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

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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 ?∈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′?k［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 ?∈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 ?∈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 ?∈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 ?∈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，μ?］have been proved by Ding，Lu and Yabuta［4］.

Lemma 3.1（see［4］）.Suppose that ?∈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，μ?，S］and［bm，μ??，λ］.

Next，we will give the corresponding result about the commutator［b，μ?］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，μ?，S］（f）（x）≤Cλ［bm，μ??，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