Commutators ofLittlewood-PaleyOperatorsonHerz Spaces with Variable Exponent

Hongbin Wangand Yihong Wu

1School of Science，Shandong University of Technology，Zibo，Shandong 255049，China

2Department of Recruitment and Employment，Zibo Normal College，Zibo，Shandong 255130，China



Commutators ofLittlewood-PaleyOperatorsonHerz Spaces with Variable Exponent

Hongbin Wang1，∗and Yihong Wu2

1School of Science，Shandong University of Technology，Zibo，Shandong 255049，China

2Department of Recruitment and Employment，Zibo Normal College，Zibo，Shandong 255130，China

.Let Ω∈L2（Sn-1）be homogeneous function of degree zero and b be BMO functions.In this paper，we obtain some boundedness of the Littlewood-Paley Operators and their higher-order commutators on Herz spaces with variable exponent.

Herz space，variable exponent，commutator，area integral，Littlewood-Paley gλ∗function.

AMS Subject Classifications:42B25，42B35，46E30

1　Introduction

The theory of function spaces with variable exponent has extensively studied by researchers since the work of Kov´aˇcik and R´akosn´ık［7］appeared in 1991.In［9］and［10］，the authors proved the boundedness of some Littlewood-Paley operators on variable Lpspaces，respectively.

Given anopenset E⊂Rn，and ameasurable function p（·）:E-→［1，∞），Lp（·）（E）denotes the set of measurable functions f on E such that for some λ＞0，

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

Define P0（E）to be set of p（·）:E-→（0，∞）such that

Define P（E）to be set of p（·）:E-→［1，∞）such that

Denote p′（x）=p（x）/（p（x）-1）.

where Br（x）=｛y∈Rn:|x-y|＜r｝.Let B（Rn）be the set of p（·）∈P（Rn）such that the Hardy-Littlewood maximal operator M is bounded on Lp（·）（Rn）.In addition，we denote the Lebesgue measure and the characteristic function of a measurable set A⊂Rnby|A| and χArespectively.The notation f≈g means that there exist constants C1，C2＞0 such that C1g≤f≤C2g.

In variable Lpspaces there are some important lemmas as follows.

Lemma 1.1.If p（·）∈P（Rn）and satisfies

and

then p（·）∈B（Rn），that is the Hardy-Littlewood maximal operator M is bounded on Lp（·）（Rn）.

Lemma 1.2（see［7］）.Let p（·）∈P（Rn）.If f∈Lp（·）（Rn）and g∈Lp′（·）（Rn），then fg is integrable on Rnand

where

This inequality is named the generalized H¨older inequality with respect to the variable Lpspaces.

Lemma 1.3（see［5］）.Let q（·）∈B（Rn）.Then there exists a positive constant C such that for all balls B in Rnand all measurable subsets S⊂B，

where δ1，δ2are constants with 0＜δ1，δ2＜1.

Throughout this paper δ1，δ2is the same as in Lemma 1.3.

Lemma 1.4（see［5］）.Suppose q（·）∈B（Rn）.Then there exists a constant C＞0 such that for all balls B in Rn，

Next we recall the definition of the Herz-type spaces with variable exponent.Let Bk=｛x∈Rn:|x|≤2k｝and Ak=BkBk-1for k∈Z.Denote Z+and N as the sets of all positive and non-negative integers，χk=χAkfor k∈Z，˜χk=χkif k∈Z+and˜χ0=χB0.

where

where

Supposethat Sn-1is the unit sphereof Rn（n≥2）equipped with normalized Lebesgue measure.Let Ω∈L1（Rn），be homogeneous function of degree zero and

and

Motivated by［8，9］，we will study the boundedness for the Littlewood-Paley operators and their commutators on the Herz space with variable exponent，where Ω∈L2（Sn-1）.

2　Estimate for the Littlewood-Paley operators

A nonnegative locally integrable function ω on Rnis said to belong to Ap（1＜p＜∞），if there is a constant C＞0 such that

where p′=p/（p-1），Q denotes a cube in Rnwith its sides parallel to the coordinate axes and|Q|denotes the Lebesgue measure of Q.

