# Morrey Meets Herz with Variable Exponent and Applications to Commutators of Homogeneous Fractional Integrals with Rough Kernels

Authors: Hongbin Wang; Jiajia Wang; Zunwei Fu
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We define the new central Morrey space with variable exponent and investigate its relation to the Morrey-Herz spaces with variable exponent. As applications, we obtain the boundedness of the homogeneous fractional integral operatorTΩ,σ and its commutator [b,TΩ,σ] on Morrey-Herz space with variable exponent, where Ω∈Ls(Sn-1) for s≥1 is a homogeneous function of degree zero, 0<σ<n, and b is a BMO function.


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## Body

## 1. Introduction

The Morrey spaces first appeared in 1938 in the work of Morrey [1] in relation to some problems in partial differential equations. In [2] the authors introduced central Morrey spaces. The Herz spaces are a class of function spaces introduced by Herz in the study of absolutely convergent Fourier transforms in 1968; see [3]. The complete theory of Herz spaces for the case of general indexes was established by Lu et al. in 2008; see [4]. Recently, Lu and Xu defined the homogeneous Morrey-Herz spaces in [5].The theory of function spaces with variable exponent was extensively studied by researchers since the work of Kováčik and Rákosník [6] appeared in 1991; see [7, 8] and the references therein. Many applications of these spaces were given, for example, in the modeling of electrorheological fluids [9], in the study of image processing [10], and in differential equations with nonstandard growth [11]. In 2009, Izuki established the Herz spaces with variable exponent and Morrey-Herz spaces with variable exponent; see [12, 13]. In [14], the authors introduced the nonhomogeneous central Morrey spaces of variable exponent.Suppose thatSn-1 denotes the unit sphere in Rn(n≥2) equipped with normalized Lebesgue measure. Let Ω∈Ls(Sn-1) for s≥1 be a homogeneous function of degree zero. For 0<σ<n, the homogeneous fractional integral operator TΩ,σ is defined by (1)TΩ,σfx=∫RnΩx-yx-yn-σfydy.Denote Tσ=TΩ,σ when Ω≡1. In 1971, Muckenhoupt and Wheeden [15] gave the weighted (Lp,Lq) boundedness of TΩ,σ with power weights. Recently, Tan and Liu [16] gave some boundedness of TΩ,σ on function spaces with variable exponent. For the applications of fractional integral operator in fractional order system, we choose to refer to [17].Letb be a locally integrable function; the commutator of homogeneous fractional integral operator [b,TΩ,σ] is defined by (2)b,TΩ,σfx=∫RnΩx-yx-yn-σbx-byfydy.Motivated by the above references, in the present paper we will study the boundedness for the homogeneous fractional integral operatorTΩ,σ and its commutator [b,TΩ,σ] on the Morrey-Herz space with variable exponent.Let us explain the outline of this article. In Section2, we first briefly recall some standard notations and lemmas in variable Lebesgue spaces. In Section 3, we will give the relation between central Morrey spaces, Herz spaces, and Morrey-Herz spaces with variable exponent. We will establish the boundedness for the homogeneous fractional integral operator TΩ,σ and its commutator [b,TΩ,σ] on the Morrey-Herz space with variable exponent in Sections 4 and 5.


## 2. Variable Lebesgue Spaces

Given an open setE⊂Rn and a measurable function p(·):E→[1,∞), Lp(·)(E) denotes the set of measurable functions f on E such that for some λ>0(3)∫Efxλpxdx<∞.This set becomes a Banach function space when equipped with the Luxemburg-Nakano norm (4)fLp·E=inf⁡λ>0:∫Efxλpxdx≤1.These spaces are referred to as variable Lp spaces, since they generalized the standard Lp spaces: if p(x)=p is constant, then Lp(·)(E) is isometrically isomorphic to Lp(E).The spaceLlocp(·)(E) is defined by (5)Llocp·E≔f:f∈Lp·F∀compact  subsetsF⊂E.Define P0(E) to be set of p(·):E→(0,∞) such that (6)p-=essinf⁡px:x∈E>0,p+=esssup⁡px:x∈E<∞.Define P(E) to be set of p(·):E→[1,∞) such that (7)p-=essinf⁡px:x∈E>1,p+=esssup⁡px:x∈E<∞.Denote p′(x)=p(x)/(p(x)-1). In addition, we denote the Lebesgue measure and the characteristic function of a measurable set A⊂Rn by A and χA, respectively. The notation f≈g means that there exist constants C1,C2>0 such that C1g≤f≤C2g.In variableLp spaces there are some important lemmas as follows.Lemma 1 (see [18]).
