Commutator of Marcinkiewicz Integral Operators on Herz-Morrey-Hardy Spaces with Variable Exponents

Omer Khalil^1,2*, Shuangping Tao^1*, Bechir Mahamat¹

¹College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China.
²Faculty of Education, Sudan University of Science and Technology, Khartoum, Khartoum State, Sudan.

DOI: 10.4236/am.2021.125030                

Abstract

In this paper, our aim is to prove the boundedness of commutators generated by the Marcinkiewicz integrals operator [b,μ_Ω] and obtain the result with Lipschitz function and BMO function f on the Herz-Morrey-Hardy spaces with variable exponents .

Keywords

Marcinkiewicz Integral Operator, Herz-Morrey-Hardy Space, Commutator, Variable Exponent, Lipschitz Space

1. Introduction

Firstly in 1938, Marcinkiewicz [1] introduced the Marcinkiewicz integral. Next, the Marcinkiewicz integral operator has been studied extensively by many mathematicians in various fields. For example, Stain in [2] introduced the Marcinkiewicz integral operator related to the littlewood-Paley $g$ function on ${ℝ}^n$ and proved that ${\mu }_{\Omega }$ is of type $\left(p\mathrm{,}p\right)$ for $1<p\le 2$ and of week type $\left(\mathrm{1,1}\right)$. In [3], Ding, Fan and Pan improved the above result and obtained the $L^p\left(1<p<\infty \right)$ and weighted $L^p\left(1<p<\infty \right)$ boundedness of the Marcinkiewicz cussed the boundedness for the commutator generated by the Marcinkiintegral $\mu$ under some weak conditions. Torchinsky and Wang in [4] discussed integral ${\mu }_{\Omega }$ and $BMO\left({ℝ}^n\right)$ function on Lebesgue spaces $L^p\left({ℝ}^n\right)$.

On the other hand, a class of functional spaces called Herz-Morrey-Hardy spaces with variable exponent has attracted great interest in recent years. We find that in successive studies in this field, in [5] [6] Xu, Yang introduced Herz-Morrey-Hardy spaces with variable exponents and their some applications. He obtained that certain singular integral operators are bounded from Herz-Morrey-Hardy spaces with variable exponents into Herz-Morrey spaces with variable exponents as an application of the atomic characterization. Also, he established their molecular decomposition, and by using their atomic and molecular decompositions, he gave the boundedness of a convolution type singular integral on Herz-Morrey-Hardy spaces with variable exponents. Omer in [7] proved the boundedness of commutators generated by the Calderón-Zygmund and used properties of variable exponent, BMO(Rⁿ) function and Lipschitz function to prove this boundedness. Also, Yang in [8] established some boundedness for $T{D}^{\gamma }-D^{\gamma }T$ and $\left(T^{\mathrm{<mo>\; *\; </mo>}}-T^{\#}\right)D^{\gamma }$ on the homogeneous Morrey-Herz-type Hardy spaces with variable exponents and studied Boundedness of Calderón-Zygmund operator on these spaces.

Suppose ${\mathbb{S}}^{n-1}\left(n\ge 2\right)$ denotes the unit sphere in ${ℝ}^n$ equipped with the normalized measure $\text{d}\sigma$. Let $\Omega$ be homogenous function of degree zero and satisfies

${\displaystyle {\int }_{{\mathbb{S}}^{n-1}}}\Omega \left(x^{\prime }\right)\text{d}\sigma \left(x^{\prime }\right)=\mathrm{0,}$ (1.1)

where $x^{\prime }=x/\left|x\right|$ for any $x\ne 0$.

Then the Marcinkiewicz integral operator ${\mu }_{\Omega }$ is defined by

${\mu }_{\Omega }\left(f\right)\left(x\right)={\left({\displaystyle {\int }_0^{\infty }}{\left|F_{\Omega \mathrm{,}t}\left(f\right)\left(x\right)\right|}^2\frac{\text{d}t}{t^3}\right)}^{1/2}\mathrm{,}$ (1.2)

where

$F_{\Omega \mathrm{,}t}\left(f\right)\left(x\right)={\displaystyle {\int }_{\left|x-y\right|\le 1}}\frac{\Omega \left(x-y\right)}{{\left|x-y\right|}^{n-1}}f\left(y\right)\text{d}y\mathrm{.}$ (1.3)

Let $b\in Li{p}_{\gamma }\left({ℝ}^n\right)$ and $b\in BMO$ be a locally integrable function on ${ℝ}^n$, the commutator generated by the Marcinkiewicz integral ${\mu }_{\Omega }$ and b is defined by

$\left[b\mathrm{,}{\mu }_{\Omega }\right]={\left({\displaystyle {\int }_0^{\infty }}{\left|{\displaystyle {\int }_{\left|x-y\right|\le t}}\frac{\Omega \left(x-y\right)}{{\left|x-y\right|}^{n-1}}\left[b\left(x\right)-b\left(y\right)\right]f\left(y\right)\text{d}y\right|}^2\frac{\text{d}t}{t^3}\right)}^{1/2}\mathrm{.}$ (1.4)

Motivated by [6] and [7], the aim of this paper is to study the boundedness for the commutator of Marcinkiewicz integral operator $\left[b\mathrm{,}{\mu }_{\Omega }\right]$ on the Herz-Morrey-Hardy space with variable exponent where $\Omega \in L^s\left(S^{n-1}\right)$ for $s\ge 1$, with BMO function and Lipschitz function, we will define The definitions of the Morrey-Herz spaces with variable exponents, the Morrey-Herz-Hardy spaces with variable exponents (which will be defined in the next section), and the preliminary lemmas are presented in Section 2. In Section 3, we will prove the boundedness of the commutator of Marcikiewicz integrals on Herz-Morrey-Hrdy spaces with variable exponent with $b\in Li{p}_{\gamma }\left({ℝ}^n\right)$. Lastly, in Section 4 we will prove the boundedness of the commutator of Marcikiewicz integrals on Herz-Morrey-Hrdy spaces with variable exponent with function $b\in BMO\left({ℝ}^n\right)$.

A given open set $\Omega \subset {ℝ}^n$ and a measurable function $p(\cdot )\mathrm{<mo>\; :\; </mo>}\Omega \to \left[\mathrm{1,}\infty \right)$, $L^{p(\cdot )}\left(\Omega \right)$ denotes the set of measurable function f on $\Omega$ such that for some $\lambda >0$,

$L^{p(\cdot )}\left(\Omega \right)=\left\{f\text{is}\text{measurable}\mathrm{<mo>\; :\; </mo>}{\displaystyle {\int }_{\Omega }}{\left(\frac{\left|f\left(x\right)\right|}{\eta }\right)}^{p\left(x\right)}\text{d}x<\infty \mathrm{<mtext>\; \hspace{0.17em}\; </mtext>}\text{for}\text{some}\text{constant}\eta >0\right\}\mathrm{,}$ (1.5)

the space $L_{Loc}^{p(\cdot )}\left(\Omega \right)$ is defined by

$L_{Loc}^{p(\cdot )}\left(\Omega \right)=\left\{f\text{ismeasurable}\mathrm{<mo>\; :\; </mo>}f\in L^{p(\cdot )}\left(K\right)\text{forallcompact}K\subset \Omega \right\}\mathrm{.}$ (1.6)

The Lebesgue spaces $L^{p(\cdot )}\left(\Omega \right)$ is Banach spaces with the norm defined by

${\Vert f\Vert }_{L^{p(\cdot )}\left(\Omega \right)}=\inf \left\{\eta >0:{\displaystyle {\int }_{\Omega }}{\left(\frac{\left|f\left(x\right)\right|}{\eta }\right)}^{p\left(x\right)}\text{d}x\le 1\right\}\mathrm{,}$ (1.7)

where $p_{-}=\text{essinf}\left\{p\left(x\right):x\in \Omega \right\}>1$, $p_{+}=\text{esssup}\left\{p\left(x\right):x\in \Omega \right\}<\infty$.

Denotes $p^{\prime }\left(x\right)=p\left(x\right)/\left(p\left(x\right)-1\right)$. Let M be the Hardy-Littlewood maximal operator. We denote $\mathcal{B}\left(\Omega \right)$ to be the set of all functions $p(\cdot )\in \mathcal{P}\left(\Omega \right)$ satisfying the M is bounded on $L^{p(\cdot )}\left(\Omega \right)$.

Definition 1.1. [6]

Let $0<q\le \infty$, $p(\cdot )\in \mathcal{P}\left({ℝ}^n\right)$, $0\le \lambda <\infty$. Let $\alpha (\cdot )$ be a bounded real-valued measurable function on ${ℝ}^n\mathrm{.}$ The nonhomogeneous Morrey-Herz space $M{K}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ and homogeneous Morrey-Herz space with variable exponents $M{\stackrel{\dot{}}{K}}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ are respectively defined by

$M{K}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}\mathrm{<mo>\; :\; </mo>}=\left\{f\in L_{loc}^{p(\cdot )}\left({ℝ}^n\backslash \left\{0\right\}\right)\mathrm{<mo>\; :\; </mo>}{\Vert f\Vert }_{M{K}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}}<\infty \right\}\mathrm{,}$ (1.8)

and

$M{\stackrel{\dot{}}{K}}_{p(\cdot ),\lambda }^{\alpha (\cdot ),q}:=\left\{f\in L_{Loc}^{p(\cdot )}\left({ℝ}^n\backslash \left\{0\right\}\right):{\Vert f\Vert }_{M{\stackrel{\dot{}}{K}}_{p(\cdot ),\lambda }^{\alpha (\cdot ),q}}<\infty \right\},$ (1.9)

where

${\Vert f\Vert }_{M{K}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}}\mathrm{<mo>\; :\; </mo>}=\underset{L\in {\mathbb{N}}_0}{\sup }{2}^{-L\lambda }{\left({\displaystyle \underset{k=0}{\overset{L}{\sum }}}{\Vert 2^{k\alpha (\cdot )}f{\tilde{\chi }}_k\Vert }_{L^{p(\cdot )}}^q\right)}^{1/q}\mathrm{,}$ (1.10)

${\Vert f\Vert }_{M{\stackrel{\dot{}}{K}}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}}\mathrm{<mo>\; :\; </mo>}=\underset{L\in \mathbb{Z}}{\sup }{2}^{-L\lambda }{\left({\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{\Vert 2^{k\alpha (\cdot )}f{\chi }_k\Vert }_{L^{p(\cdot )}}^q\right)}^{1/q}\mathrm{.}$ (1.11)

