WANG Hongbin
School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, Shandong, China; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received 16 August 2018; Revised 26 December 2018
Foundation items: Supported by Shandong Provincial Natural Science Foundation(ZR2017MA041), Shandong Province Higher Educational Science and Technology Program (J18KA225), China Postdoctoral Science Foundation (2016M601105) and National Natural Science Foundation of China(11926343, 11926342, 11761026)
Corresponding author: WANG Hongbin, E-mail: wanghb@sdut.edu.cn

Abstract: In this paper, we study the continuity of the commutators[b, μ_Ω] generated by Marcinkiewicz integral operators with rough kernels μ_Ω and Lipschitz functions b on the Herz-type space with variable exponent, where Ω∈L^S(S^n-1) for s ≥ 1.
Keywords: Herz-type spacevariable exponentMarcinkiewicz integralrough kernelcontinuity
Marcinkiewicz积分交换子在变指标Herz型空间上的连续性
王洪彬
山东理工大学数学与统计学院, 山东 淄博 255049;中国科学院大学数学科学学院, 北京 100049
摘要: 研究由带粗糙核的Marcinkiewicz积分算子μ_Ω和Lipschitz函数b生成的交换子[b, μ_Ω]在变指标Herz型空间上的连续性，其中Ω∈L^s（S^n-1）且s ≥ 1.
关键词: Herz型空间变指标Marcinkiewicz积分粗糙核连续性
Function spaces with variable exponent are being concerned with strong interest not only in harmonic analysis but also in applied mathematics. In the past 27 years, the theory of function spaces with variable exponent has made great progress since some elementary properties were given by Ková?ik and Rákosník^[1] in 1991. In Refs. [2-6], the authors proved the boundedness of some integral operators on variable L^p spaces, respectively. Lebesgue and Sobolev spaces with integrability exponent have been widely studied (see Refs. [3, 5] and the references therein). Many applications of these spaces were given, for example, in the modeling of electrorheological fluids, in the study of image processing, and in differential equations with nonstandard growth.
On the other hand, a class of function spaces called Herz-type spaces on ?ⁿ has attracted considerable attention in recent years because the interesting norm includes explicitly both local and global informations of the function. In 2011, Izuki^[7] obtained Herz spaces with variable exponent. In 2012, Wang and Liu^[8] introduced a certain Herz-type Hardy spaces with variable exponent. Subsequently, Wang^[9-12] proved the continuity of some operators on Herz-type Hardy spaces with variable exponent.
Suppose that S^n-1 denotes the unit sphere in ?ⁿ(n≥2) equipped with normalized Lebesgue measure. Let Ω∈Lip_β(S^n-1) for 0 < β≤1 be homogeneous function of degree zero and

$\int_{{S^{n - 1}}} \Omega \left( {{x^\prime }} \right){\rm{d}}\sigma \left( {{x^\prime }} \right) = 0,$

where x′=x/|x| for any x≠0. In 1958, Stein^[13] introduced the Marcinkiewicz integral related to the Littlewood-Paley g function on ?ⁿ as

${\mu _\Omega }(f)(x) = {\left( {\int_0^\infty {{{\left| {{F_{\Omega ,t}}(f)(x)} \right|}^2}} \frac{{{\rm{d}}t}}{{{t^3}}}} \right)^{1/2}},$

where

${F_{\Omega ,t}}(f)(x) = \int_{\left| {x - y} \right| \le t} {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^{n - 1}}}}f(y){\rm{d}}y} .$

It is shown that μ_Ω is of weak type (1, 1) and of type (p, p) for 1 < p≤2.
For 0 < γ≤1, the Lipschitz space, Lipγ(?ⁿ), is defined as

$\begin{array}{l} \text{Lip}{_\gamma }\left( {{\mathbb{R}^n}} \right) = \hfill \\ \left\{ {f:{{\left\| f \right\|}_{{\text{Li}}{{\text{p}}_\gamma }}} = \mathop {\sup }\limits_{x,y \in {\mathbb{R}^n};x \ne y} \frac{{\left| {f(x) - f(y)} \right|}}{{{{\left| {x - y} \right|}^\gamma }}} < \infty } \right\}. \hfill \\ \end{array}$

Let b∈Lip_γ(?ⁿ). The commutator generated by the Marcinkiewicz integral μ_Ω and b is defined by

$\begin{array}{*{20}{c}}{\left[ {b,{\mu _\Omega }} \right](f)(x) = \left( {\int_0^\infty | \int_{|x - y| \le t} {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^{n - 1}}}}} } \right.}\\{{{\left. {{{\left. {[b(x) - b(y)]f(y){\rm{d}}y} \right|}^2}\frac{{{\rm{d}}t}}{{{t^3}}}} \right)}^{1/2}}.}\end{array}$

Motivated by Refs. [14-15], we will investigate the continuity of the commutators [b, μ_Ω] generated by Marcinkiewicz integral operators with rough kernels μ_Ω and Lipschitz functions b on the Herz-type spaces with variable exponent, where Ω∈L^s(S^n-1) for s≥1.
Throughout this paper, we denote the Lebesgue measure and the characteristic function of a measurable set A??ⁿ by A and χ_A, respectively. S(?ⁿ) denotes the space of Schwartz functions, and S′ (?ⁿ)denotes the dual space of S(?ⁿ). The notation f≈g means that there exist constants C₁, C₂>0 such that C₁g≤f≤C₂g. Let B_k={x∈?ⁿ:|x|≤2^k} and A_k=B_k\B_k-1 for k∈${\mathbb{Z}}$. ${\mathbb{Z}}_ +$ and ${\mathbb{N}}$ denote the sets of all positive and non-negative integers, respectively. χ_k=χ_Ak for k∈${\mathbb{Z}}$, ${{\tilde \chi }_k} = {\chi _k}$ if k∈${\mathbb{Z}}$₊, and ${{\tilde \chi }_0} = {\chi _{{B_0}}}$. In addition, δ₂ denotes the same as in Lemma 1.3.
1 PreliminariesFirstly we give some notations and basic definitions on variable Lebesgue spaces.
Given an open set E??ⁿ and a measurable function p(·):E→[1, ∞). p′(·) is the conjugate exponent defined by p′(·)=p(·)/(p(·)-1).
The set P(E) consists of all p(·):E→[1, ∞) satisfying