Ifp(·)∈P(Rn) and satisfies (8)px-py≤C-log⁡x-y,x-y≤12,px-py≤Clog⁡x+e,y≥x,then p(·)∈B(Rn); that is, the Hardy-Littlewood maximal operator M is bounded on Lp(·)(Rn).Lemma 2 (see [6]).
Letp(·)∈P(Rn). If f∈Lp(·)(Rn) and g∈Lp′(·)(Rn), then fg is integrable on Rn and (9)∫Rnfxgxdx≤rpfLp·RngLp′·Rn,where(10)rp=1+1p--1p+.
This inequality is named the generalized Hölder inequality with respect to the variableLp spaces.Lemma 3 (see [19]).
Letp(·)∈B(Rn). Then there exists a positive constant C such that for all balls B in Rn and all measurable subsets S⊂B(11)χBLp·RnχSLp·Rn≤CBS,χSLp·RnχBLp·Rn≤CSBδ1,χSLp′·RnχBLp′·Rn≤CSBδ2,where 0<δ1,δ2<1 depend on p(·).
Throughout this paperδ2 will change with p(·).Lemma 4 (see [19]).
Supposep(·)∈B(Rn). Then there exists a constant C>0 such that for all balls B in Rn(12)1BχBLp·RnχBLp′·Rn≤C.


## 3. The Relation between Central Morrey Spaces, Herz Spaces, and Morrey-Herz Spaces with Variable Exponent

LetBk={x∈Rn:x≤2k} and Ak=Bk∖Bk-1 for k∈Z. Denote Z+ and N as the sets of all positive and nonnegative integers, χk=χAk for k∈Z,χ~k=χk if k∈Z+ and χ~0=χB0.We can give the following definitions of function spaces with variable exponent.Definition 5.
Letq(·)∈P(Rn) and λ∈R. The homogeneous central Morrey space with variable exponent B˙q(·),λ(Rn) is defined by (13)B˙q·,λRn=f∈Llocq·Rn:fB˙q·,λRn<∞,where (14)fB˙q·,λRn=supL∈Z⁡2-Lλ∑k=-∞LfχkLq·Rn.The nonhomogeneous central Morrey space with variable exponent Bq(·),λ(Rn) is defined by (15)Bq·,λRn=f∈Llocq·Rn:fBq·,λRn<∞,where (16)fBq·,λRn=supL∈Z⁡2-Lλ∑k=0Lfχ~kLq·Rn.Definition 6 (see [19]).
Letα∈R, 0<p≤∞, and q(·)∈P(Rn). The homogeneous Herz space with variable exponent K˙q(·)α,p(Rn) is defined by (17)K˙q·α,pRn=f∈Llocq·Rn∖0:fK˙q·α,pRn<∞,where (18)fK˙q·α,pRn=∑k=-∞∞2kαpfχkLq·Rnp1/p.The nonhomogeneous Herz space with variable exponent Kq(·)α,p(Rn) is defined by (19)Kq·α,pRn=f∈Llocq·Rn:fKq·α,pRn<∞,where(20)fKq·α,pRn=∑k=0∞2kαpfχ~kLq·Rnp1/p.Definition 7.