Definition 1.2. [9]

For all $0<\gamma \le 1$, the Lipschitz space $Li{p}_{\gamma }\left({ℝ}^n\right)$ is defined by

$Li{p}_{\gamma }=\left\{f\mathrm{<mo>\; :\; </mo>}{\Vert f\Vert }_{Li{p}_{\gamma }}=\underset{x\mathrm{,}y\in {ℝ}^n\mathrm{<mo>\; ;\; </mo>}x\ne y}{\sup }\frac{\left|f\left(x\right)-f\left(y\right)\right|}{{\left|x-y\right|}^{\gamma }}<\infty \right\}\mathrm{.}$ (1.12)

Definition 1.3. [5]

Let $\alpha (\cdot )\in L^{\infty }\left({ℝ}^n\right),\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>}p(\cdot )\in \mathcal{P}\left({ℝ}^n\right),\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>\; 0}<q\le \infty ,\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>\; 0}\le \lambda <\infty$ and $N>n+1$. The nonhomogeneous Herz-Morrey-Hardy space with variable exponent $HM{K}_{p(\cdot )\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ and homogeneous Herz-Morrey-Hardy space with variable exponents $HM{\stackrel{\dot{}}{K}}_{p(\cdot )\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ are respectively defined by

$HM{K}_{p(\cdot )\lambda }^{\alpha (\cdot ),q}\left({ℝ}^n\right):=\left\{f\in {\mathcal{S}}^{\prime }\left({ℝ}^n\right):{\Vert f\Vert }_{HM{K}_{p(\cdot )\lambda }^{\alpha (\cdot ),q}}:={\Vert G_{N}f\Vert }_{M{K}_{p(\cdot )\lambda }^{\alpha (\cdot ),q}}<\infty \right\},$ (1.13)

$HM{\stackrel{\dot{}}{K}}_{p(\cdot )\lambda }^{\alpha (\cdot ),q}\left({ℝ}^n\right):=\left\{f\in {\mathcal{S}}^{\prime }\left({ℝ}^n\right):{\Vert f\Vert }_{HM{\stackrel{\dot{}}{K}}_{p(\cdot )\lambda }^{\alpha (\cdot ),q}}:={\Vert G_{N}f\Vert }_{M{\stackrel{\dot{}}{K}}_{p(\cdot )\lambda }^{\alpha (\cdot ),q}}<\infty \right\}.$ (1.14)

Definition 1.4. [10] (Hölder’s inequality) Let $\alpha >1$ and $1/\alpha +1/\beta =1$. Then the discrete and integral forms of Hölder’s inequality are given as

${\displaystyle {\int }_a^b}\left|f\left(x\right)g\left(x\right)\right|\text{d}x\le {\left({\displaystyle {\int }_a^b}{\left|f\left(x\right)\right|}^{\alpha }\right)}^{1/\alpha }{\left({\displaystyle {\int }_a^b}\mathrm{<mo>\; |\; </mo>}g\mathrm{<mo\; stretchy="false">\; (\; </mo>}x{\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo>\; |\; </mo>}}^{\beta }\right)}^{1/\beta }\mathrm{,}$ (1.15)

for continuous function f and g on $\left[a\mathrm{,}b\right]$.

Definition 1.5. [10] (Minkowski’s inequality) Let $u>1$. Then the discrete and integral forms of Minkowski’s inequality are given as

${\left({\displaystyle {\int }_a^b}{\left|f\left(x\right)+g\left(x\right)\right|}^u\text{d}x\right)}^{1/u}\le {\left({\displaystyle {\int }_a^b}{\left|f\left(x\right)\right|}^u\right)}^{1/u}+{\left({\displaystyle {\int }_a^b}{\left|g\left(x\right)\right|}^u\right)}^{1/u}\mathrm{,}$ (1.16)

for continuous function f and g on $\left[a\mathrm{,}b\right]$. for more general functions can be obtained naturally. A further generalization is: If $u>1$, then

${\left({\displaystyle \int }{\left({\displaystyle \int }\left|f\left(x\mathrm{,}y\right)\right|\text{d}y\right)}^u\text{d}x\right)}^{1/u}\le {\displaystyle \int }{\left({\displaystyle \int }{\left|f\left(x\right)\right|}^u\text{d}x\right)}^{1/u}\text{d}y\mathrm{.}$ (1.17)

2. Preliminaries

In this section, we give some preliminaries which we used to prove theorems.

Lemma 2.1. [11] Let $p(\cdot )\in \mathcal{P}\left({ℝ}^n\right)$. Then for any $f\in L^{p(\cdot )}$ and $g\in L^{p^{\prime }(\cdot )}$, we have

${\displaystyle {\int }_{{ℝ}^n}}\left|f\left(x\right)g\left(x\right)\right|\text{d}x\le C_p{\Vert f\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\Vert g\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}\mathrm{,}$

where $C_p=1+\frac{1}{p_{-}}-\frac{1}{p_{+}}$.

This inequality is called the generalized Hölder inequality with respect to the variable $L^{p(\cdot )}$ spaces.

Lemma 2.2. [12] Given $p(\cdot )\mathrm{,\; <mtext>\; \hspace{0.17em}\; </mtext>}{p}_1(\cdot )\mathrm{,\; <mtext>\; \hspace{0.17em}\; </mtext>}{p}_2(\cdot )\in \mathcal{P}\left({ℝ}^n\right)$, for any $f\in L^{p_1(\cdot )}\left({ℝ}^n\right)\mathrm{,\; <mtext>\; \hspace{0.17em}\; </mtext>}g\in L^{p_2(\cdot )}\left({ℝ}^n\right)$, when $\frac{1}{p(\cdot )}=\frac{1}{p_2(\cdot )}+\frac{1}{p_1(\cdot )}$, we get

${\Vert f\left(x\right)g\left(x\right)\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\le C{\Vert f\Vert }_{L^{p_1(\cdot )}\left({ℝ}^n\right)}{\Vert g\Vert }_{L^{p_2}\left({ℝ}^n\right)}\mathrm{,}$

where $C_{p_1,\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>}{p}_2}={\left[1+\frac{1}{p_{1-}}-\frac{1}{p_{1+}}\right]}^{\frac{1}{p_{-}}}$.

Proposition 2.3. [13] If $q(\cdot )\in \mathcal{P}\left({ℝ}^n\right)$ satisfies

$\left|q\left(x\right)-q\left(y\right)\right|\le \frac{-C}{\log \left(\left|x-y\right|\right)},\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>}\left|x-y\right|\le 1/2,$

$\left|q\left(x\right)-q\left(y\right)\right|\le \frac{C}{\log \left(e+\left|x\right|\right)},\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>}\left|y\right|\ge \left|x\right|,$

then $q(\cdot )\in \mathfrak{B}\left({ℝ}^n\right)$.

Lemma 2.4. [14] Let k be a positive integer and B be a ball in ${ℝ}^n$. Then we have that for all $b\in BMO\left({ℝ}^n\right)$ and $i\mathrm{,}j\in \mathbb{Z}$ with $i<j$, we have

1) $C^{-1}{\Vert b\Vert }_{\ast }^k\le \underset{B}{\sup }\frac{1}{{\Vert {\chi }_B\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}}{\Vert \left(b-b_B\right){\chi }_B\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\le C{\Vert b\Vert }_{\ast }^k$,

2) ${\Vert {\left(b-b_{B_i}\right)}^k{\chi }_{B_j}\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\le C{\left(j-i\right)}^K{\Vert b\Vert }_{\ast }^k{\Vert {\chi }_{B_j}\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}$,

where $B_i=\left\{x\in {ℝ}^n\mathrm{<mo>\; :\; </mo>}\left|x\right|\le 2^i\right\}$ and $B_j=\left\{x\in {ℝ}^n\mathrm{<mo>\; :\; </mo>}\left|x\right|\le 2^j\right\}$.

Lemma 2.5. [15] Let $q(\cdot )\in \mathfrak{B}\left({ℝ}^n\right)$, then there exist positive constants $C>0$, such that for all balls $B\subset {ℝ}^n$ and all measurable subset $R\subset B$,

$\frac{{\Vert {\chi }_R\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}}{{\Vert {\chi }_B\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}}\le C{\left(\frac{\left|R\right|}{\left|B\right|}\right)}^{{\delta }_1}\mathrm{,\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>}\frac{{\Vert {\chi }_R\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}}{{\Vert {\chi }_B\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}}\le C{\left(\frac{\left|R\right|}{\left|B\right|}\right)}^{{\delta }_2}\mathrm{,}$

where ${\delta }_1\mathrm{,}{\delta }_2$ are constants with $0<{\delta }_1,{\delta }_2<1$.

Lemma 2.6. [16] If $q(\cdot )\in \mathfrak{B}\left({ℝ}^n\right)$, then there exists a constant $C>0$ such that for any balls B in ${ℝ}^n$,

$\frac{1}{\left|B\right|}{\Vert {\chi }_B\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_B\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}\le C\mathrm{.}$

Lemma 2.7. [6] Let $0<q<\infty ,\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>}p(\cdot )\in \mathfrak{B}\left({ℝ}^n\right),\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>\; 0}<\lambda <\infty$, and $\alpha (\cdot )\in L^{\infty }\left({ℝ}^n\right)$ be log-Hölder continuous both at the origin and infinity, $2\lambda \le \alpha (\cdot ),n{\delta }_2\le \alpha \left(0\right),{\alpha }_{\infty }<\infty$, ${\delta }_2$ as in lemma 2.4. Then $f\in HM{\stackrel{\dot{}}{K}}_{p(\cdot )\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ (or $HM{K}_{p(\cdot )\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ ) if and only if $f={\displaystyle {\sum }_{k=-\infty }^{\infty }}{\lambda }_k{f}_k$ (or $f={\displaystyle {\sum }_{k=0}^{\infty }}{\lambda }_k{f}_k$ ), in the sense of $f\in {\mathcal{S}}^{\prime }{\left(ℝ\right)}^n$, where each $a_k$ is a central $\left(\alpha (\cdot )\mathrm{,}p(\cdot )\right)$ atom with support contained in $B_k$ and

$\underset{L\le 0,L\in Z}{\sup }{2}^{-L\lambda }{\displaystyle {\sum }_{k=-\infty }^L}{\left|{\lambda }_k\right|}^q<\infty$ or $\left(\underset{L\le 0,L\in Z}{\sup }{2}^{-L\lambda }{\displaystyle {\sum }_{k=0}^L}{\left|{\lambda }_k\right|}^q\right)$,

moreover

${\Vert f\Vert }_{HM{\stackrel{\dot{}}{K}}_{p(\cdot )\lambda }^{\alpha (\cdot ),q}}\approx \inf \underset{L\le 0,L\in Z}{\sup }{2}^{-L\lambda }{\left({\displaystyle {\sum }_{k=-\infty }^L}{\left|{\lambda }_k\right|}^q\right)}^{1/q}$

or

${\Vert f\Vert }_{HM{K}_{p(\cdot )\lambda }^{\alpha (\cdot ),q}}\approx \inf \underset{L\le 0,L\in Z}{\sup }{2}^{-L\lambda }{\left({\displaystyle {\sum }_{k=0}^L}{\left|{\lambda }_k\right|}^q\right)}^{1/q}$,

where infimum is taken over all above decomposition of f.