${p^ - } = \text{essinf} \{ p(x):x \in E\} > 1,$

${p^ + } = \text{ esssup }\{ p(x):x \in E\} < \infty .$

By L^p(·)(E) we denote the space of all measurable functions f on E such that for some λ>0,

$\int_E {{{\left( {\frac{{\left| {f(x)} \right|}}{\lambda }} \right)}^{p(x)}}} {\text{d}}x < \infty .$

This is a Banach function space with respect to the Luxemburg-Nakano norm

${\left\| f \right\|_{{L^{p( \cdot )}}(E)}} = \inf \left\{ {\lambda > 0:\int_E {{{\left( {\frac{{\left| {f(x)} \right|}}{\lambda }} \right)}^{p(x)}}} {\text{d}}x \leqslant 1} \right\}.$

The space L_loc^p(·)(Ω) is defined by L_loc^p(·)(Ω):={f:f∈L^p(·)(E) for all compact subsets E?Ω}.
The set B(?ⁿ) consists of p(·)∈P(?ⁿ) satisfying the condition that the Hardy-Littlewood maximal operator M is bounded on L^p(·)(?ⁿ).
In variable L^p spaces there are some important lemmas^{[1, 3, 7]}, which are given as follows.
Lemma 1.1??If p(·)∈P(?ⁿ) and satisfies

$\left| {p(x) - p(y)} \right| \leqslant \frac{C}{{ - \log (|x - y|)}},$

$\left| {x - y} \right| \leqslant 1/2,$

(1)

and

$\left| {p\left( x \right) - p\left( y \right)} \right| \leqslant \frac{C}{{\log (|x| + e)}},\left| y \right| \geqslant \left| x \right|,$

(2)

then p(·)∈B(?ⁿ), that is, the Hardy-Littlewood maximal operator M being bounded on L^p(·)(?ⁿ).
Lemma 1.2??Suppose q(·)∈B(?ⁿ). Then there exists a constant C>0 such that for all balls B in ?ⁿ,

$\frac{1}{{\left| B \right|}}{\left\| {{\chi _B}} \right\|_{{L^{q( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}{\left\| {{\chi _B}} \right\|_{{L^{q'( \cdot )}}\left( {{\mathbb{R}^n}} \right)}} \leqslant C.$

Lemma 1.3??Let q(·)∈B(?ⁿ). Then there exists a positive constant C such that for all balls B in ?ⁿ and all measurable subsets S?B,

$\frac{{{{\left\| {{\chi _S}} \right\|}_{{L^{q( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}}}{{{{\left\| {{\chi _B}} \right\|}_{{L^{q( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}}} \leqslant C{\left( {\frac{{\left| S \right|}}{{\left| B \right|}}} \right)^{{\delta _1}}}$

and

$\frac{{{{\left\| {{\chi _S}} \right\|}_{{L^{q'( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}}}{{{{\left\| {{\chi _B}} \right\|}_{{L^{q'( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}}} \leqslant C{\left( {\frac{{\left| S \right|}}{{\left| B \right|}}} \right)^{{\delta _2}}},$

where δ₁ and δ₂ are constants with 0 < δ₁, δ₂ < 1.
Lemma 1.4??(Generalized H?lder inequality)??Let p(·)∈Ρ(?ⁿ). If f∈L^p(·)(?ⁿ)and g∈L^p′(·)(?ⁿ), then fg is integrable on ?ⁿ and

$\int_{{\mathbb{R}^n}} {\left| {f(x)g(x)} \right|{\text{d}}x} \leqslant {r_p}{\left\| f \right\|_{{L^{p( \cdot )}}\left( {{\mathbb{R}^n}} \right)}}{\left\| g \right\|_{{L^{p'( \cdot )}}\left( {{\mathbb{R}^n}} \right)}},$

where r_p=1+1/p^--1/p⁺.
Next we recall some definitions and one lemma for the Herz-type spaces with variable exponent given in Refs. [7-8].
Definition 1.1??Let α∈?, 0 < p≤∞, and q(·)∈P(?ⁿ). The homogeneous Herz space $\dot K_{q\left( \cdot \right)}^{\alpha , p}$(?ⁿ) consists of all f∈L_loc^q(·)(?ⁿ\{0}) such that

${\left\| f \right\|_{\dot K_{q( \cdot )}^{\alpha ,p}\left( {{\mathbb{R}^n}} \right)}} = {\left\{ {\sum\limits_{k = - \infty }^\infty {{2^{k\alpha p}}} \left\| {f{\chi _k}} \right\|_{{L^{q( \cdot )}}\left( {{\mathbb{R}^n}} \right)}^p} \right\}^{1/p}} < \infty .$

The non-homogeneous Herz space K_q(·)^{α, p}(?ⁿ) is defined as the set of all f∈L_loc^q(·)(?ⁿ) such that

${\left\| f \right\|_{K_{q( \cdot )}^{\alpha ,p}\left( {{\mathbb{R}^n}} \right)}} = {\left\{ {\sum\limits_{k = 0}^\infty {{2^{k\alpha p}}} \left\| {f{{\tilde \chi }_k}} \right\|_{{L^{q( \cdot )}}\left( {{\mathbb{R}^n}} \right)}^p} \right\}^{1/p}} < \infty .$