Letα∈R, 0<p≤∞, q(·)∈P(Rn), and 0≤λ<∞. The homogeneous Morrey-Herz space with variable exponent MK˙p,q(·)α,λ(Rn) (see [13]) is defined by (21)MK˙p,q·α,λRn=f∈Llocq·Rn∖0:fMK˙p,q·α,λRn<∞,where (22)fMK˙p,q·α,λRn=supL∈Z⁡2-Lλ∑k=-∞L2kαpfχkLq·Rnp1/p.The nonhomogeneous Morrey-Herz space with variable exponent MKp,q(·)α,λ(Rn) is defined by (23)MKp,q·α,λRn=f∈Llocq·Rn:fMKp,q·α,λRn<∞,where(24)fMKp,q·α,λRn=supL∈Z⁡2-Lλ∑k=0L2kαpfχ~kLq·Rnp1/p.Remark 8.
Ifα=0, p=1, then MK˙1,q(·)0,λ(Rn)=B˙q(·),λ(Rn) and MK1,q(·)0,λ(Rn)=Bq(·),λ(Rn).Remark 9.
Ifλ=0, then MK˙p,q(·)α,0(Rn)=K˙q(·)α,p(Rn) and MKp,q(·)α,0(Rn)=Kq(·)α,p(Rn).Remark 10.
Ifq(·)=q is a constant, then MK˙p,q(·)α,λ(Rn)=MK˙p,qα,λ(Rn) and MKp,q(·)α,λ(Rn)=MKp,qα,λ(Rn).


## 4. Estimate for the Homogeneous Fractional Integral Operator

In this section we will prove the boundedness of the homogeneous fractional integral operatorsTΩ,σ on Morrey-Herz spaces with variable exponent.Theorem 11.
Suppose that0<ν≤1, 0<σ<n-ν, q1(·)∈P(Rn) satisfies conditions (8) in Lemma 1 with q1+<n/σ, 1/q1(x)-1/q2(x)=σ/n, Ω∈Ls(Sn-1)(s>q1′-). Let 0<p1≤p2<∞ and 0<λ<α<nδ2-ν-n/s (or 0<λ<α2≤α1<nδ2-ν-n/s). Then TΩ,σ is bounded from MK˙q1(·)α,p1(Rn) (or MKq1(·)α1,p1(Rn)) to MK˙q2(·)α,p2(Rn) (or MKq2(·)α2,p2(Rn)).To prove Theorem11, we need the following lemmas.Lemma 12 (see [15]).
Ifa>0, 1≤s≤∞, 0<d≤s, and -n+(n-1)d/s<ν<∞, then (25)∫y≤axyνΩx-yddy1/d≤Cxν+n/dΩLsSn-1.Lemma 13 (see [20]).
Define a variable exponentq~(·) by 1/p(x)=1/q~(x)+1/q(x∈Rn). Then we have (26)fgLp·Rn≤CfLq~·RngLqRn,for all measurable functions f and g.Lemma 14 (see [8]).
Letp(·)∈P(Rn) satisfy conditions (8) in Lemma 1. Then (27)χQLp·Rn≈Q1/pxifQ≤2n,x∈Q,Q1/p∞ifQ≥1,for every cube (or ball) Q⊂Rn, where p(∞)=limx→∞p(x).Proof of Theorem11.
We only prove the homogeneous case. Similar to the method of [21], it is easy to prove that MKq2(·)α1,p2(Rn)⊂MKq2(·)α2,p2(Rn) for 0<α2≤α1. So the nonhomogeneous case can be proved in the same way. Let f∈MK˙q1(·)α,p1(Rn). Denote fj=fχj for each j∈Z; then we have f(x)=∑j=-∞∞fj(x). Note that p1≤p2; we have (28)TΩ,σfMK˙q2·α,p2Rnp1=supL∈Z⁡2-Lλp1∑k=-∞L2kαp2TΩ,σfχkLq2·Rnp2p1/p2≤supL∈Z⁡2-Lλp1∑k=-∞L2kαp1TΩ,σfχkLq2·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=-∞k-2TΩ,σfjχkLq2·Rnp1+CsupL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=k-1∞TΩ,σfjχkLq2·Rnp1≕I1+I2.
We first estimateI1. For each k∈Z, j≤k-2, anda.e. x∈Ak, using the generalized Hölder inequality, we have (29)TΩ,σfjx≤C∫BjΩx-yx-yn-σfjydy≤C2kσ-n∫BjΩx-yfjydy≤C2kσ-nΩx-·χj·Lq1′·RnfjLq1·Rn.
