# Higher Order Commutators of Fractional Integral Operator on the Homogeneous Herz Spaces with Variable Exponent

Authors: Liwei Wang; Meng Qu; Lisheng Shu
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

By decomposing functions, we establish estimates for higher order commutators generated by fractional integral with BMO functions or the Lipschitz functions on the homogeneous Herz spaces with variable exponent. These estimates extend some known results in the literatures.


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## Body

## 1. Introduction

Letb be a locally integrable function, 0<β<n, and m∈ℕ; the higher order commutators of fractional integral operator Iβ,bm are defined by
(1)Iβ,bmf(x)=∫ℝn‍[b(x)-b(y)]m|x-y|n-βf(y)dy.
Obviously, Iβ,b0=Iβ and Iβ,b1=[b,Iβ]. The famous Hardy-Littlewood-Sobolev theorem tells us that the fractional integral operator Iβ is a bounded operator from the usual Lebesgue spaces Lp1(ℝn) to Lp2(ℝn) when 0<p1<p2<∞ and 1/p1-1/p2=β/n. Also, many generalized results about Iβ and the commutator [b,Iβ] on some function spaces have been studied; see [1–3] for details.It is well known that the main motivation for studying the spaces with variable exponent arrived in the nonlinear elasticity theory and differential equations with nonstandard growth. Since the fundamental paper [4] by Kováčik and Rákosník appeared in 1991, the Lebesgue spaces with variable exponent Lp(·)(ℝn) have been extensively investigated. In the recent twenty years, boundedness of some important operators, for example, the Calderón-Zygmund operators, fractional integrals, and commutators, on Lp(·)(ℝn) has been obtained; see [5–7]. Recently, Diening [8] extended the (Lp1(ℝn),Lp2(ℝn)) boundedness of Iβ to the Lebesgue spaces with variable exponent. Izuki [7] first introduced the Herz spaces with variable exponent K˙p(·)α,q(ℝn), which is a generalized space of the Herz space K˙pα,q(ℝn); see [9, 10], and in case of b∈BMO(ℝn), he obtained the boundedness properties of the commutator [b,Iβ]. The paper [11] by Lu et al. indicates that the commutator [b,Iβ] with b∈BMO(ℝn) and with b∈Lipα(ℝn)(0<α≤1) has many different properties. In 2012, Zhou [12] studied the boundedness of Iβ on the Herz spaces with variable exponent and proved that the boundedness properties of the commutator [b,Iβ] also hold in case of b∈Lipα(ℝn)(0<α≤1). The higher order commutators Iβ,bm are recently considered by Wang et al. in the paper [13, 14]; they established the BMO and the Lipschitz estimates for Iβ,bm on the Lebesgue spaces with variable exponent Lp(·)(ℝn). Motivated by [7, 12–14], in this note, we establish the boundedness of the higher order commutators Iβ,bm on the Herz spaces with variable exponent.For brevity,|E| denotes the Lebesgue measure for a measurable set E⊂ℝn, and fE denotes the mean value of f on E(fE=(1/|E|)∫E‍f(x)dx). The exponent p′(·) means the conjugate of p(·), that is, 1/p(·)+1/p′(·)=1. C denotes a positive constant, which may have different values even in the same line. Let us first recall some definitions and notations.Definition 1.
For0<γ≤1, the Lipschitz space Lipγ(ℝn) is the space of functions f satisfying
(2)∥f∥Lipγ=supx,y∈ℝn,x≠y|f(x)-f(y)||x-y|γ<∞.Definition 2.
Forf∈Lloc1(ℝn), the bounded mean oscillation space BMO(ℝn) is the space of functions f satisfying
(3)∥f∥BMO=supB1|B|∫B‍|f(x)-fB|dx<∞,
where the supremum is taken over all balls B in ℝn.Definition 3.
Letp(·):E→[1,∞) be a measurable function.(1)
The Lebesgue space with variable exponentLp(·)(E) is defined by
(4)Lp(·)(E)={fismeasurable:∫E‍(|f(x)|λ)p(x)dx<∞forsomeconstantλ>0(|f(x)|λ)p(x)}.(2)
The space with variable exponentLlocp(·)(E) is defined by
(5)Llocp(·)(E)={f:f∈Lp(·)(K)forallcompactsubsetsK⊂E}.
