Xiao Danand Shu Li-sheng

(1.Department of Basic Courses,Anhui Professional College of Art,Hefei,230001)

(2.School of Mathematics and Statistics,Anhui Normal University,Wuhu,Anhui,241003)

Communicated by Ji You-qing

Abstract:In this paper,under natural regularity assumptions on the exponent function,we prove some boundedness results for the functions of Littlewood-Paley,Lusin and Marcinkiewicz on a new class of generalized Herz-Morrey spaces with weight and variable exponent,which essentially extend some known results.

Key words:Marcinkiewicz integral,variable exponent,Muckenhoupt weight,Herz-Morrey space

1 Introduction

Suppose that Sn?1is the unit sphere in Rn(n ≥ 2)equipped with the normalized Lebesgue measure dσ(x′).Let ? ∈ L1(Sn?1)be homogeneous of degree zero and satisfy

It is well known that by means of the real variables method,Stein[1]first proved that if ? satisfies a Lipγ(0< γ≤ 1)condition on Sn?1,that is,

then μ is of type(p,p)for 1

In recent years,following the fundamental work of Kov′aˇcik and R′akosn′?k[6],function spaces with variable exponent have attracted a great attention in connection with problems of the boundedness of classical operators on those spaces,which in turn were motivated by the treatment of recent problems arising in PDEs and the calculus of variations,see[7]–[17]and the references therein.

The classical theory of Muckenhoupt Apweights is a powerful tool in harmonic analysis,for example in the study of boundary value problems for Laplace’s equation on Lipschitz domains.Recently,the generalized Muckenhoupt Ap(·)weights with variable exponent p(·)have been intensively studied by many authors(see[18]and[19]).In particular,we note that Diening and H¨ast¨o[20]introduced the new class of Ap(·)weights and proved the equivalence between the Ap(·)condition and the boundedness of the Hardy-Littlewood maximal operator M on weighted Lp(·)spaces.Cruz-Uribe and Wang[19]extended the theory of Rubio de Francia extrapolation to the weighted variable Lebesgue spaces Lp(·)(ω).As a consequence they showed that a number of different operators,such as Calder′on-Zygmund singular integral operator,fractional integral operator and elliptic operator in divergence form etc.,are bounded on these spaces.

Motivated by the results mentioned above,the principal problem considered in this paper is to define weighted variable exponent Herz-Morrey spaces and study the boundedness of the functions of Littlewood-Paley,Lusin and Marcinkiewicz on these spaces.We note that our main results(see Theorems 3.1 and 3.2 below)improve and extend the corresponding main theorems in Torchinsky and Wang[2],where the constant exponent case was studied.

In general,by B we denote the ball with center x∈Rnand radius r>0.If E is a subset of Rn,|E|denotes its Lebesgue measure and χEits characteristic function.p′(·)denotes the conjugate exponent defined byWe use x ≈ y if there exist constants c1,c2such that c1x≤y≤c2x.The symbol C stands for a positive constant,which may vary from line to line.

2 Preliminaries and Lemmas

We begin with a brief and necessarily incomplete review of the variable exponent Lebesgue spaces Lp(·)(Rn),see[7]and[8]for more information.

2.1 Lebesgue Spaces with Variable Exponent

Let p(·):Rn→ [1,∞)be a measurable function.The set P(Rn)consists of all variable exponents p(·)satisfying

The variable exponent Lebesgue space Lp(·)(Rn)is the class of all measurable functions f on Rnsuch that

This set becomes a Banach space when it is equipped with the Luxemburg-Nakano norm

For this norm,we have the following property

Given an open set ? ? Rn,the space(?)is defined by

For our main results we need to impose some regularity on the exponent function p(·).The most important condition,one widely used in the study of variable exponent Lebesgue spaces,is so-called log-H¨older continuity.Given a measurable function p(·)with values in[1,∞),we say p(·)∈ LH0(Rn)if there exists a constant C0such that

and p(·)∈ LH∞(Rn)if there exist p∞and C∞>0 such that

The set of p(·)satisfying(2.1)and(2.2)is denoted by LH(Rn).It is easy to check that if p(·)∈ P(Rn)∩LH(Rn),then p′(·)∈ P(Rn)∩LH(Rn).A key consequence of log-H¨older continuity is the fact that if p(·)∈ P(Rn)∩LH(Rn),then the Hardy-Littlewood maximal operator M defined by

is bounded on Lp(·)(Rn)(see[26],also[27]).

2.2 Weighted Function Spaces with Variable Exponent

Let ω be a weight function on Rn,that is, ω is real-valued,non-negative and locally integrable.The weighted variable exponent Lebesgue space Lp(·)(ω)is defined by

The space Lp(·)(ω)is a Banach space with respect to the norm

A weight is said to be a Muckenhoupt A1weight if

For 1

The Muckenhoupt Apclass with constant exponent p∈(1,∞)was recently generalized by Izuki and Noi[21],[22]as follows.

Definition 2.1Let p(·) ∈ P(Rn).A weight is said to be an Ap(·)weight if

Remark 2.1If p(·)≡ p ∈ (1,∞),then we see immediately that the definition reduces to the classical Muckenhoupt Apclass.Cruz-Uribe et al.[18]showed that ω ∈ Ap(·)if and only if the Hardy-Littlewood maximal operator M is bounded on Lp(·)(ω).

Remark 2.2Suppose that p1(·),p2(·) ∈ P(Rn)∩LH(Rn)and p1(·) ≤ p2(·).Then we have A1? Ap1(·)? Ap2(·)(see[22]).