Notings>q1′-, we denote q1′~(·)>1 and 1/q1′(x)=1/q1′~(x)+1/s. By Lemmas 12 and 13 we have (30)Ωx-·χj·Lq1′·Rn≤Ωx-·χj·LsRnχj·Lq1′~·Rn≤Ωx-·χj·LsRnχBjLq1′~·Rn≤C2-jν∫AjΩx-ysysνdy1/sχBjLq1′~·Rn≤C2-jν2kν+n/sΩLsSn-1χBjLq1′~·Rn.When Bj≤2n and xj∈Bj, by Lemma 14, we have (31)χBjLq1′~·Rn≈Bj1/q1′~xj≈χBjLq1′·RnBj-1/s.When Bj≥1 we have (32)χBjLq1′~·Rn≈Bj1/q1′~∞≈χBjLq1′·RnBj-1/s.
Since(33)TσχBkx≥∫Bkdyx-yn-σχBkx≥C2kσχBkx,by Lemmas 3 and 4, we have (34)TΩ,σfjχkLq2·Rn≤C2kσ-n2-jν2kν+n/sΩLsSn-1fjLq1·RnχBjLq1′~·RnχBkLq2·Rn≤C2kσ-n2-jν2kν+n/sΩLsSn-1fjLq1·RnχBjLq1′·RnBj-1/sχBkLq2·Rn≤C2-kn2-jν2kν+n/sΩLsSn-1fjLq1·RnχBjLq1′·RnBj-1/sTσχBkLq2·Rn≤C2-kn+k-jν+n/sΩLsSn-1fjLq1·RnχBjLq1′·RnχBkLq1·Rn≤C2k-jν+n/sΩLsSn-1fjLq1·RnχBjLq1′·RnχBkLq1′·Rn≤C2j-knδ2-ν-n/sΩLsSn-1fjLq1·Rn.
So we have(35)I1=CsupL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=-∞k-2TΩ,σfjχkLq2·Rnp1≤CΩLsSn-1p1supL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=-∞k-22j-knδ2-ν-n/sfjLq1·Rnp1.When 1<p1<∞, take 1/p1+1/p1′=1. Since nδ2-ν-n/s-α>0, by the Hölder inequality, we have (36)I1≤CΩLsSn-1p1supL∈Z⁡2-Lλp1∑k=-∞L∑j=-∞k-22jαp12j-knδ2-ν-n/s-αp1/2fjLq1·Rnp1×∑j=-∞k-22j-knδ2-ν-n/s-αp1′/2p1/p1′≤CΩLsSn-1p1supL∈Z⁡2-Lλp1∑k=-∞L∑j=-∞k-22jαp12j-knδ2-ν-n/s-αp1/2fjLq1·Rnp1≤CΩLsSn-1p1supL∈Z⁡2-Lλp1∑j=-∞L-22jαp1fjLq1·Rnp1∑k=j+2L2j-knδ2-ν-n/s-αp1/2≤CΩLsSn-1p1supL∈Z⁡2-Lλp1∑j=-∞L-22jαp1fjLq1·Rnp1≤CΩLsSn-1p1fMK˙q1·α,p1Rnp1.When 0<p1≤1, we have (37)I1≤CΩLsSn-1p1supL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=-∞k-22j-knδ2-ν-n/sp1fjLq1·Rnp1=CΩLsSn-1p1supL∈Z⁡2-Lλp1∑j=-∞L-22jαp1fjLq1·Rnp1∑k=j+2L2j-knδ2-ν-n/s-αp1≤CΩLsSn-1p1supL∈Z⁡2-Lλp1∑j=-∞L-22jαp1fjLq1·Rnp1≤CΩLsSn-1p1fMK˙q1·α,p1Rnp1.
Next we estimateI2; by the (Lq1(·)(Rn),Lq2(·)(Rn))-boundedness of the commutator TΩ,σ we have (38)I2=CsupL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=k-1∞TΩ,σfjχkLq2·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=k-1∞fjLq1·Rnp1=CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=k-1∞2k-jα2jαfjLq1·Rnp1.