Let G_N(f)(x) be the grand maximal function of f(x) defined by

${G_N}(f)(x) = \mathop {\sup }\limits_{\phi \in {A_N}} \left| {\phi _\nabla ^*(f)(x)} \right|,$

where A_N={?∈S(?ⁿ):$\mathop {\sup }\limits_{\left| \alpha \right|, \left| \beta \right| \le N} \left| {{x^\alpha }{D^\beta }\phi \left( x \right)} \right|$ and $\phi _\nabla ^*(f)(x) = \mathop {{\rm{sup}}}\limits_{|y - x| < t} \left| {{\phi _t}*f(y)} \right|$ with ${\phi _t}\left( x \right) = {t^{ - n}}\phi \left( {x/t} \right)$.
Definition 1.2??Let α∈?, 0 < p < ∞, q(·)∈P(?ⁿ) and N>n+1.
(ⅰ) The homogeneous Herz-type Hardy space $H\dot K_{q\left( \cdot \right)}^{\alpha , p}$(?ⁿ) consists of all f∈S′(?ⁿ) such that G_N(f)(x)∈$\dot K_{q\left( \cdot \right)}^{\alpha , p}$(?ⁿ) and ${{\left\| f \right\|}_{H\dot{K}_{q\left( \cdot \right)}^{\alpha , p}\left( {{\mathbb{R}}^{n}} \right)}}$=${{\left\| {{G}_{N}}\left( f \right) \right\|}_{\dot{K}_{q\left( \cdot \right)}^{\alpha , p}\left( {{\mathbb{R}}^{n}} \right)}}$.
(ⅱ) The non-homogeneous Herz-type Hardy space HK_q(·)^{α, p}(?ⁿ) is defined as the set of all f∈S′(?ⁿ) such that G_N(f)(x)∈K_q(·)^{α, p}(?ⁿ) and ‖f‖_{HKq(·)^{α, p}}(?ⁿ)=‖G_N(f)‖_{Kq(·)^{α, p}}(?ⁿ).
For x∈? we denote by [x] the largest integer less than or equal to x.
Definition 1.3??Let nδ₂≤α < ∞, q(·)∈P(?ⁿ), b∈L_loc¹(?ⁿ), and integer s≥[α-nδ₂].
(ⅰ) A function a(x) on ?ⁿ is said to be a central (α, q(·))-atom, if it satisfies
1) supp a?B(0, r)={x∈?ⁿ:|x| < r},
2) ‖a‖_{L^q(·)(?ⁿ)}≤|B(0, r)|^-α/n,
3) ∫_?ⁿa(x)x^βdx=∫_?ⁿa(x)b(x)x^βdx=0, |β|≤s.
(ⅱ) A function a(x) on ?ⁿ is said to be a central (α, q(·))-atom of the restricted type, if it satisfies the above conditions 2) and 3) and

${\left. 1 \right)^\prime }{\text{supp}}a \subset B\left( {0,r} \right),r \geqslant 1.$

If r=2^k for some k∈${\mathbb{Z}}$ in Definition 1.3, then the corresponding central (α, q(·))-atom is called a dyadic central (α, q(·))-atom.
Lemma 1.5??Let nδ₂≤α < ∞, 0 < p < ∞ and q (·)∈B(?ⁿ).Then ${{\left\| f \right\|}_{H\dot{K}_{q\left( \cdot \right)}^{\alpha , p}}}\left( {{\mathbb{R}}^{n}} \right)$ (or HK_q(·)^{α, p}(?ⁿ)) if and only if

$f = \sum\limits_{k = - \infty }^\infty {{\lambda _k}} {a_k}\left( {{\text{or}}\sum\limits_{k = 0}^\infty {{\lambda _k}} {a_k}} \right),$

in the sense of S′(?ⁿ), where each a_k is a central (α, q(·))-atom(or central (α, q(·))-atom of restricted type) with support contained in B_k and $\sum\limits_{k=-\infty }^{\infty }{{{\left| {{\lambda }_{k}} \right|}^{p}}}<\infty \left( \text{ or }\sum\limits_{k=0}^{\infty }{{{\left| {{\lambda }_{k}} \right|}^{p}}}<\infty \right)$. Moreover,

${\left\| f \right\|_{H\dot K_{q\left( \cdot \right)}^{\alpha ,p}\left( {{\mathbb{R}^n}} \right)}} \approx \inf {\left( {\sum\limits_{k = - \infty }^\infty {{{\left| {{\lambda _k}} \right|}^p}} } \right)^{1/p}}$

$\left( {{\text{or}}{{\left\| f \right\|}_{HK_{q\left( \cdot \right)}^{\alpha ,p}\left( {{\mathbb{R}^n}} \right)}} \approx \inf {{\left( {\sum\limits_{k = 0}^\infty {{{\left| {{\lambda _k}} \right|}^p}} } \right)}^{1/p}}} \right),$

where the infimum is taken over all above decompositions of f.
2 Main results and their proofsLet b∈Lip_γ(?ⁿ). It is easy to know that |[b, μ_Ω]|≤C‖b‖_LipγT_{Ω, γ}, where

${{\bar T}_{\Omega ,\gamma }}f(x) = \int_{{\mathbb{R}^n}} {\frac{{\left| {\Omega (x - y)} \right|}}{{{{\left| {x - y} \right|}^{n - \gamma }}}}} \left| {f(y)} \right|{\text{d}}y.$

In Ref. [16], the authors proved that T_{Ω, γ} is bounded from L^q₁(·)(?ⁿ) to L^q₂(·)(?ⁿ) for 1/q₁(x)-1/q₂(x)=γ/n and q₁(·)∈P(?ⁿ) satisfying conditions(1) and (2) in Lemma 1.1 with q₁⁺ < n/γ. So we can get the following theorem.
Theorem 2.1??Suppose that b∈Lip_γ(?ⁿ) with 0 < γ≤1. If q₁(·)∈P(?ⁿ) satisfies conditions (1) and (2) in Lemma 1.1 with q₁⁺ < n/γ, 1/q₁(x)-1/q₂(x)=γ/n, Ω∈L^s(S^n-1)(s>q₂⁺) with 1≤s′ < q₁^-. Then [b, μ_Ω] is bounded from L^q₁(·)(?ⁿ) to L^q₂(·)(?ⁿ).
Next, we will give the continuity about the commutator [b, μ_Ω] on Herz-type Hardy spaces with variable exponent. Before stating our result, let us recall the definition of the L^s-Dini condition. We say that Ω satisfies the L^s-Dini condition if Ω(x′)∈L^s(S^n-1) with s≥1 is homogeneous of degree zero on ?ⁿ, and

$\int_0^1 {\frac{{{\omega _s}(\delta )}}{\delta }} {\text{d}}\delta < \infty ,$

where ω_s(δ) denotes the integral modulus of continuity of order s of Ω defined by