Lemma 2.8. [17] Let $q(\cdot )\in \mathcal{P}\left({ℝ}^n\right)\mathrm{,}q\in \left(\mathrm{0,}\infty \right]$ and $\lambda \in \left[\mathrm{0,}\infty \right)$. If $\alpha (\cdot )\in L^{\infty }\left({ℝ}^n\right)\cap {\mathfrak{P}}_0^{log}\left({ℝ}^n\right)\cap {\mathfrak{P}}_{\infty }^{log}\left({ℝ}^n\right)$, then

$\begin{array}{l}{\Vert f\Vert }_{M{\stackrel{\dot{}}{K}}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)}^q=\max \{\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\Vert f{\chi }_k\Vert }_{L^{p(\cdot )}}^q,\\ \underset{L\le 0,L\in Z}{\sup }{2}^{-L\lambda q}({\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\Vert f{\chi }_k\Vert }_{L^{p(\cdot )}}^q+{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\Vert f{\chi }_k\Vert }_{L^{p(\cdot )}}^q)\}.\end{array}$

Lemma 2.9. [18] Let $\Omega$ satisfies $L^r$ -Dini condition with $r\in \left[\mathrm{1,}\infty \right)$. If there exist constants $C>0$ and $R>0$ such that $\left|y\right|<R/2$, then for every $x\in {ℝ}^n$, we have

${\left({\displaystyle {\int }_{R<\left|R\right|<2R}}{\left|\frac{\Omega \left(x-y\right)}{\left|x-y\right|}-\frac{\Omega \left(x\right)}{\left|x\right|}\right|}^r\text{d}x\right)}^{1/r}\le C{R}^{\left(\frac{n}{r}-n\right)}\left\{\frac{\left|y\right|}{R}+{\displaystyle {\int }_{\left|y\right|/2R<\delta <\left|y\right|/R}}\frac{{\omega }_r\left(\delta \right)}{\delta }\text{d}\delta \right\}.$

Lemma 2.10. [15] Given E, let $q(\cdot )\in \mathcal{P}\left(E\right)\mathrm{,}f\mathrm{<mo>\; :\; </mo>}E\times E\to {ℝ}^n$ be a measurable function (with respect to product measure) such that for almost every $y\in E\mathrm{,}f\left(\mathrm{.,}y\right)\in L^{q(\cdot )}\left(E\right)$. Then

${\Vert {\displaystyle {\int }_E}f\left(\mathrm{.,}y\right)dy\Vert }_{L^{q(\cdot )}\left(E\right)}\le C{\displaystyle {\int }_E}{\Vert f\left(\mathrm{.,}y\right)\Vert }_{L^{q(\cdot )}\left(E\right)}\text{d}y\mathrm{.}$

Lemma 2.11. [19] If $a>0,\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>\; 1}\le s\le \infty ,\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>\; 0}\le d\le s$ and $-n+\left(n-1\right)d/s<v<\infty$, then

${\left({\displaystyle {\int }_{\left|y\right|\le a\left|x\right|}}{\left|y\right|}^v{\left|\Omega \left(x-y\right)\right|}^d\text{d}y\right)}^{1/d}\le C{\left|x\right|}^{\left(v+n\right)/d}{\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}\mathrm{.}$

Lemma 2.12. [19] Let $q(\cdot )\in \mathcal{P}$ satisfies Proposition 2.3. Then

${\Vert {\chi }_Q\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\approx (\begin{array}{ll}{\left|Q\right|}^{\frac{1}{q\left(x\right)}}\hfill & \text{if}\left|Q\right|\le 2^n\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>}\text{and}\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>}x\in Q\hfill \\ {\left|Q\right|}^{\frac{1}{q\left(\infty \right)}}\hfill & \text{if}\left|Q\right|\ge 1\hfill \end{array}$

for every cube (or ball) $Q\in {ℝ}^n$, where $p\left(\infty \right)={\lim }_{x\to \infty }p\left(x\right)$.

3. Lipschitz Boundedness for the Commutator of Marcikiewicz Integrals Operator

In this section, we prove the boundedness of the commutator of Marcikiewicz integrals on Herz-Morrey-Hrdy spaces with variable exponent so when $b\in Li{p}_{\gamma }\left({ℝ}^n\right)$ under some conditions.

Theorem 3.1.

Suppose that $b\in Li{p}_{\gamma }\left({ℝ}^n\right)$ with $0<\gamma \le 1$. If $q_1(\cdot )\in \mathcal{P}\left({ℝ}^n\right)$ satisfies proposition 2.3 with $q_1^{+}<n/\gamma ,1/q_1\left(x\right)-1/q_2\left(x\right)=\gamma /n$, $\Omega \in L^s\left(S^{n-1}\right)\left(s>q_2^{+}\right)$ with $1\le s^{\prime }<q_1^{-}$ and satisfies

${\displaystyle {\int }_0^1}\frac{{\Omega }_s\left(\delta \right)}{{\delta }^{1+\gamma }}\text{d}\delta <\infty ,$

let $0<p_1\le q_2<\infty$ and $n{\delta }_2\le \alpha <n{\delta }_2+\gamma$ or ( $0<\max \left(n{\delta }_2,{\alpha }_2\right)\le {\alpha }_1<n{\delta }_2+\gamma$ ). Then the commutator $\left[b\mathrm{,}{\mu }_{\Omega }\right]$ is bounded from $HM{\stackrel{\dot{}}{K}}_{p(\cdot )\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ (or $HM{K}_{p(\cdot )\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ ) to $M{\stackrel{\dot{}}{K}}_{p(\cdot )\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ (or $M{K}_{p(\cdot )\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ ).

To the proof the above theorem, we will recall the following lemma.

Lemma 3.1. [15]

Suppose that $b\in Li{p}_{\gamma }\left({ℝ}^n\right)$ with $0<\gamma \le 1$. If $q_1(\cdot )\in \mathcal{P}\left({ℝ}^n\right)$ satisfies Proposition 2.3 with $q_1^{+}<n/\gamma ,1/q_1\left(x\right)-1/q_2\left(x\right)=\gamma /n$ with $\Omega \in L^s\left(S^{n-1}\right)\left(s>q_2^{+}\right)$. Then the commutator $\left[b\mathrm{,}{\mu }_{\Omega }\right]$ is bounded from $L^{q_1(\cdot )}\left({ℝ}^n\right)$ to $L^{q_1(\cdot )}\left({ℝ}^n\right)$.

Next, we will give the Lipschitz estimate about the commutator $\left[b\mathrm{,}{\mu }_{\Omega }\right]$ on Herz-Morrey-Hardy spaces with variable exponent.

Proof Theorem 3.1:

To prove this theorem, we only prove the homogeneous case. Let $f\in HM{K}_{p(\cdot )\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$. By lemma 2.6 we have $f={\displaystyle {\sum }_{j=-\infty }^{\infty }}{\lambda }_j{f}_j$ converged in ${\mathcal{S}}^{\prime }{\left(ℝ\right)}^n$, where each $b_j$ is a central $\left(\alpha (\cdot )\mathrm{,}p(\cdot )\right)$ atom with support contained in $B_j$ and

${\Vert f\Vert }_{HM{K}_{p(\cdot )\lambda }^{\alpha (\cdot )\mathrm{,}q}}\approx \inf \underset{L\le \mathrm{0,}L\in Z}{{\displaystyle \sup }}{2}^{-L\lambda }{\left({\displaystyle \underset{j=-\infty }{\overset{L}{\sum }}}{\left|{\lambda }_j\right|}^q\right)}^{1/q}\mathrm{.}$

Here we denote $\Delta =\underset{L\le \mathrm{0,}L\in Z}{{\displaystyle \sup }}{2}^{-L\lambda }{\displaystyle {\sum }_{k=-\infty }^L}{\left|{\lambda }_k\right|}^q$. By lemma 2.8 we have

$\begin{array}{c}{\Vert \left[b\mathrm{,}{\mu }_{\Omega }\right]\left(f\right)\Vert }_{M{K}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)}^q=\max \{\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q,\\ \underset{L\le 0,L\in Z}{{\displaystyle \sup }}{2}^{-L\lambda q}({\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q\\ +{\displaystyle \underset{k\mathrm{<mo>\; =\; </mo>0}}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\Vert \left[b\mathrm{,}{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q)\}\mathrm{.}\end{array}$

$\begin{array}{l}I=\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q,\\ II={\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q,\\ III=\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q.\end{array}$

In beginning, we examine a function which we will use in proving

$\begin{array}{c}\left|\left[b\mathrm{,}{\mu }_{\Omega }\right]\left(b_j\right)\left(x\right)\right|\le {\left\{{\displaystyle {\int }_0^{\left|x\right|}}{\left|{\displaystyle {\int }_{\left|x-y\right|\le t}}\frac{\left|\Omega \left(x-y\right)\right|}{{\left|x-y\right|}^{n-1}}\left[b\left(x\right)-b\left(y\right)\right]b_j\left(y\right)\text{d}y\right|}^2\frac{\text{d}t}{t^3}\right\}}^{1/2}\\ +{\left\{{\displaystyle {\int }_{\left|x\right|}^{\infty }}{\left|{\displaystyle {\int }_{\left|x-y\right|\le t}}\frac{\left|\Omega \left(x-py\right)\right|}{{\left|x-y\right|}^{n-1}}\left[b\left(x\right)-b\left(y\right)\right]b_j\left(y\right)\text{d}y\right|}^2\frac{\text{d}t}{t^3}\right\}}^{1/2}\\ \mathrm{<mo>\; :\; </mo>}\simeq {\Upsilon }_1+{\Upsilon }_2\mathrm{.}\end{array}$