If0<p1≤1, then we have (39)I2≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=k-1∞2k-jαp12jαp1fjLq1·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=k-1L-12k-jαp12jαp1fjLq1·Rnp1+CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp12jαp1fjLq1·Rnp1≕I21+I22.For I21, we have (40)I21≤CsupL∈Z⁡2-Lλp1∑j=-∞L-12jαp1fjLq1·Rnp1∑k=-∞j+12k-jαp1≤CsupL∈Z⁡2-Lλp1∑j=-∞L-12jαp1fjLq1·Rnp1≤CfMK˙q1·α,p1Rnp1.For I22, by 0<λ<α we have (41)I22=CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp12jαp1fjLq1·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp12jλp12-jλp1∑m=-∞j2mαp1fmLq1·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp12jλp1fMK˙q1·α,p1Rnp1=CfMK˙q1·α,p1Rnp1supL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=L∞2jλ-αp1≤CfMK˙q1·α,p1Rnp1supL∈Z⁡2-Lλp12Lαp12Lλ-αp1=CfMK˙q1·α,p1Rnp1.
If1<p1<∞, noting λ<α, we can take a constant η>1 so that λ-α/η<0. By the Hölder inequality we have (42)I2≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=k-1∞2k-jαp1/η2jαp1fjLq1·Rnp1∑j=k-1∞2k-jαp1′η-1/ηp1/p1′≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=k-1∞2k-jαp1/η2jαp1fjLq1·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=k-1L-12k-jαp1/η2jαp1fjLq1·Rnp1+CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp1/η2jαp1fjLq1·Rnp1≕I23+I24.For I23, we have (43)I23≤CsupL∈Z⁡2-Lλp1∑j=-∞L-12jαp1fjLq1·Rnp1∑k=-∞j+12k-jαp1/η≤CsupL∈Z⁡2-Lλp1∑j=-∞L-12jαp1fjLq1·Rnp1≤CfMK˙q1·α,p1Rnp1.For I24, by 0<λ<α/η we have (44)I24=CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp1/η2jαp1fjLq1·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp1/η2jλp1fMK˙q1·α,p1Rnp1=CfMK˙q1·α,p1Rnp1supL∈Z⁡2-Lλp1∑k=-∞L2kαp1/η∑j=L∞2jλ-α/ηp1≤CfMK˙q1·α,p1Rnp1supL∈Z⁡2-Lλp12Lαp1/η2Lλ-α/ηp1=CfMK˙q1·α,p1Rnp1.
Thus, by (28), (37), and (39)–(44) we complete the proof of Theorem 11.Remark 15.
In [22], Abdalmonem et al. considered the boundedness of fractional integral with variable kernel on variable exponent Herz-Morrey spaces (see Theorem  A, pp. 790). It is different from Theorem 11. Theorem 11 is under the weaker conditions than Theorem  A. For example, the conditions of Theorem A are Ω∈Lr(Sn-1)(r>p2+) and satisfy ∫01ωr(δ)/δ1+βdδ<∞. But we do not need the condition ∫01ωr(δ)/δ1+βdδ<∞ in Theorem 11.


## 5. BMO Estimate for the Commutator of Homogeneous Fractional Integral Operator

Let us first recall that the spaceBMO(Rn) consists of all locally integrable functions f such that (45)f∗=supQ⁡1Q∫Qfx-fQdx<∞,where fQ=Q-1∫Qf(y)dy, the supremum is taken over all cubes Q⊂Rn with sides parallel to the coordinate axes, and Q denotes the Lebesgue measure of Q.A nonnegative locally integrable functionω(x) on Rn is said to belong to A(p,q)(1<p,q<∞), if there is a constant C>0 such that (46)supQ⁡1Q∫Qωxqdx1/q1Q∫Qωx-p′dx1/p′≤C<∞,where p′=p/(p-1).Letb∈BMO(Rn). The weighted (Lp,Lq) boundedness of [b,TΩ,σ] has been proved by Segovia and Torrea [23], Ding [24], and Ding and Lu [25], respectively.Lemma 16 (see [24]).