${\omega _s}(\delta ) = \mathop {\sup }\limits_{\left\| \rho \right\| \le \delta } {\left( {\int_{{S^{n - 1}}} {{{\left| {\Omega \left( {\rho x'} \right) - \Omega \left( {x'} \right)} \right|}^s}} {\rm{d}}\sigma \left( {x'} \right)} \right)^{1/s}}$

and ρ is a rotation on S^n-1 and $\left\| \rho \right\| = \mathop {\sup }\limits_{x' \in {S^{n - 1}}} \left| {\rho x' - x'} \right|$.
Theorem 2.2 Suppose that b∈Lip_γ(?ⁿ) with 0 < γ≤1. If q₁(·)∈P(?ⁿ) satisfies conditions (1) and (2) in Lemma 1.1 with q₁⁺ < n/γ, 1/q₁(x)-1/q₂(x)=γ/n, Ω∈L^s(S^n-1)(s>q₂⁺) with 1≤s′ < q₁^- and satisfies

$\int_0^1 {\frac{{{\omega _s}(\delta )}}{{{\delta ^{1 + \gamma }}}}} {\text{d}}\delta < \infty .$

Let 0 < p1≤p2 < ∞ and nδ2≤α < nδ2+γ(or 0 < max(nδ2, α2)≤α1 < nδ2+γ). Then [b, μΩ] maps $H\dot{K}_{{{q}_{1}}\left( \cdot \right)}^{\alpha , {{p}_{1}}}\left( {{\mathbb{R}}^{n}} \right)$(or HKα1, p1q1(·)(?n)) continuously into $\dot{K}_{{{q}_{2}}\left( \cdot \right)}^{\alpha , {{p}_{2}}}\left( {{\mathbb{R}}^{n}} \right)$(or ${K}_{{{q}_{2}}\left( \cdot \right)}^{{\alpha}_{2} , {{p}_{2}}}\left( {{\mathbb{R}}^{n}} \right)$)).
In the proof of Theorem 2.2, we need the following lemmas in Refs. [3, 5, 17-18].
Lemma 2.1??Given E and p(·)∈P(E), let f:E×E→? be a measurable function (with respect to product measure) such that for almost every y∈E, f(·, y)∈Lp(·)(E). Then

${\left\| {\int_E f ( \cdot ,y){\text{d}}y} \right\|_{{L^{p( \cdot )}}(E)}} \leqslant C\int_E {{{\left\| {f\left( { \cdot ,y} \right)} \right\|}_{{L^p}( \cdot )(E)}}{\text{d}}y} .$

Lemma 2.2??Define a variable exponent $\tilde{q}(\cdot )$ by $\frac{1}{p(x)}=\frac{1}{\tilde{q}(x)}+\frac{1}{q}$(x∈?ⁿ). Then we have

${\left\| {fg} \right\|_{{L^p}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \leqslant C{\left\| f \right\|_{{L^{\tilde q( \cdot )}}}}_{\left( {{\mathbb{R}^n}} \right)}{\left\| g \right\|_{{L^q}\left( {{\mathbb{R}^n}} \right)}}$

for all measurable functions f and g.
Lemma 2.3??Let p(·)∈P(?ⁿ) satisfies conditions (1) and (2) in Lemma 1.1. Then

${\left\| {{\chi _Q}} \right\|_{{L^{p( \cdot )}}\left( {{\mathbb{R}^n}} \right)}} \approx \left\{ {\begin{array}{*{20}{c}} {{{\left| Q \right|}^{\frac{1}{{p\left( x \right)}}}}}&{{\text{if}}\left| Q \right| \leqslant {2^n}\;{\text{and}}\;x \in Q,} \\ {{{\left| Q \right|}^{\frac{1}{{p\left( \infty \right)}}}}}&{{\text{if}}\left| Q \right| \geqslant 1} \end{array}} \right.$

for every cube (or ball) Q??ⁿ, where p(∞)=$\underset{x\to \infty }{\mathop{\lim }}\, p\left( x \right)$.
Lemma 2.4??Suppose that Ω satisfies the L^s-Dini condition (1≤s < ∞). Then for any R>0 and x∈?ⁿ, when |y| < R/2, there is a constant C>0 such that

$\begin{array}{*{20}{c}} {{{\left( {\int_{R < \left| x \right| < 2R} {{{\left| {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^n}}} - \frac{{\Omega (x)}}{{{{\left| x \right|}^n}}}} \right|}^s}} {\text{d}}x} \right)}^{1/s}} \leqslant } \\ {C{R^{\frac{n}{s} - n}}\left\{ {\frac{{\left| y \right|}}{R} + \int_{\left| y \right|/2R < \delta < \left| y \right|/R} {\frac{{{\omega _s}(\delta )}}{\delta }{\text{d}}\delta } } \right\}.} \end{array}$

Proof of Theorem 2.2??We only prove homogeneous case. In Ref. [19], the authors proved K_q(·)^{α₁, p₂}(?ⁿ)?K_q(·)^{α₂, p₂}(?ⁿ) for 0 < α₂≤α₁. So the non-homogeneous case can be proved in the same way. Let $f\in H\dot{K}_{{{q}_{1}}\left( \cdot \right)}^{\alpha , {{p}_{1}}}\left( {{\mathbb{R}}^{n}} \right)$ and b∈Lip_γ(?ⁿ). By Lemma 1.5 we get $f=\sum\limits_{j=-\infty }^{\infty }{{{\lambda }_{j}}}{{a}_{j}}$, where ${{\left\| f \right\|}_{H\dot{K}_{{{q}_{1}}\left( \cdot \right)}^{\alpha , {{p}_{1}}}\left( {{\mathbb{R}}^{n}} \right)}}$≈$\inf {{\left( \sum\limits_{j=-\infty }^{\infty }{{{\left| {{\lambda }_{j}} \right|}^{{{p}_{1}}}}} \right)}^{1/{{p}_{1}}}}$ (the infimum is taken over above decompositions of f), and a_j is a dyadic central (α, q₁(·))-atom with the support B_j. We have