The Lebesgue spaceLp(·)(E) is a Banach space with the Luxemburg norm
(6)∥f∥Lp(·)(E)=inf{λ>0:∫E‍(|f(x)|λ)p(x)dx≤1}.
We denote(7)p-=essinf{p(x):x∈E},p+=esssup{p(x):x∈E},𝒫(E)={p(·):p->1,p+<∞},ℬ(E)={Lp(·)p(·):p(·)∈𝒫(E),MisboundedonLp(·)(E)},
where the Hardy-Littlewood maximal operator M is defined by
(8)Mf(x)=supr>0r-n∫B(x,r)∩E‍|f(y)|dy,
where B(x,r)={y∈ℝn:|x-y|<r}.Proposition 4 (see [15]).
Ifp(·)∈𝒫(E) satisfies
(9)|p(x)-p(y)|≤-Clog(|x-y|),|x-y|≤12,|p(x)-p(y)|≤Clog(e+|x|),|y|≤|x|,
then one has p(·)∈ℬ(E).LetBk={x∈ℝn:|x|⩽2k}, Rk=Bk∖Bk-1, and χk=χRk be the characteristic function of the set Rk for k∈ℤ. For m∈ℕ, we denote χm~=χRm if m≥1, and χ0~=χB0.Definition 5 (see [7]).
Forα∈ℝ,0<q≤∞ and p(·)∈𝒫(ℝn).(1)
The homogeneous Herz spacesK˙p(·)α,q(ℝn) are defined by
(10)K˙p(·)α,q(ℝn)={f∈Llocp(·)(ℝn∖{0}):∥f∥K˙p(·)α,q(ℝn)<∞},
where
(11)∥f∥K˙p(·)α,q(ℝn)=∥{2αk∥fχk∥Lp(·)(ℝn)}k=-∞∞∥ℓq(ℤ).(2)
The nonhomogeneous Herz spacesKp(·)α,q(ℝn) are defined by
(12)Kp(·)α,q(ℝn)={f∈Llocp(·)(ℝn):∥f∥Kp(·)α,q(ℝn)<∞},
where
(13)∥f∥Kp(·)α,q(ℝn)=∥{2αm∥fχm~∥Lp(·)(ℝn)}m=0∞∥ℓq(ℕ).In this note, we obtain the following results.Theorem 6.
Suppose thatb∈Lipβ1(ℝn)(0<β1<1), p2(·)∈𝒫(ℝn) satisfies conditions (9) in Proposition 4. If 0<r<min{1/(p1)+,1/(p2′)+}, 0<β+mβ1<nr, 0<α<nr-β-mβ1, 0<q1≤q2<∞, and 1/p1(x)-1/p2(x)=(β+mβ1)/n, then the higher order commutators Iβ,bm are bounded from K˙p1(·)α,q1(ℝn) to K˙p2(·)α,q2(ℝn).Theorem 7.
Suppose thatb∈BMO(ℝn), p2(·)∈𝒫(ℝn) satisfies conditions (9) in Proposition 4. If 0<r<min{1/(p1)+,1/(p2′)+}, 0<β<nr, 0<α<nr-β, 0<q1≤q2<∞, and 1/p1(x)-1/p2(x)=β/n, then the higher order commutators Iβ,bm are bounded from K˙p1(·)α,q1(ℝn) to K˙p2(·)α,q2(ℝn).Remark A.
The previous main results generalize the(Lp(·)(ℝn),Lq(·)(ℝn)) boundedness of the higher order commutators Iβ,bm in [13] to the case of the Herz spaces with variable exponent. If m=1, our conclusions coincide with the corresponding results in [7, 12]. Moreover, the same boundedness also holds for the nonhomogeneous case.


## 2. Proof of Theorems6 and 7

To prove our main results, we need the following lemmas.Lemma 8 (see [4]).