When $x\in A_k$ and $\left|x-y\right|\le t$ with $t\le \left|x\right|$, it follows from $j\le k-2$ that $\left|x-y\right|\sim \left|x\right|$. We have

$\left|\frac{1}{{\left|x-y\right|}^2}-\frac{1}{{\left|x\right|}^2}\right|\le \frac{\left|y\right|}{{\left|x-y\right|}^3}\mathrm{.}$ (3.1)

Then by (3.1), the Minkowski’s inequality, the generalized Hölder’s inequality and the vanishing of the moment of $b_j$ we have

$\begin{array}{c}{\Upsilon }_1\le C{\displaystyle {\int }_{{ℝ}^n}}\left|\frac{\left|\Omega \left(x-y\right)\right|}{{\left|x-y\right|}^2}-\frac{\left|\Omega \left(x\right)\right|}{{\left|x\right|}^2}\right|\left|b\left(x\right)-b\left(y\right)\right|\left|b_j\left(y\right)\right|{\left({\displaystyle {\int }_{\left|x-y\right|}^{\left|x\right|}}\frac{\text{d}t}{t^3}\right)}^{1/2}\text{d}y\\ \le C{\displaystyle {\int }_{{ℝ}^n}}\left|\frac{\left|\Omega \left(x-y\right)\right|}{{\left|x-y\right|}^2}-\frac{\left|\Omega \left(x\right)\right|}{{\left|x\right|}^2}\right|\left|b\left(x\right)-b\left(y\right)\right|\left|b_j\left(y\right)\right|{\left|\frac{1}{{\left|x-y\right|}^2}-\frac{1}{{\left|x\right|}^2}\right|}^{1/2}\text{d}y\end{array}$

$\begin{array}{c}\le C{\displaystyle {\int }_{{ℝ}^n}}\left|\frac{\left|\Omega \left(x-y\right)\right|}{{\left|x-y\right|}^2}-\frac{\left|\Omega \left(x\right)\right|}{{\left|x\right|}^2}\right|\left|b\left(x\right)-b\left(y\right)\right|\left|b_j\left(y\right)\right|\frac{{\left|y\right|}^{1/2}}{{\left|x-y\right|}^{3/2}}\text{d}y\\ \le C{2}^{\left(j-k\right)/2}{\displaystyle {\int }_{B_j}}\left|\frac{\left|\Omega \left(x-y\right)\right|}{{\left|x-y\right|}^2}-\frac{\left|\Omega \left(x\right)\right|}{{\left|x\right|}^2}\right|\left|b\left(x\right)-b\left(y\right)\right|\left|b_j\left(y\right)\right|\text{d}y.\end{array}$

Similarly, we consider ${\Upsilon }_2$. Noting that $\left|x-y\right|\sim \left|x\right|$. By the Minkowski’s inequality, the generalized Hölder’s inequality and the vanishing moments of $b_j$ we have

$\begin{array}{c}{\Upsilon }_2\le C{\displaystyle {\int }_{{ℝ}^n}}\left|\frac{\left|\Omega \left(x-y\right)\right|}{{\left|x-y\right|}^{n-1}}-\frac{\left|\Omega \left(x\right)\right|}{{\left|x\right|}^{n-2}}\right|\left|b\left(x\right)-b\left(y\right)\right|\left|b_j\left(y\right)\right|{\left({\displaystyle {\int }_{\left|x\right|}^{\infty }}\frac{\text{d}t}{t^3}\right)}^{1/2}\text{d}y\\ \le C{\displaystyle {\int }_{B_j}}\left|\frac{\left|\Omega \left(x-y\right)\right|}{{\left|x-y\right|}^2}-\frac{\left|\Omega \left(x\right)\right|}{{\left|x\right|}^2}\right|\left|b\left(x\right)-b\left(y\right)\right|\left|b_j\left(y\right)\right|\text{d}y.\end{array}$

So we have

$\left|\left[b\mathrm{,}{\mu }_{\Omega }\right]\left(b_j\right)\left(x\right)\right|\le C{\displaystyle {\int }_{B_j}}\left|\frac{\left|\Omega \left(x-y\right)\right|}{{\left|x-y\right|}^n}-\frac{\left|\Omega \left(x\right)\right|}{{\left|x\right|}^n}\right|\left|b\left(x\right)-b\left(y\right)\right|\left|b_j\left(y\right)\right|\text{d}y\mathrm{.}$

From lemma 2.10 and the Minkowski’s inequality we have

$\begin{array}{l}{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\\ \le C{\displaystyle {\int }_{B_j}}{\Vert \left|\frac{\left|\Omega \left(x-y\right)\right|}{{\left|x-y\right|}^n}-\frac{\left|\Omega \left(x\right)\right|}{{\left|x\right|}^n}\right|\left|b\left(x\right)-b\left(y\right)\right|{\chi }_k(\cdot )\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\left|b_j\left(y\right)\right|\text{d}y\\ \le C{\displaystyle {\int }_{B_j}}{\Vert \left|\frac{\left|\Omega \left(\cdot -y\right)\right|}{{\left|\cdot -y\right|}^n}-\frac{\left|\Omega (\cdot )\right|}{{\left|\cdot \right|}^n}\right|\left|b(\cdot )-b\left(0\right)\right|{\chi }_k(\cdot )\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\left|b_j\left(y\right)\right|\\ +C{\displaystyle {\int }_{B_j}}{\Vert \left|\frac{\left|\Omega \left(\cdot -y\right)\right|}{{\left|\cdot -y\right|}^n}-\frac{\left|\Omega (\cdot )\right|}{{\left|\cdot \right|}^n}\right|{\chi }_k(\cdot )\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\left|b\left(0\right)-b\left(y\right)\right|\left|b_j\left(y\right)\right|\\ :={\Upsilon }_1^{\ast }+{\Upsilon }_2^{\ast }.\end{array}$

For ${\Upsilon }_1^{\ast }$, noting $s>p^{\prime }$, we denote ${\tilde{p}}^{\prime }(\cdot )>1$ and $\frac{1}{p\left(x\right)}=\frac{1}{{\tilde{p}}^{\prime }\left(x\right)}+\frac{1}{s}$. By lemma 2.2 we have

$\begin{array}{l}{\Vert \left|\frac{\left|\Omega \left(\cdot -y\right)\right|}{{\left|\cdot -y\right|}^n}-\frac{\left|\Omega (\cdot )\right|}{{\left|\cdot \right|}^n}\right|\left|b(\cdot )-b\left(0\right)\right|{\chi }_k(\cdot )\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\\ \le {\Vert \left|\frac{\left|\Omega \left(\cdot -y\right)\right|}{{\left|\cdot -y\right|}^n}-\frac{\left|\Omega (\cdot )\right|}{{\left|\cdot \right|}^n}\right|{\chi }_k(\cdot )\Vert }_{L^{s(\cdot )}\left({ℝ}^n\right)}{\Vert \left|b\left(0\right)-b\left(y\right)\right|\left|b_j\left(y\right)\right|\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}{2}^{k\gamma }{\Vert \left|\frac{\left|\Omega \left(\cdot -y\right)\right|}{{\left|\cdot -y\right|}^n}-\frac{\left|\Omega (\cdot )\right|}{{\left|\cdot \right|}^n}\right|{\chi }_k(\cdot )\Vert }_{L^{s(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_{B_K}\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}.\end{array}$

When $\left|B_k\right|\le 2^n$ and $x_k\in B_k$, by Lemma 2.12 we have

${\Vert {\chi }_{B_K}\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}\simeq {\left|B_k\right|}^{\frac{1}{p^{\prime }\left(x_k\right)}}\approx {\Vert {\chi }_{B_K}\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\left|B_k\right|}^{\frac{-1}{s}-\frac{-\gamma }{n}}\mathrm{.}$

When $\left|B_k\right|\ge 1$ we have

${\Vert {\chi }_{B_K}\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}\simeq {\left|B_k\right|}^{\frac{1}{p^{\prime }\left(\infty \right)}}\approx {\Vert {\chi }_{B_K}\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\left|B_k\right|}^{\frac{-1}{s}-\frac{-\gamma }{n}}\mathrm{.}$

So we obtain

${\Vert {\chi }_{B_K}\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}\approx {\Vert {\chi }_{B_K}\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\left|B_k\right|}^{\frac{-1}{s}-\frac{-\gamma }{n}}\mathrm{.}$

By lemma 2.9 we have

$\begin{array}{l}{\Vert \left|\frac{\left|\Omega \left(\cdot -y\right)\right|}{{\left|\cdot -y\right|}^n}-\frac{\left|\Omega (\cdot )\right|}{{\left|\cdot \right|}^n}\right|{\chi }_k(\cdot )\Vert }_{L^{s(\cdot )}\left({ℝ}^n\right)}\\ \le 2^{\left(k-1\right)\left(\frac{n}{s}-n\right)}\left\{\frac{\left|y\right|}{2^k}+{\displaystyle {\int }_{\frac{\left|y\right|}{2^k}}^{\frac{\left|y\right|}{2^{k-1}}}}\frac{{\omega }_s\left(\delta \right)}{\delta }\text{d}\delta \right\}\\ \le 2^{\left(k-1\right)\left(\frac{n}{s}-n\right)}\left\{2^{j-k+1}+2^{\left(j-k+1\right)\gamma }{\displaystyle {\int }_0^1}\frac{{\omega }_s\left(\delta \right)}{\delta }\text{d}\delta \right\}\\ \le 2^{\left(k-1\right)\left(\frac{n}{s}-n\right)}{2}^{\left(j-k\right)\gamma }\mathrm{.}\end{array}$

Now, by using the generalized Hölder’s inequality we get:

