weak-L¹ space by L¹,00. We introduce weighted integrable functions with a weight we(x) = (1+ Ixl)e,c >O.

For c > 0, denote by L¹(we ) the space of all measurable functions I

defined on Rn for which

( I/(x)1^We(x) dx < 00.

J~.n

Then we note that L¹(w_{e )} decreases in terms of inclusion as c >0 in- creases. Even though we take the union on c > 0 of such spacesL¹(we ),

the space U_e>oL1(we ) is still a proper subspace of L1(Rn) because a function

where B(O;1) denotes the unit ball with center 0 E Rn belongs to L1(Rn)but not toU_e>oL1(Rn). Inwhat follows, we denote byLl(w*) = U_e>oL1(w_{e ),} which we call the maximal weighted Ll-space. Our main result is to obtain the weak type L1(Rn)-estimate for certain maximal operator to be defined in the following theorem.

THEOREM 1.1. Suppose that JC be a function defined on Rn such that

1K:(x) I ::; Jt(lxIQ)

Weak type £1(Rn )-estimate for certain maximal operators 1031

wbereJ't(t) is a bounded and decreasing function defined on lR+ and J't⁰ I. IQ E L1(w*). ForI E Ll(lRn ), define

SJR/(x) = supIKt *l(x)l, t>o

wbere Kt(x) = rVK(A1/t x) for t >o. Tben SJRis a bounded operator ofL1(lRn ) into L1,OO(lRn ); tbat is, tbere is a constant G = G(n) sucb tbat for anylELl(lRn ),

IISJR/II£1,oo ::; G11/11£1·

REMARK. We see that the maximal operator associated with isotropic dilation on the kernel which is in L1(lRn ) is dominated by the Hardy- Littlewood maximal operator. However, this is no longer true for anisotropic cases. If1<^{p ::; 00}and the kernel has a quasiradial majo- rant that is bounded, decreasing, and integrable, then it is well-known ( see [4] ) that the maximal operator SJRis bounded on LP(lRn ).

2. Preliminary estimates

Md(x) = supl1lt*l(x)1 t>o

where 1lt (x) = rV1l"(A~/tx) for t > o. Then we need the following elementary lemma in order to prove Lemma 2.2 below.

LEMMA 2.1. IfJ't0I·IQ E Ll(weJ fore> 0, then2:~12k(n+e)Jt(2k-l)

< 00.

Proof It easily follows from (1.1) and simple computation. 0

LEMMA 2.2. IfJ't0 I .IQ E L1(we) for e > 0, then tbere exists a constantG = G(e) >0such tbat SJR/(x) ::; G Mel(x) for anyx E lRn.

1032 Yong-Cheol Kim

Proof For k E Z, set Ak = {y E Rⁿ12k- ¹ ::; /ylQ ::; 2k} and U= {y E RnllylQ ::; 1}. By simple computation, we have that

IK*f(x)1 ::; ~[(Jto ^1'!Q)xAk]*/fl(x)

kEZ

00

~ ^{[(Jto1'!Q)xu]}*If/(x)+~[(Jtol'IQ)xAk *Ifl(x) k=l

00

~ CXU *Ifl(x) +C~Jt(2k-l)xAk *If/(x) k=l

00 . 1

::; CXu*Ifl(x) +C~2k(n+e)Jt(2k-l)2k(n+e)XAk *Ifl(X) k=l

where XE denotes the characteristic function on a set E. Note that there is a constant C > 0 such that for anyk = 1,2,3,'" ,

1 C

2k(n+e)XAk(X) ~ ⁽¹+ IxIQ)n+e

because IxlQ ^rv 2k for x E Ak. Thus it follows from Lemma 2.1 that for any x ERn,

IK*f(x)1 ::; C11£C*f(x)l·

Hence we conclude thatrotf(x) ::; CMef(x) for any x ERn. 0

3. Weak type Ll(~n)-estimate

In this section, we introduce a Vitali family [5] and the result of Stein and Weiss [7] on adding weak type functions.

Let {UsI^S > ^o} be a family of open subsets of Rn whose closure is compact. Then we say that {UsI^S > o} is a Vitali family with constant A > 0, if the followings are satisfied; (i) Us C US' for s ::; Sf

and ns>ous ⁼ {O}, (ii) IUs - Usl ::; AIUsI for all s > 0, and (iii) limk.-+oo IUSkI^{= IUs}Iwhen limk.-+ooSk = s.

LEMMA 3.1. Suppose that {hj } isa sequence of nonnegative nmc- tions on a measure space for which

Weak typeLl(JRn)-estimatefor certain maximal operators 1033

where A > 0is a constant. Let{Oj} be a sequence of positive numbers withE_j OJ = 1. Then we have that

11L ^Ojhj !l£1,oo ::; 2A(N+2)

j

where N =E_j OJ log(ljOj).

LEMMA 3.2. Let g be a function defined on Rn such that (1 + I . IQPg E Loo(lRn) for, > n. For I E L1(lRn), define

MI(x) =supIgt*l(x)1

t>O

where gt(x) = rvg(Ai/tx) for t > O. Then there isa constant C =

C(n) > 0 such that for any lE Ll(lRⁿ),

IIM/II£1,OO ::; CII/II£1'

Prool. Without loss of generality, we may assume that

where (!j(r) = (1 +1r)"Y for r > 0 and, > n. For k E Z, let At =

{y E lRⁿl2^{k- 1 ::;} IAi/tylQ ::; 2^k}, and for k = 1,2,3,·'" letUt = {yE

lRnllAi/tylQ ::;2^k}. From simple computation, we have that

Igt*l(x)1 ::;L(lgtIXA;)*1/1(x)

kEZ

00

::; L(lgtIXu;)*1/1(x)

k=l

00 1

::; L ^tv<!5(2^k- 1

hu; *1/1(x)

k=l

= ~_L 2kn(!j(2k-l)-2kl_ntvXu^k *1/1(x)

t

k=l

::; L

k=l

1034 where

Yong-Cheol Kim

Mkf(x) = sup - 2k1 ^Xuk *Ifl(x).