Suppose that0<σ<n, s′<p<n/σ, 1/q=1/p-σ/n, Ω∈Ls(Sn-1), and ωs′∈A(p/s′,q/s′), then for b∈BMO(Rn), m∈Z, there is a constant C, independent of f, such that (47)∫Rnb,TΩ,σmfxωxqdx1/q≤C∫Rnfxωxpdx1/p.
We say thatω∈A1 if Mω(x)≤Cω(x) for a.e. x.Lemma 17 (see [26]).
Given a familyF and an open set E⊂Rn, assume that, for some p0 and q0, 0<p0≤q0<∞, and every weight ω∈A1(48)∫Efxq0ωxdx1/q0≤C0∫Egxp0ωxp0/q0dx1/p0,f,g∈F.Given p(·)∈P0(E) such that p0<p-≤p+<p0q0/(q0-p0), define the function q(·) by (49)1px-1qx=1p0-1q0,x∈E.If p(·) satisfies (8) in Lemma 1, then for all (f,g)∈F such that f∈Lq(·)(E)(50)fLq·E≤CgLp·E.Similar to the method of [16, Theorem  1.9], by Lemmas 16 and 17, it is easy to get the (Lp(·)(Rn),Lq(·)(Rn))-boundedness of the commutator [b,TΩ,σ].Next, we will give the corresponding result about the commutator[b,TΩ,σ] on Morrey-Herz spaces with variable exponent.Theorem 18.
Suppose thatb∈BMO(Rn), 0<ν≤1, 0<σ<n-ν, q1(·)∈P(Rn) satisfy conditions (8) in Lemma 1 with q1+<n/σ, 1/q1(x)-1/q2(x)=σ/n, and Ω∈Ls(Sn-1)(s>q1′-). Let 0<p1≤p2<∞ and 0<λ<α<nδ2-ν-n/s (or 0<λ<α2≤α1<nδ2-ν-n/s). Then [b,TΩ,σ] is bounded from MK˙q1(·)α,p1(Rn) (or MKq1(·)α1,p1(Rn)) to MK˙q2(·)α,p2(Rn) (or MKq2(·)α2,p2(Rn)).In the proof of Theorem18, we also need the following lemmas.Lemma 19 (see [27]).
Letp(·)∈B(Rn), k 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(51)1Cb∗k≤supB⁡1χBLp·Rnb-bBkχBLp·Rn≤Cb∗k,b-bBikχBjLp·Rn≤Cj-ikb∗kχBjLp·Rn,where Bi={x∈Rn:x≤2i} and Bj={x∈Rn:x≤2j}.Proof of Theorem18.
We only prove the homogeneous case. The nonhomogeneous case can be proved in the same way. Letf∈MK˙q1(·)α,p1(Rn) and b∈BMO(Rn). Denote fj=fχj for each j∈Z; then we have f(x)=∑j=-∞∞fj(x). Note that p1≤p2; we have (52)b,TΩ,σfMK˙q2·α,p2Rnp1=supL∈Z2-Lλp1∑k=-∞L2kαp2b,TΩ,σfχkLq2·Rnp2p1/p2≤supL∈Z2-Lλp1∑k=-∞L2kαp1b,TΩ,σfχkLq2·Rnp1≤CsupL∈Z2-Lλp1∑k=-∞L2kαp1∑j=-∞k-2b,TΩ,σfjχkLq2·Rnp1+CsupL∈Z2-Lλp1∑k=-∞L2kαp1∑j=k-1∞b,TΩ,σfjχkLq2·Rnp1≕J1+J2.
We first estimateJ1. For each k∈Z, j≤k-2, anda.e.x∈Ak, using the generalized Hölder inequality we have (53)b,TΩ,σfjx≤C∫BjΩx-yx-yn-σbx-byfjydy≤C2kσ-n∫BjΩx-ybx-byfjydy≤C2kσ-nbx-bBj∫BjΩx-yfjydy+∫BjΩx-ybBj-byfjydy≤C2kσ-nbx-bBjΩx-·χj·Lq1′·RnfjLq1·Rn+Ωx-·bBj-b·χj·Lq1′·RnfjLq1·Rn.