$\begin{array}{c}{\Upsilon }_1^{*}\le {\displaystyle {\int }_{B_j}}{\Vert \left|\frac{\left|\Omega \left(\cdot -y\right)\right|}{{\left|\cdot -y\right|}^n}-\frac{\left|\Omega (\cdot )\right|}{{\left|\cdot \right|}^n}\right|\left|b(\cdot )-b\left(y\right)\right|{\chi }_k(\cdot )\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\left|b_j\left(y\right)\right|\text{d}y\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}{2}^{-kn+\left(j-k\right)\gamma -k\gamma }{\Vert {\chi }_{B_K}\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\left|B_k\right|}^{\frac{-1}{s}-\frac{-\gamma }{n}}{\displaystyle {\int }_{B_j}}\left|b_j\left(y\right)\right|\text{d}y\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}{2}^{-kn+\left(j-k\right)\gamma -k\gamma }{\Vert {\chi }_{B_K}\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\Vert b_j\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_{B_j}\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}.\end{array}$ (3.2)

For ${\Upsilon }_2^{\mathrm{<mo>\; *\; </mo>}}$ similar to the method of ${\Upsilon }_1^{\mathrm{<mo>\; *\; </mo>}}$ we have

$\begin{array}{l}{\Vert \left|\frac{\left|\Omega \left(\cdot -y\right)\right|}{{\left|\cdot -y\right|}^n}-\frac{\left|\Omega (\cdot )\right|}{{\left|\cdot \right|}^n}\right|{\chi }_k(\cdot )\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\\ \le {\Vert \left|\frac{\left|\Omega \left(\cdot -y\right)\right|}{{\left|\cdot -y\right|}^n}-\frac{\left|\Omega (\cdot )\right|}{{\left|\cdot \right|}^n}\right|{\chi }_k(\cdot )\Vert }_{L^{s(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_k(\cdot )\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}\\ \le {\Vert \left|\frac{\left|\Omega \left(\cdot -y\right)\right|}{{\left|\cdot -y\right|}^n}-\frac{\left|\Omega (\cdot )\right|}{{\left|\cdot \right|}^n}\right|{\chi }_k(\cdot )\Vert }_{L^{s(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_{B_k}(\cdot )\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}\\ \le 2^{\left(k-1\right)\left(\frac{n}{s}-n\right)}{2}^{\left(j-k\right)\gamma }{\Vert {\chi }_{B_K}\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}\\ \le 2^{-kn+\left(j-k\right)\gamma -k\gamma }{\Vert {\chi }_{B_k}\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}.\end{array}$

Now, by using the generalized Hölder’s inequality we get:

$\begin{array}{c}{\Upsilon }_2^{*}\le {\displaystyle {\int }_{B_j}}{\Vert \left|\frac{\left|\Omega \left(\cdot -y\right)\right|}{{\left|\cdot -y\right|}^n}-\frac{\left|\Omega (\cdot )\right|}{{\left|\cdot \right|}^n}\right|{\chi }_k(\cdot )\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\left|b\left(0\right)-b\left(y\right)\right|\left|b_j\left(y\right)\right|\text{d}y\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}{2}^{-kn+2\left(j-k\right)\gamma }{\Vert {\chi }_{B_k}\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\Vert b_j\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_{B_j}\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}{2}^{-kn+\left(j-k\right)\gamma }{\Vert {\chi }_{B_k}\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\Vert b_j\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_{B_j}\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}.\end{array}$ (3.3)

Now by (3.3), (3.4), and lemmas 2.5 and 2.6, we have

$\begin{array}{l}{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}{2}^{-kn+\left(j-k\right)\gamma }{\Vert {\chi }_{B_k}\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\Vert b_j\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_{B_j}\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}{2}^{\left(j-k\right)\gamma }{\Vert b_j\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\frac{{\Vert {\chi }_{B_j}\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}}{{\Vert {\chi }_{B_k}\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}}\\ \le C{2}^{-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)}{\Vert b\Vert }_{Li{p}_{\gamma }}.\end{array}$

Firstly we estimate I. We need to show that there exists a positive constant C, such that $I\le C\Delta$, we consider

$\begin{array}{c}I=\underset{L\ge 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}^q\\ \le \underset{L\ge 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=K}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]{\chi }_k\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\right)}^q\\ +\underset{L\ge 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]{\chi }_k\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\right)}^q\\ :=I_1+I_2.\end{array}$

By the $\left(L^{p(\cdot )}\left({ℝ}^n\right)\mathrm{,}{L}^{q(\cdot )}\left({ℝ}^n\right)\right)$, bounbedness of the commutator $\left[b\mathrm{,}{\mu }_{\Omega }\right]$ on $L^{p(\cdot )}$ (see [15] ), we have the following. Therefore, when $0<q\le 1$

$\begin{array}{c}{I}_1=\underset{L\ge 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=K}{\overset{\infty }{\sum }}}{\Vert \left[b,{\mu }_{\Omega }\right]{\chi }_k\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\right)}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|2^{-j{\alpha }_j}\right)}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}\left({\displaystyle \underset{j=k}{\overset{-1}{\sum }}}\left|{\lambda }_j\right|2^{-j\alpha \left(0\right)jq}+{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|2^{-j{\alpha }_{\infty }q}\right)\\ \le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{\displaystyle \underset{j=k}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\alpha \left(0\right)\left(k-j\right)q}\\ +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-j{\alpha }_{\infty }q}\end{array}$

$\begin{array}{l}\le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q}\\ +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{-j\lambda q}{\left|{\lambda }_j\right|}^q{2}^{\left(\lambda -{\alpha }_{\infty }\right)jq}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha (\; 0\; )}\end{array}$

$\begin{array}{l}\le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{L}{\sum }}}{\left|{\lambda }_j\right|}^q+\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{j=L}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q}\\ +\Delta \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{\left(\lambda -{\alpha }_{\infty }\right)jq}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{\left(\alpha \left(0\right)k-L\lambda \right)q}\\ \le \Delta +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{\displaystyle \underset{j=L}{\overset{-1}{\sum }}}{2}^{-j\lambda q}{\left|{\lambda }_j\right|}^q{2}^{\left(j-L\right)\lambda q}{\displaystyle \underset{k=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q}+\Delta \\ \le \Delta +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{\displaystyle \underset{j=L}{\overset{-1}{\sum }}}{2}^{-j\lambda q}{\left|{\lambda }_j\right|}^q{2}^{\left(j-L\right)\lambda q}{\displaystyle \underset{k=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q}\\ \le \Delta .\end{array}$ (3.4)

When $0<q<\infty$, let $1/q+1/q^{\prime }=1$ we have

$\begin{array}{c}{I}_1=\underset{L\ge 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=K}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]{\chi }_k\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\right)}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|2^{-j\alpha }\right)}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{\left({\displaystyle \underset{j=k}{\overset{-1}{\sum }}}\left|{\lambda }_j\right|2^{\alpha \left(0\right)\left(k-j\right)}\right)}^q\\ +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|2^{-j{\alpha }_{\infty }}\right)}^q\end{array}$

$\begin{array}{l}\le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}\left({\displaystyle \underset{j=k}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\alpha \left(0\right)\left(k-j\right)q/2}\right){\left({\displaystyle \underset{j=k}{\overset{-1}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q^{\prime }/2}\right)}^{q/q^{\prime }}\\ +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}\left({\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-j{\alpha }_{\infty }q/2}\right){\left({\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{-j{\alpha }_{\infty }{q}^{\prime }/2}\right)}^{q/q^{\prime }}\\ \le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{\displaystyle \underset{j=k}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left(j-k\right)q/2}\\ +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-j{\alpha }_{\infty }q/2}\end{array}$

$\begin{array}{l}\le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(j-k\right)q/2}\\ +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{j\lambda q}{\left|{\lambda }_j\right|}^q{2}^{\left(\lambda -{\alpha }_{\infty }/2\right)jq}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}\\ \le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{L}{\sum }}}{\left|{\lambda }_j\right|}^q+\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{j=L}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{j=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q/2}\\ +\Delta \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{\left(\lambda -{\alpha }_{\infty }/2\right)j/2q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)-L\lambda q}\end{array}$

$\begin{array}{l}\le \Delta +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{\displaystyle \underset{j=L}{\overset{-1}{\sum }}}{2}^{-j\lambda q}{\left|{\lambda }_j\right|}^q{2}^{\left(j-L\right)\lambda q}{\displaystyle \underset{j=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q/2}+\Delta \\ \le \Delta +\Delta \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{\displaystyle \underset{j=L}{\overset{-1}{\sum }}}{2}^{\left(j-L\right)\lambda q}{\displaystyle \underset{j=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q/2}\\ \le \Delta .\end{array}$ (3.5)

We estimate $I_2$ by lemma 2.1 when $0<q\le 1$ by $n{\delta }_2\le \alpha \left(0\right)<\gamma +n{\delta }_2$, we get

$\begin{array}{c}{I}_2=\underset{L\ge 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]{\chi }_k\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\right)}^q\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\ge 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}{2}^{-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)}\left|{\lambda }_j\right|\right)}^p\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\ge 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}{\left|{\lambda }_j\right|}^p{2}^{\left(-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right)q}\right)\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\ge 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{L}{\sum }}}{\left|{\lambda }_j\right|}^p{\displaystyle \underset{k=j+1}{\overset{-1}{\sum }}}{2}^{q\left(j-k\right)\left[\gamma +n{\delta }_2-\alpha \left(0\right)\right]}\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\Delta .\end{array}$ (3.6)

When $0<q<\infty$, let $1/q+1/q^{\prime }=1$. Since $n{\delta }_2\le \alpha \left(0\right)<\gamma +n{\delta }_2$, by Hölder’s inequality, we have

$\begin{array}{c}{I}_2=\underset{L\ge 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]{\chi }_k\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\right)}^q\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\ge 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}{2}^{-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)}\left|{\lambda }_j\right|\right)}^p\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\ge 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}{\left|{\lambda }_j\right|}^p{2}^{\left(-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right)q/2}\right)\\ \times {\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}{2}^{\left(-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right)q^{\prime }/2}\right)}^{q/q^{\prime }}\end{array}$

$\begin{array}{l}\le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\ge 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}{\left|{\lambda }_j\right|}^p{2}^{\left(-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right)q/2}\right)\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\ge 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{L}{\sum }}}{\left|{\lambda }_j\right|}^p{\displaystyle \underset{k=j+1}{\overset{-1}{\sum }}}{2}^{q/2\left(j-k\right)\left[\gamma +n{\delta }_2-\alpha \left(0\right)\right]}\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>\; <mtext>\; \hspace{0.17em}\; </mtext>}\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\Delta .\end{array}$ (3.7)