$\begin{array}{*{20}{c}} {\left\| {\left[ {b,{\mu _\Omega }} \right](f)} \right\|_{\dot K_{{q_2}\left( \cdot \right)}^{\alpha ,{p_2}}\left( {{\mathbb{R}^n}} \right)}^{{p_1}}} \\ { = {{\left\{ {\sum\limits_{k = - \infty }^\infty {{2^{k\alpha {p_2}}}} \left\| {\left[ {b,{\mu _\Omega }} \right](f){\chi _k}} \right\|_{{L^{{q_2}}}( \cdot )\left( {{\mathbb{R}^n}} \right)}^{{p_2}}} \right\}}^{{p_1}/{p_2}}}} \\ { \leqslant \sum\limits_{k = - \infty }^\infty {{2^{k\alpha {p_1}}}} \left\| {\left[ {b,{\mu _\Omega }} \right](f){\chi _k}} \right\|_{{L^{{q_2}}}( \cdot )\left( {{\mathbb{R}^n}} \right)}^{{p_1}} \leqslant C\sum\limits_{k = - \infty }^\infty {{2^{k\alpha {p_1}}}} } \\ {{{\left( {\sum\limits_{j = - \infty }^{k - 2} {\left| {{\lambda _j}} \right|} {{\left\| {\left[ {b,{\mu _\Omega }} \right]\left( {{a_j}} \right){\chi _k}} \right\|}_{{L^{{q_2}}}( \cdot )\left( {{\mathbb{R}^n}} \right)}}} \right)}^{{p_1}}} + C\sum\limits_{k = - \infty }^\infty {{2^{k\alpha {p_1}}}} } \\ {{{\left( {\sum\limits_{j = k - 1}^\infty {\left| {{\lambda _j}} \right|} {{\left\| {\left[ {b,{\mu _\Omega }} \right]\left( {{a_j}} \right){\chi _k}} \right\|}_{{L^{{q_2}}}( \cdot )\left( {{\mathbb{R}^n}} \right)}}} \right)}^{{p_1}}}} \\ { = :{U_1} + {U_2}.} \end{array}$

(3)

We first estimate U₁. Note that

$\begin{array}{*{20}{c}} {\left| {\left[ {b,{\mu _\Omega }} \right]\left( {{a_j}} \right)(x)} \right| \leqslant } \\ {\left( {\int_0^{|x|} | \int_{|x - y| \leqslant t|} {\frac{{\Omega (x - y)}}{{|x - y{|^{n - 1}}}}} [b(x) - b(y)]} \right.} \\ {{{\left. {{{\left. {{a_j}(y){\text{d}}y} \right|}^2}\frac{{{\text{d}}t}}{{{t^3}}}} \right)}^{1/2}} + } \\ {\left( {\int_{|x|}^\infty | \int_{|x - y| \leqslant t} {\frac{{\Omega (x - y)}}{{|x - y{|^{n - 1}}}}} [b(x) - b(y)]} \right.} \\ {{{\left. {{{\left. {{a_j}(y){\text{d}}y} \right|}^2}\frac{{{\text{d}}t}}{{{t^3}}}} \right)}^{1/2}}} \\ { = :{U_{11}} + {U_{12}}.} \end{array}$

When x∈A_k and |x-y|≤t with t≤|x|, it follows from j≤k-2 that |x-y|~|x|. By mean value theorem we have

$\left| {\frac{1}{{|x - y{|^2}}} - \frac{1}{{|x{|^2}}}} \right| \leqslant \frac{{|y|}}{{|x - y{|^3}}}.$

(4)

Then by (4), the Minkowski inequality, the generalized H?lder inequality and the vanishing moments of a_j we have

$\begin{gathered} {U_{11}} \leqslant C\int_{{\mathbb{R}^n}} {\left| {\frac{{\Omega (x - y)}}{{|x - y{|^{n - 1}}}} - \frac{{\Omega (x)}}{{|x{|^{n - 1}}}}} \right|} \hfill \\ \;\;\;\;\;\;\;\;\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|{\left( {\int_{|x - y|}^{|x|} {\frac{{{\text{d}}t}}{{{t^3}}}} } \right)^{1/2}}{\text{d}}y \hfill \\ \;\;\;\;\; \leqslant C\int_{{\mathbb{R}^n}} {\left| {\frac{{\Omega (x - y)}}{{|x - y{|^{n - 1}}}} - \frac{{\Omega (x)}}{{|x{|^{n - 1}}}}} \right|} \hfill \\ \;\;\;\;\;\;\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|{\left| {\frac{1}{{|x - y{|^2}}} - \frac{1}{{|x{|^2}}}} \right|^{1/2}}{\text{d}}y \hfill \\ \;\;\;\;\; \leqslant C\int_{{\mathbb{R}^n}} {\left| {\frac{{\Omega (x - y)}}{{|x - y{|^{n - 1}}}} - \frac{{\Omega (x)}}{{|x{|^{n - 1}}}}} \right|} \hfill \\ \;\;\;\;\;\;\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|\frac{{|y{|^{1/2}}}}{{|x - y{|^{3/2}}}}{\text{d}}y \hfill \\ \;\;\;\;\; \leqslant C{2^{(j - k)/2}}\int_{{A_j}} {\left| {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^n}}} - \frac{{\Omega (x)}}{{{{\left| x \right|}^n}}}} \right|} \hfill \\ \;\;\;\;\;\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|{\text{d}}y. \hfill \\ \end{gathered} $

Similarly, we consider U₁₂. Noting that |x-y|~|x|, by the Minkowski inequality and the vanishing moments of a_j we have

$\begin{gathered} {U_{12}} \leqslant C\int_{{\mathbb{R}^n}} {\left| {\frac{{\Omega (x - y)}}{{|x - y{|^{n - 1}}}} - \frac{{\Omega (x)}}{{|x{|^{n - 1}}}}} \right|} \hfill \\ \;\;\;\;\;\;\;\;\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|{\left( {\int_{|x|}^\infty {\frac{{{\text{d}}t}}{{{t^3}}}} } \right)^{1/2}}{\text{d}}y \hfill \\ \;\;\;\;\;\; \leqslant C\int_{{A_j}} {\left| {\frac{{\Omega (x - y)}}{{|x - y{|^n}}} - \frac{{\Omega (x)}}{{|x{|^n}}}} \right|} \hfill \\ \;\;\;\;\;\;\;\;\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|{\text{d}}y. \hfill \\ \end{gathered} $