Notings>q1′-, we denote q1′~(·)>1 and 1/q1′(x)=1/q1′~(x)+1/s. By Lemmas 12 and 13 we have (54)Ωx-·χj·Lq1′·Rn≤Ωx-·χj·LsRnχj·Lq1′~·Rn≤Ωx-·χj·LsRnχBjLq1′~·Rn≤C2-jν∫AjΩx-ysysνdy1/sχBjLq1′~·Rn≤C2-jν2kν+n/sΩLsSn-1χBjLq1′~·Rn.When Bj≤2n and xj∈Bj, by Lemma 14 we have (55)χBjLq1′~·Rn≈Bj1/q1′~xj≈χBjLq1′·RnBj-1/s.When Bj≥1 we have (56)χBjLq1′~·Rn≈Bj1/q1′~∞≈χBjLq1′·RnBj-1/s.So we obtain χBjLq1′~(·)(Rn)≈χBjLq1′(·)(Rn)Bj-1/s.
So we have(57)Ωx-·χj·Lq1′·Rn≤C2k-jν+n/sΩLsSn-1χBjLq1′·Rn.
Similarly, by Lemma19 we have (58)Ωx-·bBj-b·χj·Lq1′·Rn≤Ωx-·χj·LsRnbBj-b·χj·Lq1′~·Rn≤Cb∗χBjLq1′~·RnΩx-·χj·LsRn≤Cb∗2k-jν+n/sΩLsSn-1χBjLq1′·Rn.
By (57), (58), and Lemmas 3, 4, and 19 we have (59)b,TΩ,σfjχkLq2·Rn≤C2kσ-n2k-jν+n/sΩLsSn-1χBjLq1′·Rnb·-bBjχk·Lq2·RnfjLq1·Rn+b∗2k-jν+n/sΩLsSn-1χBjLq1′·RnχkLq2·RnfjLq1·Rn≤C2kσ-nk-jb∗2k-jν+n/sΩLsSn-1χBjLq1′·RnχBkLq2·RnfjLq1·Rn+b∗2k-jν+n/sΩLsSn-1χBjLq1′·RnχBkLq2·RnfjLq1·Rn≤Ck-jb∗2kσ-n2k-jν+n/sΩLsSn-1χBjLq1′·RnχBkLq2·RnfjLq1·Rn≤Ck-jb∗2-kn2k-jν+n/sΩLsSn-1χBjLq1′·RnTσχBkLq2·RnfjLq1·Rn≤Ck-jb∗2-kn2k-jν+n/sΩLsSn-1fjLq1·RnχBjLq1′·RnχBkLq1·Rn≤Ck-jb∗2k-jν+n/sΩLsSn-1fjLq1·RnχBjLq1′·RnχBkLq1′·Rn≤Cb∗k-j2j-knδ2-ν-n/sΩLsSn-1fjLq1·Rn.
So we have(60)J1=CsupL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=-∞k-2b,TΩ,σfjχkLq2·Rnp1≤Cb∗p1ΩLsSn-1p1supL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=-∞k-2k-j2j-knδ2-ν-n/sfjLq1·Rnp1.When 1<p1<∞, take 1/p1+1/p1′=1. Since nδ2-ν-n/s-α>0, by the Hölder inequality, we have (61)J1≤Cb∗p1ΩLsSn-1p1supL∈Z⁡2-Lλp1∑k=-∞L∑j=-∞k-22jαp12j-knδ2-ν-n/s-αp1/2fjLq1·Rnp1×∑j=-∞k-22j-knδ2-ν-n/s-αp1′/2k-jp1′p1/p1′≤Cb∗p1ΩLsSn-1p1supL∈Z⁡2-Lλp1∑k=-∞L∑j=-∞k-22jαp12j-knδ2-ν-n/s-αp1/2fjLq1·Rnp1≤Cb∗p1ΩLsSn-1p1supL∈Z⁡2-Lλp1∑j=-∞L-22jαp1fjLq1·Rnp1∑k=j+2L2j-knδ2-ν-n/s-αp1/2≤Cb∗p1ΩLsSn-1p1supL∈Z⁡2-Lλp1∑j=-∞L-22jαp1fjLq1·Rnp1≤Cb∗p1ΩLsSn-1p1fMK˙q1·α,p1Rnp1.When 0<p1≤1, we have(62)J1≤Cb∗p1ΩLsSn-1p1supL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=-∞k-22j-knδ2-ν-n/sp1k-jp1fjLq1·Rnp1=Cb∗p1ΩLsSn-1p1supL∈Z⁡2-Lλp1∑j=-∞L-22jαp1fjLq1·Rnp1∑k=j+2L2j-knδ2-ν-n/s-αp1k-jp1≤Cb∗p1ΩLsSn-1p1supL∈Z⁡2-Lλp1∑j=-∞L-22jαp1fjLq1·Rnp1≤Cb∗p1ΩLsSn-1p1fMK˙q1·α,p1Rnp1.