Letp(·)∈𝒫(ℝn); if f∈Lp(·)(ℝn) and g∈Lp′(·)(ℝn), then
(14)∫ℝn‍|f(x)g(x)|dx≤rp∥f∥Lp(·)(ℝn)∥g∥Lp′(·)(ℝn),
whererp=1+1/p--1/p+.Lemma 9 (see [7]).
Letp(·)∈ℬ(ℝn); then for all balls B in ℝn,
(15)1|B|∥χB∥Lp(·)(ℝn)∥χB∥Lp′(·)(ℝn)≤C.Lemma 10 (see [7]).
Letp2′(·)∈ℬ(ℝn); then for all balls B in ℝn and all measurable subsets S⊂B, one can take a constant 0<r<1/(p2′)+, so that
(16)∥χS∥Lp2′(·)(ℝn)∥χB∥Lp2′(·)(ℝn)≤C(|S||B|)r.Lemma 11 (see [8]).
Suppose thatp1(·)∈𝒫(ℝn) satisfies conditions (9) in Proposition 4, 0<β<n/(p1)+ and 1/p1(x)-1/p2(x)=β/n; then
(17)∥Iβ(f)∥Lp2(·)(ℝn)≤C∥f∥Lp1(·)(ℝn).Lemma 12 (see [13]).
Suppose thatp1(·), p2(·)∈𝒫(ℝn).(1)
Let0<β<n/(p1)+, b∈BMO(ℝn). If p2(·) satisfies conditions (9) in Proposition 4 and 1/p1(x)-1/p2(x)=β/n, then
(18)∥Iβ,bm(f)∥Lp2(·)(ℝn)≤C∥b∥BMOm∥f∥Lp1(·)(ℝn).(2)
Let0<β+mβ1<n/(p1)+, b∈Lipβ1(ℝn)(0<β1<1). If p2(·) satisfies conditions (9) in Proposition 4 and 1/p1(x)-1/p2(x)=(β+mβ1)/n, then
(19)∥Iβ,bm(f)∥Lp2(·)(ℝn)≤C∥b∥Lipβ1m∥f∥Lp1(·)(ℝn).Lemma 13 (see [16]).
Letb∈BMO(ℝn), k>j(k,j∈ℕ); one has(1)
C-1||b||BMOm≤supB⊂ℝn(1/||χB||Lp(·)(ℝn))||(b-bB)mχB||Lp(·)(ℝn)≤C||b||BMOm;(2)
||(b-bBj)mχBk||Lp(·)(ℝn)≤C(k-j)m||b||BMOm  ×||χBk||Lp(·)(ℝn).Proof of Theorem6.
Letf∈K˙p1(·)α,q1(ℝn); we can write
(20)f(x)=∑j=-∞∞‍f(x)χj(x)=∑j=-∞∞‍fj(x).
For0<q1/q2≤1, applying the inequality
(21)(∑i=1∞‍ai)q1/q2≤∑i=1∞‍aiq1/q2(ai>0,i=1,2…),
we obtain
(22)∥Iβ,bm(f)∥K˙p2(·)α,q2(ℝn)q1=C(∑k=-∞∞‍2αq2k∥Iβ,bm(f)χk∥Lp2(·)(ℝn)q2)q1/q2≤C∑k=-∞∞‍2αq1k(∑j=-∞k-2‍∥Iβ,bm(fj)χk∥Lp2(·)(ℝn))q1+C∑k=-∞∞‍2αq1k(∑j=k-1∞‍∥Iβ,bm(fj)χk∥Lp2(·)(ℝn))q1=U1+U2.
We first estimateU1. Noting that if x∈Rk,y∈Rj, and j≤k-2, then |x-y|~|x|~2k, we get
(23)|Iβ,bm(fj)χk|≤∫Rj‍|b(x)-b(y)|m|x-y|n-β|fj(y)|dy·χk(x)≤C2k(β-n)∫Rj‍|b(x)-b(y)|m|fj(y)|dy·χk(x)≤C2k(β+mβ1-n)∥b∥Lipβ1m∫Rj‍|fj(y)|dy·χk(x).