Secondly we estimate $II$. We need to show that there exists a positive constant C, such that $II\le C\Delta$, we consider

$\begin{array}{c}II={\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q\\ \le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}\right)}^q\\ +{\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}\right)}^q\\ :=I{I}_1+I{I}_2.\end{array}$

When $0<q\le 1$, we get

$\begin{array}{c}I{I}_1={\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}\right)}^q\\ \le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}{2}^{-j{\alpha }_j}\left|{\lambda }_j\right|\right)}^p\end{array}$

$\begin{array}{l}\le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}\left({\displaystyle \underset{j=k}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-jq\alpha \left(0\right)}+{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-jq{\alpha }_{\infty }}\right)\\ \le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{\displaystyle \underset{j=k}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\alpha \left(0\right)\left(k-j\right)q}+{\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-j{\alpha }_{\infty }q}\end{array}$

$\begin{array}{l}\le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=\infty }{\overset{j}{\sum }}}{2}^{q\alpha \left(0\right)\left(k-j\right)}+{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-jq{\alpha }_{\infty }}{\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}\\ \le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q+{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{-j\lambda q}{\left|{\lambda }_j\right|}^q{2}^{-jq{\alpha }_{\infty }}{\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}\\ \le \Delta +\Delta {\displaystyle \underset{i=-\infty }{\overset{j}{\sum }}}{\left|{\lambda }_i\right|}^q{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{\left(\lambda -{\alpha }_{\infty }\right)jq}{\displaystyle \underset{k=-\infty }{\overset{j}{\sum }}}{2}^{kq\alpha \left(0\right)}\\ \le \Delta .\end{array}$ (3.8)

When $0<q<\infty$, let $1/q_1+1/{q^{\prime }}_1=1$ we have

$\begin{array}{c}I{I}_1={\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}\right)}^q\\ \le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|2^{-j\alpha }\right)}^q\\ \le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{\left({\displaystyle \underset{j=k}{\overset{-1}{\sum }}}\left|{\lambda }_j\right|2^{\alpha \left(0\right)\left(j-k\right)}\right)}^q+{\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|2^{-j{\alpha }_{\infty }}\right)}^q\\ \le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}\left({\displaystyle \underset{j=k}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{q/2\alpha \left(0\right)\left(j-k\right)}\right){\left({\displaystyle \underset{j=k}{\overset{-1}{\sum }}}{2}^{\alpha \left(0\right)\left(j-k\right)q^{\prime }/2}\right)}^{q/q^{\prime }}\\ +{\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}\left({\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-q/2j{\alpha }_{\infty }}\right){\left({\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{-q^{\prime }/2j{\alpha }_{\infty }}\right)}^{q/q^{\prime }}\end{array}$

$\begin{array}{l}\le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=-\infty }{\overset{j}{\sum }}}{2}^{q/2\alpha \left(0\right)\left(j-k\right)}+{\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-q/2j{\alpha }_{\infty }}\\ \le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q+{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{\left(\lambda -{\alpha }_{\infty }/2\right)jq}{2}^{-\lambda jq}{\displaystyle \underset{i=-\infty }{\overset{j}{\sum }}}{\left|{\lambda }_i\right|}^q{\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}\\ \le \Delta +\Delta {\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{\left(\lambda -{\alpha }_{\infty }/2\right)jq}{\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}\\ \le \Delta .\end{array}$ (3.9)

For $I{I}_2$, when $0<q\le 1$, by $n{\delta }_2\le \alpha \left(0\right)<\gamma +n{\delta }_2$ we get

$\begin{array}{c}I{I}_2={\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=\infty }{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}\right)}^q\\ \le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left(C{\Vert b\Vert }_{Li{p}_{\gamma }}{\displaystyle \underset{j=\infty }{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|2^{-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)}\right)}^q\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q{\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}\left({\displaystyle \underset{j=\infty }{\overset{k-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left[-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right]q}\right)\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q{\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=j+1}{\overset{-1}{\sum }}}{2}^{\left[-j{\alpha }_{\infty }+\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right]q}\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\Delta .\end{array}$ (3.10)

When $1<q<\infty$, let $1/q+1/q^{\prime }=1$. Since $n{\delta }_2\le \alpha \left(0\right)<\gamma +n{\delta }_2$, by Hölder’s inequality, we have

$\begin{array}{c}I{I}_2={\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=\infty }{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}\right)}^q\\ \le {\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left(C{\Vert b\Vert }_{Li{p}_{\gamma }}{\displaystyle \underset{j=\infty }{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|2^{-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)}\right)}^q\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q{\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}\left({\displaystyle \underset{j=\infty }{\overset{k-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left[-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right]q/2}\right)\\ \times {\left({\displaystyle \underset{j=\infty }{\overset{k-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left[-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right]q^{\prime }/2}\right)}^{q/q^{\prime }}\end{array}$

$\begin{array}{l}\le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q{\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}\left({\displaystyle \underset{j=\infty }{\overset{k-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left[-j\alpha +\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right]q/2}\right)\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q{\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=j+1}{\overset{-1}{\sum }}}{2}^{\left(j-k\right)\left[\gamma +n{\delta }_2-\alpha \left(0\right)\right]q/2}\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\Delta .\end{array}$ (3.11)

Thirdly, we estimate $III$, we need to show that there exists a positive constant C, such that $III\le C\Delta$

$\begin{array}{c}III=\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}\right)}^q\\ +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\left({\displaystyle \underset{j=\infty }{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]L^{p(\cdot )}{\chi }_k\Vert }_{L^{p(\cdot )}}\right)}^q\\ :=II{I}_1+II{I}_2.\end{array}$

When $0<q\le 1$, by the boundedness of $\left[b\mathrm{,}{\mu }_{\Omega }\right]$ in $L^{p(\cdot )}$ ( [20] ), we have

$\begin{array}{c}II{I}_1=\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq{\alpha }_{\infty }}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}\right)}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq{\alpha }_{\infty }}{\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq{\alpha }_{\infty }}{\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-{\alpha }_{j}jq}\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq{\alpha }_{\infty }}{\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-{\alpha }_{\infty }jq}\\ =\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=0}{\overset{L}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=0}{\overset{j}{\sum }}}{2}^{{\alpha }_{\infty }\left(k-j\right)q}+\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=L}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{{\alpha }_{\infty }\left(k-j\right)q}\end{array}$

$\begin{array}{l}\le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{\left|{\lambda }_j\right|}^q+\underset{L\le 0,L\in \mathbb{Z}}{\sup }{\displaystyle \underset{j=L}{\overset{\infty }{\sum }}}{2}^{\left(j-L\right)q\lambda }{2}^{-j\lambda q}{\displaystyle \underset{i=-\infty }{\overset{j}{\sum }}}{\left|{\lambda }_i\right|}^q{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{{\alpha }_{\infty }\left(k-j\right)q}\\ \le \Delta +\Delta \underset{L\le 0,L\in \mathbb{Z}}{\sup }{\displaystyle \underset{j=L}{\overset{\infty }{\sum }}}{2}^{\left(j-L\right)q\lambda }{2}^{-j\lambda q}{2}^{{\alpha }_{\infty }\left(L-j\right)q}\\ \le \Delta +\Delta \underset{L\le 0,L\in \mathbb{Z}}{\sup }{\displaystyle \underset{j=L}{\overset{\infty }{\sum }}}{2}^{\left(j-L\right)q\left(\lambda -{\alpha }_{\infty }\right)}\\ \le \Delta .\end{array}$ (3.12)

When $0<q\le \infty$, by $n{\delta }_2\le \alpha \left(0\right),{\alpha }_{\infty }<\gamma +n{\delta }_2$ and the boundedness of $\left[b\mathrm{,}{\mu }_{\Omega }\right]$ in $L^{p(\cdot )}$ ( [20] ) and Hölder’s inequality, we get

$\begin{array}{c}II{I}_1=\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}\right)}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq{\alpha }_{\infty }}\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}^{q/2}\right)\\ \times {\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}^{q^{\prime }/2}\right)}^{q/q^{\prime }}\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq{\alpha }_{\infty }}\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{\Vert b_j\Vert }_{L^{p(\cdot )}}^{q/2}\right){\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}{\Vert b_j\Vert }_{L^{p(\cdot )}}^{q^{\prime }/2}\right)}^{q/q^{\prime }}\end{array}$

$\begin{array}{l}\le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq{\alpha }_{\infty }}\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{\left|B_j\right|}^{-{\alpha }_{j}q/\left(2n\right)}\right){\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}{\left|B_j\right|}^{-{\alpha }_j{q}^{\prime }/\left(2n\right)}\right)}^{q/q^{\prime }}\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq{\alpha }_{\infty }}\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{\left|B_j\right|}^{-{\alpha }_{j}q/\left(2n\right)}\right)\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{{\alpha }_{\infty }kq/2}\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{\left|B_j\right|}^{-{\alpha }_{j}q/\left(2n\right)}\right)\\ =\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=0}{\overset{L}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=0}{\overset{j}{\sum }}}{2}^{\left(k-j\right){\alpha }_{\infty }q/2}+\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=L}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{\left(k-j\right){\alpha }_{\infty }q/2}\end{array}$

$\begin{array}{l}\le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=0}{\overset{L}{\sum }}}{\left|{\lambda }_j\right|}^q+\underset{L\le 0,L\in \mathbb{Z}}{\sup }{\displaystyle \underset{j=L}{\overset{\infty }{\sum }}}{2}^{\left(j-L\right)\lambda q}{2}^{-j\lambda q}{\displaystyle \underset{i=-\infty }{\overset{j}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{\left(k-j\right){\alpha }_{\infty }q/2}\\ \le \Delta +\Delta \underset{L\le 0,L\in \mathbb{Z}}{\sup }{\displaystyle \underset{j=L}{\overset{\infty }{\sum }}}{2}^{\left(j-L\right)\lambda q}{2}^{\left(L-j\right){\alpha }_{\infty }q/2}\\ \le \Delta +\Delta \underset{L\le 0,L\in \mathbb{Z}}{\sup }{\displaystyle \underset{j=L}{\overset{\infty }{\sum }}}{2}^{\left(j-L\right)q\left(\alpha -{\alpha }_{\infty }/2\right)}\\ \le \Delta .\end{array}$ (3.13)