Next we estimateJ2; by the (Lq1(·)(Rn),Lq2(·)(Rn))-boundedness of the commutator [b,TΩ,σ] we have (63)J2=CsupL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=k-1∞b,TΩ,σfjχkLq2·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=k-1∞fjLq1·Rnp1=CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=k-1∞2k-jα2jαfjLq1·Rnp1.If 0<p1≤1, then we have (64)J2≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=k-1∞2k-jαp12jαp1fjLq1·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=k-1L-12k-jαp12jαp1fjLq1·Rnp1+CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp12jαp1fjLq1·Rnp1≕J21+J22.For J21, we have (65)J21≤CsupL∈Z⁡2-Lλp1∑j=-∞L-12jαp1fjLq1·Rnp1∑k=-∞j+12k-jαp1≤CsupL∈Z⁡2-Lλp1∑j=-∞L-12jαp1fjLq1·Rnp1≤CfMK˙q1·α,p1Rnp1.For J22, by 0<λ<α we have (66)J22=CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp12jαp1fjLq1·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp12jλp12-jλp1∑m=-∞j2mαp1fmLq1·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp12jλp1fMK˙q1·α,p1Rnp1=CfMK˙q1·α,p1Rnp1supL∈Z⁡2-Lλp1∑k=-∞L2kαp1∑j=L∞2jλ-αp1≤CfMK˙q1·α,p1Rnp1supL∈Z⁡2-Lλp12Lαp12Lλ-αp1=CfMK˙q1·α,p1Rnp1.
If1<p1<∞, noting λ<α, we can take a constant η>1 so that λ-α/η<0. By the Hölder inequality we have (67)J2≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=k-1∞2k-jαp1/η2jαp1fjLq1·Rnp1∑j=k-1∞2k-jαp1′η-1/ηp1/p1′≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=k-1∞2k-jαp1/η2jαp1fjLq1·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=k-1L-12k-jαp1/η2jαp1fjLq1·Rnp1+CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp1/η2jαp1fjLq1·Rnp1≕J23+J24.For J23, we have (68)J23≤CsupL∈Z⁡2-Lλp1∑j=-∞L-12jαp1fjLq1·Rnp1∑k=-∞j+12k-jαp1/η≤CsupL∈Z⁡2-Lλp1∑j=-∞L-12jαp1fjLq1·Rnp1≤CfMK˙q1·α,p1Rnp1.For J24, by 0<λ<α/η we have (69)J24=CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp1/η2jαp1fjLq1·Rnp1≤CsupL∈Z⁡2-Lλp1∑k=-∞L∑j=L∞2k-jαp1/η2jλp1fMK˙q1·α,p1Rnp1=CfMK˙q1·α,p1Rnp1supL∈Z⁡2-Lλp1∑k=-∞L2kαp1/η∑j=L∞2jλ-α/ηp1≤CfMK˙q1·α,p1Rnp1supL∈Z⁡2-Lλp12Lαp1/η2Lλ-α/ηp1=CfMK˙q1·α,p1Rnp1.
Thus, by (52), (61), (62), and (64)–(69) we complete the proof of Theorem 18.Remark 20.
In [28], the authors pointed out BMO⊂⋂q>1CMOq, so we will consider the corresponding results on CMO space in the following paper.



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