So we have

$\begin{array}{*{20}{c}} {\left| {\left[ {b,{\mu _\Omega }} \right]\left( {{a_j}} \right)(x)} \right| \leqslant C\int_{{A_j}} {\left| {\frac{{\Omega (x - y)}}{{|x - y{|^n}}} - \frac{{\Omega (x)}}{{|x{|^n}}}} \right|} } \\ {\left| {b(x) - b(y)} \right|\left| {{a_j}(y)} \right|{\text{d}}y.} \end{array}$

Using Lemma 2.1 and the Minkowski inequality we have

$\begin{align} & {{\left\| \left[ b,{{\mu }_{\Omega }} \right]\left( {{a}_{j}} \right){{\chi }_{k}} \right\|}_{L{{q}_{2}}(\cdot )\left( {{\mathbb{R}}^{n}} \right)}} \\ & \le \int_{{{B}_{j}}}{\left\| \left| \frac{\Omega (\cdot -y)}{|\cdot -y{{|}^{n}}}-\frac{\Omega (\cdot )}{|\cdot {{|}^{n}}} \right| \right.} \\ & {{\left. (b(\cdot )-b(y)){{\chi }_{k}}(\cdot ) \right\|}_{L{{q}_{2}}(\cdot )\left( {{\mathbb{R}}^{n}} \right)}}\left| {{a}_{j}}(y) \right|\text{d}y \\ & \le \int_{{{B}_{j}}}{\left\| \left| \frac{\Omega (\cdot -y)}{|\cdot -y{{|}^{n}}}-\frac{\Omega (\cdot )}{|\cdot {{|}^{n}}} \right| \right.} \\ & b(\cdot )-b(0)|{{\chi }_{k}}(\cdot )|{{|}_{L{{q}_{2}}(\cdot )\left( {{\mathbb{R}}^{n}} \right)}}\left| {{a}_{j}}(y) \right|\text{d}y+ \\ & \int_{{{B}_{j}}}{\left| \frac{\Omega (\cdot -y)}{|\cdot -y{{|}^{n}}}-\frac{\Omega (\cdot )}{|\cdot {{|}^{n}}} \right|{{\chi }_{k}}(}\cdot )|{{|}_{{{L}{{{q}_{2}}}}(\cdot )\left( {{\mathbb{R}}^{n}} \right)}} \\ & |b(0)-b(y)|\left| {{a}_{j}}(y) \right|\text{d}y \\ & =:{{U}_{13}}+{{U}_{14}}. \\ \end{align}$

For U₁₃, noting s>q₂⁺, we denote ${{{\tilde{q}}}_{2}}(\cdot )>1$ and $\frac{1}{{{q}_{2}}(x)}=\frac{1}{{{{\tilde{q}}}_{2}}(x)}+\frac{1}{s}$ By Lemma 2.2 we have

$\begin{gathered} {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right||b( \cdot ) - b(0)|{\chi _k}( \cdot )} \right\|_{{L^{{q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|{\chi _k}( \cdot )} \right\|_{{L^s}\left( {{\mathbb{R}^n}} \right)}} \hfill \\ {\left\| {|b( \cdot ) - b(0)|{\chi _k}( \cdot )} \right\|_{L{{\bar q}_2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant C{\left\| b \right\|_{Li{p_\gamma }}}{2^{k\gamma }}{\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|{\chi _k}( \cdot )} \right\|_{{L^s}\left( {{\mathbb{R}^n}} \right)}} \hfill \\ {\left\| {{\chi _{{B_k}}}} \right\|_{L_2^{\bar q}( \cdot )\left( {{\mathbb{R}^n}} \right)}}. \hfill \\ \end{gathered} $

When |B_k|≤2ⁿ and x_k∈B_k, by Lemma 2.3 we have

$\begin{gathered} {\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{\tilde q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \approx {\left| {{B_k}} \right|^{\frac{1}{{{{\tilde q}_2}\left( {{x_k}} \right)}}}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \approx {\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}\left( \cdot \right)\left( {{\mathbb{R}^n}} \right)}}{\left| {{B_k}} \right|^{ - \frac{1}{s} - \frac{\gamma }{n}}}. \hfill \\ \end{gathered} $

When |B_k|≥1 we have

$\begin{gathered} {\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{\tilde q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \approx {\left| {{B_k}} \right|^{\frac{1}{{{{\tilde q}_2}\left( \infty \right)}}}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \approx {\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}\left( \cdot \right)\left( {{\mathbb{R}^n}} \right)}}{\left| {{B_k}} \right|^{ - \frac{1}{s} - \frac{\gamma }{n}}}. \hfill \\ \end{gathered} $

So we obtain

${\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{\tilde q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \approx {\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}\left( \cdot \right)\left( {{\mathbb{R}^n}} \right)}}{\left| {{B_k}} \right|^{ - \frac{1}{s} - \frac{\gamma }{n}}}.$

Meanwhile, by Lemma 2.4 we have

$\begin{gathered} {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{{{\left| \cdot \right|}^n}}}} \right|{\chi _k}( \cdot )} \right\|_{{L^s}\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant {2^{(k - 1)\left( {\frac{n}{s} - n} \right)}}\left\{ {\frac{{|y|}}{{{2^k}}} + \int_{|y|/{2^k}}^{|y|/{2^{k - 1}}} {\frac{{{\omega _s}(\delta )}}{\delta }{\text{d}}\delta } } \right\} \hfill \\ \leqslant {2^{(k - 1)\left( {\frac{n}{s} - n} \right)}}\left( {{2^{j - k + 1}} + {2^{(j - k + 1)\gamma }}\int_0^1 {\frac{{{\omega _s}(\delta )}}{\delta }{\text{d}}\delta } } \right) \hfill \\ \leqslant C{2^{\left( {k - 1} \right)\left( {\frac{n}{s} - n} \right)}}{2^{\left( {j - k} \right)\gamma }}. \hfill \\ \end{gathered} $