By Hölder’s inequality, Lemmas9 and 10, we have
(24)∥Iβ,bm(fj)χk∥Lp2(·)(ℝn)≤C2k(β+mβ1-n)∥b∥Lipβ1m∥fj∥Lp1(·)(ℝn)≤×∥χBk∥Lp2(·)(ℝn)∥χBj∥Lp1′(·)(ℝn)≤C2k(β+mβ1)∥b∥Lipβ1m∥fj∥Lp1(·)(ℝn)≤×∥χBj∥Lp1′(·)(ℝn)∥χBk∥Lp2′(·)(ℝn)-1≤C2k(β+mβ1)∥b∥Lipβ1m∥fj∥Lp1(·)(ℝn)≤×∥χBj∥Lp1′(·)(ℝn)∥χBj∥Lp2′(·)(ℝn)-1∥χBj∥Lp2′(·)(ℝn)∥χBk∥Lp2′(·)(ℝn)≤C2k(β+mβ1)2nr(j-k)∥b∥Lipβ1m∥fj∥Lp1(·)(ℝn)≤×∥χBj∥Lp1′(·)(ℝn)∥χBj∥Lp2′(·)(ℝn)-1.
Note that(25)Iβ+mβ1(χBj)(x)≥Iβ+mβ1(χBj)(x)·χBj(x)=∫Bj‍dy|x-y|n-β-mβ1·χBj(x)≥C2j(β+mβ1)·χBj(x).
By Lemmas 8 and 11, we obtain
(26)∥χBj∥Lp2′(·)(ℝn)-1≤C2-nj∥χBj∥Lp2(·)(ℝn)≤C2-nj2-j(β+mβ1)∥Iβ+mβ1(χBj)∥Lp2(·)(ℝn)≤C2-j(β+mβ1)2-nj∥χBj∥Lp1(·)(ℝn)≤C2-j(β+mβ1)∥χBj∥Lp1′(·)(ℝn)-1.
Combining (24) and (26), we have the estimate
(27)∥Iβ,bm(fj)χk∥Lp2(·)(ℝn)≤C2(k-j)(β+mβ1-nr)≤×∥b∥Lipβ1m∥fj∥Lp1(·)(ℝn).
Thus,
(28)U1≤C∥b∥Lipβ1mq1×∑k=-∞∞‍(∑j=-∞k-2‍2αj∥fj∥Lp1(·)(ℝn)2(k-j)(β+mβ1-nr+α))q1.
If1<q1<∞, noting that β+mβ1-nr+α<0, by Hölder’s inequality, we have
(29)U1≤C∥b∥Lipβ1mq1×∑k=-∞∞‍(∑j=-∞k-2‍2αjq1∥fj∥Lp1(·)(ℝn)q12(k-j)(β+mβ1-nr+α)q1/2)×(∑j=-∞k-2‍2(k-j)(β+mβ1-nr+α)q1′/2)q1/q1′≤C∥b∥Lipβ1mq1∑j=-∞∞‍2αjq1∥fj∥Lp1(·)(ℝn)q1∑k=j+2∞‍2(k-j)(β+mβ1-nr+α)q1/2≤C∥b∥Lipβ1mq1∥f∥K˙p1(·)α,q1(ℝn)q1.
If0<q1≤1, by inequality (21), we have
(30)U1≤C∥b∥Lipβ1mq1×∑k=-∞∞‍∑j=-∞k-2‍2αjq1∥fj∥Lp1(·)(ℝn)q12(k-j)(β+mβ1-nr+α)q1≤C∥b∥Lipβ1mq1∑j=-∞∞‍2αjq1∥fj∥Lp1(·)(ℝn)q1∑k=j+2∞‍2(k-j)(β+mβ1-nr+α)q1≤C∥b∥Lipβ1mq1∥f∥K˙p1(·)α,q1(ℝn)q1.
Next, we estimateU2. By Lemma 12(2), we obtain
(31)U2≤C∥b∥Lipβ1mq1∑k=-∞∞‍(∑j=k-1∞‍2αj∥fj∥Lp1(·)(ℝn)2α(k-j))q1.