Let Bk={x ∈ Rn:|x| ≤ 2k},Rk=BkBk?1and χk= χRkbe the characteristic function of the set Rkfor k∈Z.

Definition 2.2Suppose that p(·)∈ P(Rn),0

with the usual modi fi cation when q=∞.

Definition 2.3Suppose that 0≤ σ< ∞,p(·)∈ P(Rn),0

with the usual modification when q=∞.

Remark 2.3It obviously follows thatIn the particular case p(·)≡p is constant and ω ≡1,thencoincide with the classical Herz spaces and Herz-Morrey spaces in[28]and[29],respectively.

2.3 Key Lemmas

In order to prove our main results,we need the following lemmas.

Lemma 2.1Let X be a Banach function space on Rn.If f∈ X and g∈ X′,then we have

where X′denotes the associated space of X.

We remark that Lemma 2.1 is the well-known generalized H¨older’s inequality.As a direct application of Lemma 2.1,we get

Lemma 2.2If X is a Banach function space on Rn,then we have

Lemma 2.3[22]Let p(·) ∈ P(Rn)∩LH(Rn)and ωThen there exists a constant 0<δ<1 such that for all k,j∈Z,

Torchinsky and Wang[2]proved the following boundedness of the Marcinkiewicz integral operator μ on the classical weighted Lebesgue spaces.

Lemma 2.4Let 1

The next extrapolation theorem on weighted Lebesgue spaces has recently been proved by Cruz-Uribe and Wang[19](see also Theorem 4 of[22]).

Lemma 2.5Suppose that there exists a constant 1

holds for all f ∈ Lp0(ω0)and all measurable functions g.Let p(·)∈ P(Rn)and ω be a weight.If the Hardy-Littlewood maximal operator M is bounded on Lp(·)(ω)and on Lp′(·)(ω?p(·1)?1),then the inequality

holds for all f ∈ Lp(·)(ω)and all measurable functions g.

Remark 2.4Suppose p(·)∈ P(Rn)∩LH(Rn).Then the following three conditions are equivalent:

(a) ω ∈ Ap(·);

(b)M is bounded on Lp(·)(ω);

(c)M is bounded on Lp′(·)

Combining Lemmas 2.4 and 2.5 above,we have the following boundedness of the Marcinkiewicz integral μ on weighted variable exponent Lebesgue spaces.

Corollary 2.1Let p(·) ∈ P(Rn)∩ LH(Rn)and ω ∈ Ap(·).Then the Marcinkiewicz integral operator μ is bounded on Lp(·)(ω).

Remark 2.5Clearly,Corollary 2.1 is a generalized version of Torchinsky and Wang’s result in[2],and is also a generalization of Lemma 7 in[30],Page 1096.

3 Main Results and Their Proofs

In this section,we prove the boundedness of the functions of Littlewood-Paley,Lusin and Marcinkiewicz on the homogeneous weighted Herz-Morrey spacesWe consider

only 0<σ<∞,the arguments are similar in the case σ=0.

Our main results can be stated as follows.

Theorem 3.1Suppose that ? ∈ Lipγ(Sn?1)(0< γ ≤ 1)for n>2 is homogeneous of degree zero and satisfies(1.1).Let 0≤ σ< ∞,p(·)∈ P(Rn)∩LH(Rn),0

Next we consider the Marcinkiewicz integralcorresponding to the Littlewood-Paleyfunction,which is defined by

It is well known that if λ >2 and ω ∈ Apfor 1

Theorem 3.2Suppose that λ >2 and ? ∈ Lipγ(Sn?1)(0< γ ≤ 1)for n>2 is homogeneous of degree zero and satisfies(1.1).Let 0≤ σ< ∞,p(·)∈P(Rn)∩LH(Rn),and ?nδ+σ < α

Remark 3.1The above result also holds for the Lusin’s area integral μSdefined by

Remark 3.2We would like to point out that even in the special case σ=0,namely,in the framework of weighted variable exponent Herz spaces(ω),Theorems 3.1 and 3.2 are completely new.

Proof of Theorem 3.1Let f∈We decompose

For I1,it is clear that if x∈Rk,y∈Rjand j≤k?2,then

Then by Minkowski’s inequality,we obtain that

From(3.1),Lemmas 2.1 and 2.2,we get

which combined with Lemma 2.3 yields

On the other hand,we see that

Hence,in view of n(1?r)+σ?α>0,combining(3.3)and(3.4),we have

For I2,according to Corollary 2.1,we arrive at the estimate

We proceed now to estimate I3.Similarly to(3.1),we deduce that

As argued before,by applying Lemmas 2.1 and 2.2,we obtain

An application of Lemma 2.3 gives

From(3.4)and(3.7),in view of α +nδ? σ >0,we conclude that

Consequently,the proof of Theorem 3.1 is completed.

Proof of Theorem 3.2As in the proof of Theorem 3.1,we have

It follows from Minkowski’s inequality that

For λ >2,we can take 0< β <(λ?2)n.Noticing that

Straightforward calculations show that

For J2,using the boundedness ofon Lp(·)(ω),we derive the estimate

For J1,note that if x∈Rk,z∈Rjand j≤k?2,then

From(3.8)–(3.10),arguing as in(3.2),we get

For J3,note that if x∈Rk,z∈Rjand j≥k+2,then

Similarly to(3.11)and(3.6),we arrive at

Then by the same arguments as in the estimation of I1and I3,one can immediately complete the proof of Theorem 3.2.For brevity,we omit the details.