So by the generalized H?lder inequality we have

$\begin{gathered} {U_{13}} = \int_{{B_j}} {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|} \right.} \hfill \\ \;\;\;\;\;\;\;\;\;\;\left| {b( \cdot ) - b(0)} \right|{\chi _k}( \cdot )\left\| {_{_{{L^{{q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}}} \right.\left| {{a_j}(y)} \right|{\text{d}}y \hfill \\ \;\;\;\;\;\;\;\;\;\; \leqslant C{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}{2^{k\gamma }}{2^{(k - 1)\left( {\frac{n}{s} - n} \right)}}{2^{(j - k)\gamma }} \hfill \\ \;\;\;\;\;\;\;\;\;\;{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}{\left| {{B_k}} \right|^{ - \frac{1}{s} - \frac{\gamma }{n}}}\int_{{B_j}} {\left| {{a_j}(y)} \right|{\text{d}}y} \hfill \\ \;\;\;\;\;\;\;\;\;\; \leqslant C{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}{2^{ - kn + \left( {j - k} \right)\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \;\;\;\;\;\;\;\;\;\;{\left\| {{a_j}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}{\left\| {{\chi _{{B_j}}}} \right\|_{{L^{{q'}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}. \hfill \\ \end{gathered} $

(5)

For U₁₄, similar to the method of U₁₃ we have

$\begin{gathered} {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|{\chi _k}( \cdot )} \right\|_{{L^{{q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|{\chi _k}( \cdot )} \right\|_{{L^s}\left( {{\mathbb{R}^n}} \right)}} \hfill \\ {\left\| {{\chi _k}( \cdot )} \right\|_{L{{\tilde q}_2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant {\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|{\chi _k}( \cdot )} \right\|_{{L^s}\left( {{\mathbb{R}^n}} \right)}} \hfill \\ {\left\| {{\chi _{{B_k}}}} \right\|_{L{{\tilde q}_2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant C{2^{(k - 1)\left( {\frac{n}{s} - n} \right)}}{2^{(j - k)\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{\tilde q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant C{2^{ - kn + (j - k)\gamma - k\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}. \hfill \\ \end{gathered} $

So by the generalized H?lder inequality we have

$\begin{gathered} {U_{14}} = \int_{{B_j}} {{{\left\| {\left| {\frac{{\Omega ( \cdot - y)}}{{| \cdot - y{|^n}}} - \frac{{\Omega ( \cdot )}}{{| \cdot {|^n}}}} \right|{\chi _k}( \cdot )} \right\|}_{{L^{{q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}}} \hfill \\ \;\;\;\;\;\;\;\;\;|b(0) - b(y)|\left| {{a_j}(y)} \right|{\text{d}}y \hfill \\ \;\;\;\;\;\;\;\;\; \leqslant C{2^{ - kn + (j - k)\gamma - k\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \;\;\;\;\;\;\int_{{B_j}} {\left| {b(0) - b(y)} \right|} \left| {{a_j}(y)} \right|{\text{d}}y \hfill \\ \;\;\;\;\;\;\;\;\; \leqslant C{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}{2^{ - kn + 2(j - k)\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \;\;\;\;\;\;\;\;\;{\left\| {{\chi _{{B_j}}}} \right\|_{{L^{{q'}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}{\left\| {{a_j}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \;\;\;\;\;\;\;\;\; \leqslant C{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}{2^{ - kn + 2(j - k)\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \;\;\;\;\;\;\;\;\;{\left\| {{\chi _{{B_j}}}} \right\|_{{L^{{q'}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}{\left\| {{a_j}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}. \hfill \\ \end{gathered} $

(6)

By (5), (6), Lemma 1.2 and Lemma 1.3 we have

$\begin{gathered} {\left\| {\left[ {b,{\mu _\Omega }} \right]\left( {{a_j}} \right){\chi _k}} \right\|_{{L^{{q}}2}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant C{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}{2^{ - kn + (j - k)\gamma }}{\left\| {{\chi _{{B_k}}}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ {\left\| {{a_j}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}{\left\| {{X_{{B_j}}}} \right\|_{{L^{{q'}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}} \hfill \\ \leqslant C{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}{2^{(j - k)\gamma }}{\left\| {{a_j}} \right\|_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}\frac{{{{\left\| {{\chi _{{B_j}}}} \right\|}_{{L^{{q'}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}}}{{{{\left\| {{\chi _{{B_k}}}} \right\|}_{{L^{{q'}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}}} \hfill \\ \leqslant C{2^{ - j\alpha + (j - k)\left( {\gamma + n{\delta _2}} \right)}}{\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}}. \hfill \\ \end{gathered} $

So we have

$\begin{gathered} {U_1} \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{k = - \infty }^\infty {{2^{k\alpha {p_1}}}} \hfill \\ \;\;\;\;\;\;{\left( {\sum\limits_{j = - \infty }^{k - 2} {\left| {{\lambda _j}} \right|} {2^{ - j\alpha + (j - k)\left( {\gamma + n{\delta _2}} \right)}}} \right)^{{p_1}}} \hfill \\ \;\;\; = C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\;\;\sum\limits_{k = - \infty }^\infty {{{\left( {\sum\limits_{j = - \infty }^{k - 2} {\left| {{\lambda _j}} \right|} {2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right)}}} \right)}^{{p_1}}}} . \hfill \\ \end{gathered} $

When 1 < p₁ < ∞, take 1/p₁+1/p₁′=1. Since γ+nδ₂-α>0, by the H?lder inequality we have

$\begin{array}{l} {U_1} \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\;\;\;\;\sum\limits_{k = - \infty }^\infty {\left( {\sum\limits_{j = - \infty }^{k - 2} {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} {2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right){p_1}/2}}} \right)} \hfill \\ \;\;\;\;\;\;{\left( {\sum\limits_{j = - \infty }^{k - 2} {{2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right)p_1^\prime /2}}} } \right)^{{p_1}/p_1^\prime }} \hfill \\ \;\;\;\; \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\;\;\sum\limits_{k = - \infty }^\infty {\left( {\sum\limits_{j = - \infty }^{k - 2} {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} {2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right){p_1}/2}}} \right)} \hfill \\ \;\;\; = C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} \left( {\sum\limits_{k = j + 2}^\infty {{2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right){p_1}/2}}} } \right) \hfill \\ \;\;\; \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} . \hfill \\ \end{array}$

(7)

When 0 < p₁≤1, we have

$\begin{array}{l} {U_1} \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\;\;\;\sum\limits_{k = - \infty }^\infty {\left( {\sum\limits_{j = - \infty }^{k - 2} {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} {2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right){p_1}}}} \right)} \hfill \\ \;\;\;\;\; = C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\;\;\;\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} \left( {\sum\limits_{k = j + 1}^\infty {{2^{(j - k)\left( {\gamma + n{\delta _2} - \alpha } \right){p_1}}}} } \right) \hfill \\ \;\;\;\;\; \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} . \hfill \\ \end{array}$