Lemma2.1（see［3］）.Suppose that Ω∈Ls（Sn-1）（s＞1）satisfying（1.3）.Ifω∈Ap/β，max｛s′，2｝= β＜p＜∞，then there is a constant C，independent of f，such that

Now we give the main theorem in this section.

we have

Now we estimate I1.By the Minkowski inequality we have

Note that z∈Ajand|y-z|＜t.So we know that|y-z|～|y|，then for Ω∈L2（Sn-1）we have

For λ＞2，we take 0＜θ＜（λ-2）n.Since|x-z|≤|x-y|+|y-z|≤|x-y|+t，by（2.4）we have

Similarly，noting that|y-z|～|y|，by（2.4）we have

Note that x∈Ak，z∈Ajand j≤k-2.By（2.5），（2.6）and the generalized H¨older inequality we have

By Lemma 1.3 and Lemma 1.4，we have

Thus we obtain

If 1＜p＜∞，take 1/p+1/p′=1.Since nδ2-α＞0，by the H¨older inequality we have

If 0＜p≤1，then we have

Let us now estimate I3.Note that x∈Ak，y∈Ajand j≥k+2，so we have|y-z|～|y|. By（2.3）-（2.6）and the generalized H¨older inequality we have

By Lemma 1.3 and Lemma 1.4，we have

Thus we obtain

If 1＜p＜∞，take 1/p+1/p′=1.Since nδ1+α＞0，by the H¨older inequality we have

If 0＜p≤1，then we have

Therefore，by（2.1），（2.2），（2.8），（2.9），（2.11）and（2.12），we complete the proof of Theorem 2.1.

3　BMO estimate for the commutators of Littlewood-Paley operators

Let us first recall that the space BMO（Rn）consists of all locally integrable functions f such that

where fQ=|Q|-1RQf（y）dy，the supremum is taken over all cubes Q⊂Rnwith sides parallel to the coordinate axes and|Q|denotes the Lebesgue measure of Q.

Let b∈BMO（Rn）.The weighted（Lp，Lp）boundedness of［b，µΩ］have been proved by Ding，Lu and Yabuta［4］.

Lemma 3.1（see［4］）.Suppose that Ω∈Ls（Sn-1）（s＞1）satisfying（1.3）.For an integer m≥1，if b∈BMO（Rn）and ω∈Ap/β，max｛s′，2｝=β＜p＜∞，then there is a constant C，independent of f，such thatZ

and

By Lemma 3.1 and Lemma 2.2，it is easy to get the（Lp（·）（Rn），Lp（·）（Rn））-boundedness of the commutators［bm，µΩ，S］and［bm，µ∗Ω，λ］.

Next，we will give the corresponding result about the commutator［b，µΩ］on Herztype Hardy spaces with variable exponent.

In the proof of Theorem 3.1，we also need the following lemma.

Lemma 3.2（see［6］）.Let p（·）∈B（Rn），m be a positive integer and B be a ball in Rn.Then we have that for all b∈BMO（Rn）and all j，i∈Z with j＞i，

where Bi=｛x∈Rn:|x|≤2i｝and Bj=｛x∈Rn:|x|≤2j｝.

Then we have

Now we estimate J1.By the Minkowski inequality we have

Note that x∈Ak，z∈Ajand j≤k-2.By（2.5），（2.6）and the generalized H¨older inequality we have

By Lemma 1.3，Lemma 1.4 and Lemma 3.2 we have

Thus we obtain

If 1＜p＜∞，take 1/p+1/p′=1.Since nδ2-α＞0，by the H¨older inequality we have

If 0＜p≤1，then we have

Let us now estimate J3.Note that x∈Ak，y∈Ajand j≥k+2，so we have|y-z|～|y|.

Similar to（3.4），we get

By Lemma 1.3，Lemma 1.4 and Lemma 3.2，we have

Thus we obtain

If 1＜p＜∞，take 1/p+1/p′=1.Since nδ1+α＞0，by the H¨older inequality we have

If 0＜p≤1，then we have

Therefore，by（3.1），（3.2），（3.5），（3.6），（3.8），（3.9），we complete the proof of Theorem 3.1.

Since［bm，µΩ，S］（f）（x）≤Cλ［bm，µ∗Ω，λ］（f）（x），we easily obtain the following theorem.

Acknowledgments

The authors are very grateful to the referees for their valuable comments.

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.Email addresses:wanghb@sdut.edu.cn（H.Wang），wfapple123456@163.com（Y.Wu）

11 December 2015；Accepted（in revised version）11 April 2016