If1<q1<∞, by Hölder’s inequality, we have
(32)U2≤C∥b∥Lipβ1mq1∑k=-∞∞‍(∑j=k-1∞‍2αjq1∥fj∥Lp1(·)(ℝn)q12α(k-j)q1/2)×(∑j=k-1∞‍2α(k-j)q1′/2)q1/q1′≤C∥b∥Lipβ1mq1∑j=-∞∞‍2αjq1∥fj∥Lp1(·)(ℝn)q1∑k=-∞j+1‍2α(k-j)q1/2≤C∥b∥Lipβ1mq1∥f∥K˙p1(·)α,q1(ℝn)q1.
If0<q1≤1, by inequality (21), we have
(33)U2≤C∥b∥Lipβ1mq1∑k=-∞∞‍∑j=k-1∞‍2αjq1∥fj∥Lp1(·)(ℝn)q12α(k-j)q1U2≤C∥b∥Lipβ1mq1∑j=-∞∞‍2αjq1∥fj∥Lp1(·)(ℝn)q1∑k=-∞j+1‍2α(k-j)q1U2≤C∥b∥Lipβ1mq1∥f∥K˙p1(·)α,q1(ℝn)q1.
Combining the estimates forU1 and U2, the proof of Theorem 6 is completed.Proof of Theorem7.
Letf∈K˙p1(·)α,q1(ℝn); we can write
(34)f(x)=∑j=-∞∞‍f(x)χj(x)=∑j=-∞∞‍fj(x).
By inequality (21), we obtain
(35)∥Iβ,bm(f)∥K˙p2(·)α,q2(ℝn)q1=C(∑k=-∞∞‍2αq2k∥Iβ,bm(f)χk∥Lp2(·)(ℝn)q2)q1/q2≤C∑k=-∞∞‍2αq1k(∑j=-∞k-2‍∥Iβ,bm(fj)χk∥Lp2(·)(ℝn))q1≤+C∑k=-∞∞‍2αq1k(∑j=k-1∞‍∥Iβ,bm(fj)χk∥Lp2(·)(ℝn))q1=V1+V2.
ForV1, using Hölder’s inequality and Lemma 8, we have
(36)|Iβ,bm(fj)χk|≤C2k(β-n)∫Rj‍|b(x)-b(y)|m|fj(y)|dy·χk(x)≤C2k(β-n)∑i=0m‍Cmi|b(x)-bBj|m-i≤×∫Rj‍|bBj-b(y)|i|fj(y)|dy≤C2k(β-n)∥fj∥Lp1(·)(ℝn)≤×∑i=0m‍Cmi|b(x)-bBj|m-i∥(bBj-b)iχj∥Lp1′(·)(ℝn).
By Lemmas 9, 10, and 13, we have
(37)∥Iβ,bm(fj)χk∥Lp2(·)(ℝn)≤C2k(β-n)∥fj∥Lp1(·)(ℝn)≤×∑i=0m‍Cmi∥(b(x)-bBj)m-iχk∥Lp2(·)(ℝn)×∥(bBj-b)iχj∥Lp1′(·)(ℝn)≤C2k(β-n)∥fj∥Lp1(·)(ℝn)≤×∑i=0m‍Cmi(k-j)m-i∥b∥BMOm-i∥χBk∥Lp2(·)(ℝn)×∥b∥BMOi∥χBj∥Lp1′(·)(ℝn)=C(k-j+1)m∥b∥BMOm∥fj∥Lp1(·)(ℝn)=×2k(β-n)∥χBk∥Lp2(·)(ℝn)∥χBj∥Lp1′(·)(ℝn)≤C(k-j+1)m∥b∥BMOm∥fj∥Lp1(·)(ℝn)=×2kβ∥χBj∥Lp1′(·)(ℝn)∥χBk∥Lp2′(·)(ℝn)-1≤C(k-j+1)m∥b∥BMOm∥fj∥Lp1(·)(ℝn)=×2kβ∥χBj∥Lp1′(·)(ℝn)∥χBk∥Lp2′(·)(ℝn)-1∥χBj∥Lp2′(·)(ℝn)∥χBk∥Lp2′(·)(ℝn)≤C(k-j+1)m∥b∥BMOm∥fj∥Lp1(·)(ℝn)≤×2kβ2nr(j-k)∥χBj∥Lp1′(·)(ℝn)∥χBj∥Lp2′(·)(ℝn)-1.