When $0<q\le 1$, by $n{\delta }_2\le \alpha \left(0\right),{\alpha }_{\infty }<\gamma +n{\delta }_2$ we get

$\begin{array}{c}II{I}_2=\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq{\alpha }_{\infty }}{\left({\displaystyle \underset{j=\infty }{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}\right)}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq{\alpha }_{\infty }}\left(C{\Vert b\Vert }_{Li{p}_{\gamma }}^q{\displaystyle \underset{j=\infty }{\overset{k-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left[-j{\alpha }_j+\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right]q}\right)\\ =C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq{\alpha }_{\infty }}\left({\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left[-j\alpha \left(0\right)+\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right]q}\right)\\ +C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha (\infty )}\left({\displaystyle \underset{j=0}{\overset{k-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left[-j{\alpha }_{\infty }+\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right]q}\right)\end{array}$

$\begin{array}{l}\le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\left[{\alpha }_{\infty }+\gamma +n{\delta }_2\right]}{\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left[\gamma +n{\delta }_2+\alpha \left(0\right)\right]jq}\\ +C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=0}{\overset{L}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=j+1}{\overset{\infty }{\sum }}}{2}^{\left[\gamma +n{\delta }_2-{\alpha }_{\infty }\right]\left(j-k\right)q}\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q+C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=0}{\overset{L-1}{\sum }}}{\left|{\lambda }_j\right|}^q\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\Delta .\end{array}$ (3.14)

When $1<q<\infty$, let $1/q+1/q^{\prime }=1$. Since $n{\delta }_2\le \alpha \left(0\right),{\alpha }_{\infty }<\gamma +n{\delta }_2$, and by Hölder’s inequality, we have

$\begin{array}{c}II{I}_2=\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}\right)}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\left({\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\left|{\lambda }_j\right|2^{\left[-j{\alpha }_j+\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right]}\right)}^q\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\left({\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}\left|{\lambda }_j\right|2^{\left[-j\alpha \left(0\right)+\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right]}\right)}^q\\ +C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\left({\displaystyle \underset{j=0}{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|2^{\left[-j{\alpha }_{\infty }+\left(j-k\right)\left(\gamma +n{\delta }_2\right)\right]}\right)}^q\end{array}$

$\begin{array}{l}\le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\left[{\alpha }_{\infty }-\left(\gamma +n{\delta }_2\right)\right]}{\left({\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}\left|{\lambda }_j\right|2^{\left[\left(\gamma +n{\delta }_2\right)-\alpha \left(0\right)\right]j}\right)}^q\\ +C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{\left({\displaystyle \underset{j=0}{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|2^{\left(j-k\right)\left[\gamma +n{\delta }_2-{\alpha }_{\infty }\right]}\right)}^q\\ \le \left(C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left[\left(\gamma +n{\delta }_2\right)-\alpha \left(0\right)\right]jq/2}\right)\\ \times {\left({\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left[\left(\gamma +n{\delta }_2\right)-\alpha \left(0\right)\right]j{q}^{\prime }/2}\right)}^{q/q^{\prime }}\end{array}$

$\begin{array}{l}+C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}\left({\displaystyle \underset{j=0}{\overset{k-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left(j-k\right)\left[\gamma +n{\delta }_2-{\alpha }_{\infty }\right]q/2}\right)\\ \times {\left({\displaystyle \underset{j=0}{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|2^{\left(j-k\right)\left[\gamma +n{\delta }_2-{\alpha }_{\infty }\right]q^{\prime }/2}\right)}^{q\mathrm{<mo>\; /\; </mo>}{q}^{\prime }}\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left[\left(\gamma +n{\delta }_2\right)-\alpha \left(0\right)\right]jq/2}\\ +C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{\displaystyle \underset{j=0}{\overset{k-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left(j-k\right)\left[\gamma +n{\delta }_2-{\alpha }_{\infty }\right]q/2}\end{array}$

$\begin{array}{l}\le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q\\ +C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=0}{\overset{L-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=j+1}{\overset{L}{\sum }}}{2}^{\left(j-k\right)\left[\gamma +n{\delta }_2-{\alpha }_{\infty }\right]q/2}\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q+C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=0}{\overset{L-1}{\sum }}}{\left|{\lambda }_j\right|}^q\\ \le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q\Delta .\end{array}$ (3.15)

Joint the estimates for I, II and III, we obtain

${\Vert \left[b\mathrm{,}{\mu }_{\Omega }\right]\left(f\right)\Vert }_{M{K}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)}^q\le C{\Vert b\Vert }_{Li{p}_{\gamma }}^q{\Vert f\Vert }_{HM{K}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}}\mathrm{.}$

Then we complete the proof of Theorem 3.1.

4. BMO Boundedness for the Commutator of Marcikiewicz Integrals Operator

In this section, we prove the boundedness of the commutator of Marcikiewicz integrals on Herz-Morrey-Hrdy spaces with variable exponent with function $b\in BMO\left({ℝ}^n\right)$.

Theorem 4.1.

Suppose that $b\in BMO\left({ℝ}^n\right)$ with $0<\gamma \le 1$. If $p(\cdot )\in \mathcal{P}\left({ℝ}^n\right)$ satisfies proposition 2.3 and $\Omega \in L^s\left(S^{n-1}\right)\left(s>{q^{\prime }}^{-}\right)$. Let $0<p_1\le p_2<\infty$ and

$0<\lambda <\alpha <n{\delta }_2-\gamma -\frac{n}{s}$ (or $0<\lambda <{\alpha }_1\le {\alpha }_1<n{\delta }_2-\gamma -\frac{n}{s}$ ). Then $\left[b\mathrm{,}{\mu }_{\Omega }\right]$ is bounded from $HM{\stackrel{\dot{}}{K}}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ (or $HM{K}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ ) to $M{\stackrel{\dot{}}{K}}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ (or $M{K}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$ ).

proof:

In a way similar to theorem (3.2) we only prove the homogeneous case. Let $b\in BMO\left({ℝ}^n\right)$ and $f\in HM{K}_{p(\cdot ),\lambda }^{\alpha (\cdot )\mathrm{,}q}\left({ℝ}^n\right)$. Let us write

$f\left(x\right)={\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}f\left(x\right){\chi }_j\left(x\right)={\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{f}_j\left(x\right).$

Then we have

$\begin{array}{c}{\Vert \left[b\mathrm{,}{\mu }_{\Omega }\right]\left(f\right)\Vert }_{M{K}_{p(\cdot ),\lambda }^{\alpha (\cdot ),q}\left({ℝ}^n\right)}^q=\max \{\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q,\\ \underset{L\le 0,L\in Z}{{\displaystyle \sup }}{2}^{-L\lambda q}({\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q\\ +{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q)\}\\ :=\max \left\{H,HH+HHH\right\}.\end{array}$

$\begin{array}{l}H=\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q,\\ HH={\displaystyle \underset{k=-\infty }{\overset{-1}{\sum }}}{2}^{kq\alpha \left(0\right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q,\\ HHH=\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=0}{\overset{L}{\sum }}}{2}^{kq\alpha \left(\infty \right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q.\end{array}$

From the Hölder’s inequality, we have

$\begin{array}{l}{\left|\left[b,{\mu }_{\Omega }\right]\left(b_j\right){\chi }_k\right|}_{L^{p(\cdot )}\left({ℝ}^n\right)}\\ \le C{\displaystyle {\int }_{B_j}}\frac{\left|\Omega \left(x-y\right)\right|}{{\left|x-y\right|}^n}\left|b\left(x\right)-b\left(y\right)\right|\left|f_j\left(y\right)\right|\text{d}y\\ \le C{2}^{-kn}{\displaystyle {\int }_{B_j}}\left|\Omega \left(x-y\right)\right|\left|b\left(x\right)-b\left(y\right)\right|\left|f_j\left(y\right)\right|\text{d}y\end{array}$

$\begin{array}{l}\le C{2}^{-kn}(\left|b\left(x\right)-b_{B_j}\right|{\displaystyle {\int }_{B_j}}\left|\Omega \left(x-y\right)\right|\left|f_j\left(y\right)\right|\text{d}y\\ +{\displaystyle {\int }_{B_j}}\left|\Omega \left(x-y\right)\right|\left|b_{B_j}-b\left(y\right)\right|\left|f_j\left(y\right)\right|\text{d}y)\\ \le C{2}^{-kn}(\left|b\left(x\right)-b_{B_j}\right|{\Vert \Omega \left(x-\cdot \right){\chi }_j(\cdot )\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}{\Vert f_j\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\text{d}y\\ +{\Vert \Omega \left(x-\cdot \right)\left(b_{B_j}-b(\cdot )\right){\chi }_j(\cdot )\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}{\Vert f_j\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\text{d}y).\end{array}$

Noting $s>{q^{\prime }}^{-}$, we denote ${\tilde{q}}^{\prime }(\cdot )>1$ and $\frac{1}{q^{\prime }\left(x\right)}=\frac{1}{{\tilde{q}}^{\prime }\left(x\right)}+\frac{1}{s}$. By lemmas 3.2, 3.10 we have

$\begin{array}{c}{\Vert \Omega \left(x-\cdot \right){\chi }_j(\cdot )\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}\le {\Vert \Omega \left(x-\cdot \right){\chi }_j(\cdot )\Vert }_{L^{s(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_j(\cdot )\Vert }_{L^{{{\tilde{q}}^{\prime }}_2(\cdot )}\left({ℝ}^n\right)}\\ \le {\Vert \Omega \left(x-\cdot \right){\chi }_j(\cdot )\Vert }_{L^{s(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_{B_j}(\cdot )\Vert }_{L^{{{\tilde{q}}^{\prime }}_2(\cdot )}\left({ℝ}^n\right)}\\ \le 2^{-j\gamma }{\left({\displaystyle {\int }_{A_j}}{\left|y\right|}^{s\gamma }{\left|\Omega \left(x-y\right)\right|}^s\text{d}y\right)}^{\frac{1}{s}}{\Vert {\chi }_{B_j}(\cdot )\Vert }_{L^{{{\tilde{q}}^{\prime }}_2(\cdot )}\left({ℝ}^n\right)}\\ \le C{2}^{-j\gamma }{\left|2\right|}^{k\left(\gamma +\frac{n}{s}\right)}{\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}{\Vert {\chi }_{B_j}(\cdot )\Vert }_{L^{{{\tilde{q}}^{\prime }}_2(\cdot )}\left({ℝ}^n\right)}.\end{array}$