t>O ntv t

Then it is not hard to show that for eachk,{U;lt >O} is a Vitali family with constant2ⁿ;for, limt--7oo IAi'/t^ylQ = 0 andlimt--7ooIAi'/t^ylQ =00,

A;y is continuous in t and y, U; - ~k C 2~k, and lutI⁼ wn2kntv , where Wn = I{x E lRnllxlQ ::::; 1}1. Thus by the maximal theorem [5]

on a Vitali family, we have that for each k 2:1,

Set (3k = 2kn ®(2k-¹) and set (3 = L:~=l2kn ®(2k-¹) < ^00. IfOk = {3k/{3, thenL:~=l(Xk = 1. Then by Lemma 3.1, we get that

00

11L okMkfll£l,oo ::::; 2n

+l(K+2)lIfll£l,

k=l

where K = L:~=lOklog(1/Ok) < 00. Also we have that

00

supIQt *f(x)1 ::::; (3L okMkf(x).

t>O k=l

Therefore, this implies thatM is bounded from Ll(lRn)toL1,00(lRn).D

Proof of Theorem1.1. Assume thatita\·IQ ^E Ll(w*),and letEO> 0 be the supremum of numbers^E >0 so that itaI·IQE Ll(we} ^Applying Lemma 3.2 to the maximal operator M_eo that we defined in Section 2, it follows that M_eo is a bounded operator ofLl(lRn )intoLl,oo(lRn).

Therefore, by Lemma 2.2, we conclude that 9Jt is a bounded operator

ofLl(lRn)into Ll,oo(lRn). 0

4. Application

Inthis section, we give an application of Theorem 1.1 to the maximal Bochner-Riesz operator. Inorder to do that, we first ofallintroduce quasi-homogeneous distance functions defined onlRn.

Weak type £1(JRn)-estimate for certain maximal operators 1035

We say that a function e ^E coo(Rn\{o}) is an Arhomogeneous distance function defined on Rn, if e(~) > 0 for ~ E Rn\{O}, and e(At~) ⁼ te(~) for t > 0 and ~ ^E Rn. For I E Ll(lRn), let j(~) =

Je-i(x,~)I(x)dxdenote the Fourier transform. We consider quasiradial Bochner-lliesz means of index8> 0defined by

B6 I(x) = _1_ ( ei(x,~)^{(1 -} e(~)/t)6 j(~)^d1;,

e,t (21l")n JJRn ⁺

and the associated maximal operator

9J1~/(x)^{= sup}IB~,d(x)l.

t>O

In what follows, denote by E" = _{~ E Rnle(~) = I} the unit sphere with respect to the distance function e.

COROLLARY 4.1. Suppose that the Gaussian curvature of the unit sphere El! does not vanish. If8 > (n - 1)/2, then there is a constant C =C(n) >0 such that for any lE L¹(Rn),

119J1~/IIL1,oo ~ C 11/11£1.

REMARK. For another proof, the reader can refer to the result of Dappa and Trebels [1]. Ifthe sphere E" satisfies a finite type condition, then we also have sharp weak type (1, I)-estimate for the maximal Bochner-lliesz operator ( see [2] and [3J for related results ).

Proof It easily follows from Theorem 1.1 and the decay estimate of the Bochner-lliesz kernel by using the stationary phase method. 0

In order to apply the Marcinkiewicz interpolation theorem, we use the standard linearization technique of the maximal Bochner-lliesz op- erator. Itthus suffices to consider the analytic family of linear opera- tors given by

z~ ^{_1_ {}(21l")n JJRn ei(x'~)(I-e(~)/t(x))Z j(~)d1;,+ ,

wherex ~t(x) is an arbitrary measurable function defined onRn with values in R+. Combining this with Corollary 4.1, we obtain the LP- estimate for _9J1~ nearp =1. By the complex interpolation method, we interpolate this estimate with the L2-estimate for 9J1~ on 8> O. Then we easily get the following corollary.

1036 Yong-Cheol Kiln

COROLLARY4.2. Suppose that the unit sphere E_ehas nonvanishing Gaussian curvature. If8 >(n -1) (~ ^- ~) ^{and 1 < p :s 2, then there}

is a constant C = C(n, p) >0 such that for anyf E LP(Rn),

References

[1] H. Dappa and W. Trebels, On maximal junctions genemted byFou'T"ier multi- plier,Ark. Mat. 23 (1985), 241-259.

[2] Y.-C. Kim, Quasimdial Bochner-Riesz means for some nonsmooth distance functions, Indiana Univ. Math.J. (to appear).

[3] Y. Kim and A. Seeger, A note on pointwise convergence of quasiradial Riesz means, Acta Sci. Math. (Szeged) 62 (1996), 187-199.

[4] W. Madych, On Littlewood-Paley functions, Studia Math. 50 (1974), 43-63.

[5] N. M. Riviere, Singular integmls amd multiplier opemtors, Arkiv for Mat. 9 (1971), 243-278.

[6] E. M. Stein and S. Wainger, Problems in harmonic analysis related to curva- ture, Bull. Amer. Math. Soc.84 (1978), 1239-1295.

[7] E. M. Stein and N. J. Weiss, On the convergence of Poisson integmls, Trans.

Amer. Math. Soc. 140 (1969), 34-54.

Department of Mathematics Korea University

Seoul 136-701, Korea