(8)

Next we estimate U₂, by the (L^q₁(·)(?ⁿ), L^q₂(·)(?ⁿ))-boundedness of the commutator [b, μ_Ω] we have

$\begin{gathered} {U_2} \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}} \hfill \\ \;\;\;\;\;\;\;\;\sum\limits_{k = - \infty }^\infty {{2^{k\alpha {p_1}}}} {\left( {\sum\limits_{j = k - 1}^\infty {\left| {{\lambda _j}} \right|} {{\left\| {{a_j}} \right\|}_{{L^{{q}}1}( \cdot )\left( {{\mathbb{R}^n}} \right)}}} \right)^{{p_1}}} \hfill \\ \;\;\;\;\;\;\;\; \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{k = - \infty }^\infty {{{\left( {\sum\limits_{j = k - 1}^\infty {\left| {{\lambda _j}} \right|} {2^{(k - j)\alpha }}} \right)}^{{p_1}}}} . \hfill \\ \end{gathered} $

If 0<p₁≤1, then we have

$\begin{gathered} {U_2} \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} \sum\limits_{k = - \infty }^{j + 1} {{2^{(k - j)\alpha {p_1}}}} \hfill \\ \;\;\;\;\; \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} . \hfill \\ \end{gathered} $

(9)

If 1 < p₁ < ∞, by the H?lder inequality we have

$\begin{gathered} {U_2} \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{k = - \infty }^\infty {\left( {\sum\limits_{j = k - 1}^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} {2^{(k - j)\alpha {p_1}/2}}} \right)} \hfill \\ \;\;\;\;\;\;\;{\left( {\sum\limits_{j = k - 1}^\infty {{2^{(k - j)\alpha {p^\prime }1/2}}} } \right)^{{p_1}/{p^\prime }_1}} \hfill \\ \;\;\;\;\; \leqslant C\left\| b \right\|_{{\text{Li}}{{\text{p}}_\gamma }}^{{p_1}}\sum\limits_{j = - \infty }^\infty {{{\left| {{\lambda _j}} \right|}^{{p_1}}}} . \hfill \\ \end{gathered} $

(10)

Thus, by (3), (7)-(10) we complete the proof of Theorem 2.2.
References

[1]	Ková?ik O, Rákosník J. On spaces Lp(x) and Wk, p(x)[J]. Czechoslovak Math J, 1991, 41(4): 592-618.
[2]	Capone C, Cruz-Uribe D, Fiorenza A. The fractional maximal operators and fractional integrals on variable Lp spaces[J]. Rev Mat Iberoamericana, 2007, 23(3): 743-770.
[3]	Cruz-Uribe D, Fiorenza A. Variable Lebesgue spaces:foundations and harmonic analysis (applied and numerical harmonic analysis)[M]. Heidelberg: Springer, 2013.
[4]	Cruz-Uribe D, Fiorenza A, Martell J M, et al. The boundedness of classical operators on variable Lp spaces[J]. Ann Acad Sci Fen Math, 2006, 31(1): 239-264.
[5]	Diening L, Harjulehto P, H?st? P, et al. Lebesgue and Sobolev spaces with variable exponents[M]//Lecture Notes in Math, vol.2017. Heidelberg: Springer, 2011.
[6]	Wang H, Fu Z, Liu Z. Higher-order commutators of Marcinkiewicz integrals and fractional integrals on variable Lebesgue spaces[J]. Acta Math Sci Ser A Chin Ed, 2012, 32(6): 1092-1101.
[7]	Izuki M. Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization[J]. Anal Math, 2010, 36(1): 33-50.
[8]	Wang H, Liu Z. The Herz-type Hardy spaces with variable exponent and their applications[J]. Taiwanese J Math, 2012, 16(4): 1363-1389. DOI:10.11650/twjm/1500406739
[9]	Wang H. The continuity of commutators on Herz-type Hardy spaces with variable exponent[J]. Kyoto J Math, 2016, 56(3): 559-573. DOI:10.1215/21562261-3600175
[10]	Wang H. Boundedness of commutators on Herz-type Hardy spaces with variable exponent[J]. Jordan J Math Stat, 2016, 9(1): 17-30.
[11]	Wang H. Commutators of singular integral operator on Herz-type Hardy spaces with variable exponent[J]. J Korean Math Soc, 2017, 54(3): 713-732. DOI:10.4134/JKMS.j150771
[12]	Wang H. Commutators of homogeneous fractional integrals on Herz-type Hardy spaces with variable exponent[J]. J Contemp Math Anal, 2017, 52(3): 134-143.
[13]	Stein E M. On the function of Littlewood-Paley, Lusin and Marcinkiewicz[J]. Trans Amer Math Soc, 1958, 88(2): 430-466. DOI:10.1090/S0002-9947-1958-0112932-2
[14]	Liu Z, Wang H. Boundedness of Marcinkiewicz integrals on Herz spaces with variable exponent[J]. Jordan J Math Stat, 2012, 5(4): 223-239.
[15]	Tan J, Liu Z. Some boundedness of homogeneous fractional integrals on variable exponent function spaces[J]. Acta Math Sinica (Chin Ser), 2015, 58(2): 309-320.
[16]	Wu H, Lan J. Lipschitz estimates for fractional multilinear singular integral on variable exponent Lebesgue spaces[J/OL]. Abstract and Applied Analysis, 2013, Article ID 632384. http://dx.doi.org/10.1155/2013/632384.
[17]	Nakai E, Sawano Y. Hardy spaces with variable exponents and generalized Campanato spaces[J]. J Funct Anal, 2012, 262(9): 3665-3748. DOI:10.1016/j.jfa.2012.01.004
[18]	Lu S, Ding Y, Yan D. Singular integrals and related topics[M]. Beijing: World Scientific Press, 2011.
[19]	Wang H, Liu Z. Some characterizations of Herz-type Hardy spaces with variable exponent[J]. Ann Funct Anal, 2015, 6(2): 224-233. DOI:10.15352/afa/06-2-19