Note that(38)Iβ(χBj)(x)≥∫Bj‍dy|x-y|n-β·χBj(x)≥C2jβ·χBj(x).
By Lemmas 8 and 11, we obtain
(39)∥χBj∥Lp2′(·)(ℝn)-1≤C2-nj∥χBj∥Lp2(·)(ℝn)≤C2-nj2-jβ∥Iβ(χBj)∥Lp2(·)(ℝn)≤C2-jβ2-nj∥χBj∥Lp1(·)(ℝn)≤C2-jβ∥χBj∥Lp1′(·)(ℝn)-1.
Combining (37) and (39), we have the estimate
(40)∥Iβ,bm(fj)χk∥Lp2(·)(ℝn)≤C(k-j+1)m≤×∥b∥BMOm2(k-j)(β-nr)∥fj∥Lp1(·)(ℝn).
Thus,
(41)V1≤C∥b∥BMOmq1∑k=-∞∞‍(∑j=-∞k-2‍2αj∥fj∥Lp1(·)(ℝn)C∥b∥BMOm∑k=-∞∞‍×(k-j+1)m2(k-j)(β-nr+α)∑j=-∞k-2‍)q1.
In case of1<q1<∞, noting that β-nr+α<0, by Hölder’s inequality, we have
(42)V1≤C∥b∥BMOmq1≤×∑k=-∞∞‍(∑j=-∞k-2‍2αjq1∥fj∥Lp1(·)(ℝn)q12(k-j)(β-nr+α)q1/2)×(∑j=-∞k-2‍(k-j+1)mq′2(k-j)(β-nr+α)q1′/2)q1/q1′≤C∥b∥BMOmq1∑j=-∞∞‍2αjq1∥fj∥Lp1(·)(ℝn)q1∑k=j+2∞‍2(k-j)(β-nr+α)q1/2≤C∥b∥BMOmq1∥f∥K˙p1(·)α,q1(ℝn)q1.
In case of0<q1≤1, by inequality (21), we have
(43)V1≤C∥b∥BMOmq1≤×∑k=-∞∞‍∑j=-∞k-2‍2αjq1∥fj∥Lp1(·)(ℝn)q1≤×(k-j+1)mq12(k-j)(β-nr+α)q1≤C∥b∥BMOmq1∑j=-∞∞‍2αjq1∥fj∥Lp1(·)(ℝn)q1≤×∑k=j+2∞‍(k-j+1)mq12(k-j)(β-nr+α)q1≤C∥b∥BMOmq1∥f∥K˙p1(·)α,q1(ℝn)q1.
ForV2, by Lemma 12(1), we obtain
(44)V2≤C∥b∥BMOmq1∑k=-∞∞‍(∑j=k-1∞‍2αj∥fj∥Lp1(·)(ℝn)2α(k-j))q1.
If1<q1<∞, by Hölder’s inequality, we have
(45)V2≤C∥b∥BMOmq1∑k=-∞∞‍(∑j=k-1∞‍2αjq1∥fj∥Lp1(·)(ℝn)q12α(k-j)q1/2)×(∑j=k-1∞‍2α(k-j)q1′/2)q1/q1′≤C∥b∥BMOmq1∑j=-∞∞‍2αjq1∥fj∥Lp1(·)(ℝn)q1∑k=-∞j+1‍2α(k-j)q1/2≤C∥b∥BMOmq1∥f∥K˙p1(·)α,q1(ℝn)q1.
If0<q1≤1, by inequality (21), we have
(46)V2≤C∥b∥BMOmq1∑k=-∞∞‍∑j=k-1∞‍2αjq1∥fj∥Lp1(·)(ℝn)q12α(k-j)q1≤C∥b∥BMOmq1∑j=-∞∞‍2αjq1∥fj∥Lp1(·)(ℝn)q1∑k=-∞j+1‍2α(k-j)q1≤C∥b∥BMOmq1∥f∥K˙p1(·)α,q1(ℝn)q1.
Combining the estimates forV1 and V2, consequently, we have proved Theorem 7.



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