By lemma (2.12), when $\left|B_j\right|\le 2^n\mathrm{,}{x}_j\in B_j$ and when $\left|B_k\right|\ge 1$ respectively we have

${\Vert {\chi }_{B_j}\Vert }_{L^{{\tilde{q}}^{\prime }(\cdot )}\left({ℝ}^n\right)}\approx {\left|B_k\right|}^{\frac{1}{{\tilde{q}}^{\prime }\left(x_k\right)}}\approx {\Vert {\chi }_{B_j}\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}{\left|B_j\right|}^{\frac{-1}{s}}$,

and

${\Vert {\chi }_{B_j}\Vert }_{L^{{\tilde{q}}^{\prime }(\cdot )}\left({ℝ}^n\right)}\approx {\left|B_j\right|}^{\frac{1}{{\tilde{q}}^{\prime }\left(\infty \right)}}\approx {\Vert {\chi }_{B_j}\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}{\left|B_j\right|}^{\frac{-1}{s}},$

we obtain ${\Vert {\chi }_{B_j}\Vert }_{L^{{\tilde{q}}^{\prime }(\cdot )}\left({ℝ}^n\right)}\approx {\Vert {\chi }_{B_j}\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}{\left|B_j\right|}^{\frac{-1}{s}}$.

So we have

${\Vert \Omega \left(x-\cdot \right){\chi }_j(\cdot )\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}\le C{2}^{\left(k-j\right)\left(\gamma +\frac{n}{s}\right)}{\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}{\Vert {\chi }_{B_j}(\cdot )\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}\mathrm{.}$ (4.1)

Similarly by lemma 2.4 we have

$\begin{array}{l}{\Vert \Omega \left(x-\cdot \right)\left(b_{B_j}-b(\cdot )\right){\chi }_j(\cdot )\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}\\ \le {\Vert \Omega \left(x-\cdot \right){\chi }_j(\cdot )\Vert }_{L^s\left({ℝ}^n\right)}{\Vert \left(b_{B_j}-b(\cdot )\right){\chi }_j(\cdot )\Vert }_{L^{{\tilde{q}}^{\prime }(\cdot )}\left({ℝ}^n\right)}\\ \le C{\Vert b\Vert }_{\ast }{\Vert {\chi }_{B_j}(\cdot )\Vert }_{L^{{{\tilde{q}}^{\prime }}_2(\cdot )}\left({ℝ}^n\right)}{\Vert \Omega \left(x-\cdot \right){\chi }_j(\cdot )\Vert }_{L^s\left({ℝ}^n\right)}\\ \le C{\Vert b\Vert }_{\ast }{2}^{\left(k-j\right)\left(\gamma +\frac{n}{s}\right)}{\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}{\Vert {\chi }_{B_j}\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}.\end{array}$ (4.2)

Now, by (4.1), (4.2), lemmas 2.4, 2.5 and 2.3, we have

$\begin{array}{l}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f_j\right){\chi }_k\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\\ \le C{2}^{-kn}(2^{\left(k-j\right)\left(\gamma +\frac{n}{s}\right)}{\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}{\Vert {\chi }_{B_j}\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}{\Vert \left(b(\cdot )-b_{B_j}\right){\chi }_k(\cdot )\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}{\Vert f_j\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\\ +{\Vert b\Vert }_{\ast }{2}^{\left(k-j\right)\left(\gamma +\frac{n}{s}\right)}{\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}{\Vert {\chi }_{B_j}\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_k\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}{\Vert f_j\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\text{d}y)\\ \le C{2}^{-kn}(\left(k-j\right){\Vert b\Vert }_{\ast }{2}^{\left(k-j\right)\left(\gamma +\frac{n}{s}\right)}{\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}{\Vert {\chi }_{B_j}\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_{B_k}\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}{\Vert f_j\Vert }_{L^{q(\cdot )}(\; ℝ\; n\; )}\end{array}$

$\begin{array}{l}+{\Vert b\Vert }_{\ast }{2}^{\left(k-j\right)\left(\gamma +\frac{n}{s}\right)}{\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}{\Vert {\chi }_{B_j}\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_{B_k}\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}{\Vert f_j\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\text{d}y)\\ \le C\left(k-j\right){\Vert b\Vert }_{\ast }{2}^{-kn}{2}^{\left(k-j\right)\left(\gamma +\frac{n}{s}\right)}{\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}{\Vert {\chi }_{B_j}\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}{\Vert {\chi }_{B_k}\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}{\Vert f_j\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\\ \le C\left(k-j\right){\Vert b\Vert }_{\ast }{2}^{\left(k-j\right)\left(\gamma +\frac{n}{s}\right)}{\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}{\Vert f_j\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\frac{{\Vert {\chi }_{B_j}\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}}{{\Vert {\chi }_{B_k}\Vert }_{L^{q^{\prime }(\cdot )}\left({ℝ}^n\right)}}\\ \le C{\Vert b\Vert }_{\ast }\left(k-j\right)2^{\left(k-j\right)\left(n{\delta }_2-\gamma -\frac{n}{s}\right)}{\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}{\Vert f_j\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}.\end{array}$ (4.3)

By the boundedness of ${\mu }_{\Omega }$ in $L^{p(\cdot )}$ see [7], we have

${\Vert \left({\mu }_{\Omega }{f}_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}\le {\Vert f_j\Vert }_{L^{p(\cdot )}}\le {\left|B_j\right|}^{-{\alpha }_j/n}=2^{-j{\alpha }_j}.$

So we have

${\Vert \left[b,{\mu }_{\Omega }\right]\left(f_j\right){\chi }_k\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\le {\Vert b\Vert }_{\ast }\left(k-j\right){\Vert \Omega \Vert }_{L^s\left(S^{n-1}\right)}{2}^{\left(k-j\right)\left(n{\delta }_2-\gamma -\frac{n}{s}\right)-j{\alpha }_j}.$

Firstly we estimate H. We need to show that there exists a positive constant C, such that $H\le C\Delta$ Consider

$\begin{array}{c}H=\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\Vert \left[b,{\mu }_{\Omega }\right]\left(f\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(f_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q\right)}^q\\ +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=-\infty }{\overset{k-1}{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(f_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q\right)}^q\\ :=H_1+H_2.\end{array}$

By boundedness of $\left[b,{\mu }_{\Omega }\right]$ in $L^{p(\cdot )}$, see ( [20] ), when $0<q\le 1$ we have

$\begin{array}{c}{H}_1=\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(f_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q\right)}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|2^{-j{\alpha }_j}\right)}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}\left({\displaystyle \underset{j=k}{\overset{-1}{\sum }}}\left|{\lambda }_j\right|2^{-j\alpha \left(0\right)jq}+{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|2^{-j{\alpha }_{\infty }q}\right)\\ \le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{\displaystyle \underset{j=k}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\alpha \left(0\right)\left(k-j\right)q}\\ +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-j{\alpha }_{\infty }q}\end{array}$

$\begin{array}{l}\le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q}\\ +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{-j\lambda q}{\left|{\lambda }_j\right|}^q{2}^{\left(\lambda -{\alpha }_{\infty }\right)jq}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha (\; 0\; )}\end{array}$

$\begin{array}{l}\le \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{L}{\sum }}}{\left|{\lambda }_j\right|}^q+\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{2}^{-L\lambda q}{\displaystyle \underset{j=L}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q}\\ +\Delta \underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{\left(\lambda -{\alpha }_{\infty }\right)jq}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{\left(\alpha \left(0\right)k-L\lambda \right)q}\\ \le \Delta +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{\displaystyle \underset{j=L}{\overset{-1}{\sum }}}{2}^{-j\lambda q}{\left|{\lambda }_j\right|}^q{2}^{\left(j-L\right)\lambda q}{\displaystyle \underset{k=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q}+\Delta \\ \le \Delta +\underset{L\le 0,L\in \mathbb{Z}}{{\displaystyle \sup }}{\displaystyle \underset{j=L}{\overset{-1}{\sum }}}{2}^{-j\lambda q}{\left|{\lambda }_j\right|}^q{2}^{\left(j-L\right)\lambda q}{\displaystyle \underset{k=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q}\\ \le \Delta .\end{array}$ (4.4)

When $1<q<\infty$ and $1/q+1/q^{\prime }=1$, and let $\gamma +n{\delta }_2-\alpha >0$, we have

$\begin{array}{c}{H}_1=\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|{\Vert \left[b,{\mu }_{\Omega }\right]\left(f_j\right){\chi }_k\Vert }_{L^{p(\cdot )}}^q\right)}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=k}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|2^{-j\alpha }\right)}^q\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{\left({\displaystyle \underset{j=k}{\overset{-1}{\sum }}}\left|{\lambda }_j\right|2^{\alpha \left(0\right)\left(k-j\right)}\right)}^q\\ +\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\left({\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}\left|{\lambda }_j\right|2^{-j{\alpha }_{\infty }}\right)}^q\end{array}$

$\begin{array}{l}\le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}\left({\displaystyle \underset{j=k}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\alpha \left(0\right)\left(k-j\right)q/2}\right){\left({\displaystyle \underset{j=k}{\overset{-1}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q^{\prime }/2}\right)}^{q/q^{\prime }}\\ +\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}\left({\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-j{\alpha }_{\infty }q/2}\right){\left({\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{-j{\alpha }_{\infty }{q}^{\prime }/2}\right)}^{q/q^{\prime }}\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{\displaystyle \underset{j=k}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{\left(j-k\right)q/2}\\ +\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{\left|{\lambda }_j\right|}^q{2}^{-j{\alpha }_{\infty }q/2}\end{array}$

$\begin{array}{l}\le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{k=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(j-k\right)q/2}\\ +\underset{L\le 0,L\in \mathbb{Z}}{\sup }{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{j\lambda q}{\left|{\lambda }_j\right|}^q{2}^{\left(\lambda -{\alpha }_{\infty }/2\right)jq}{2}^{-L\lambda q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)}\\ \le \underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=-\infty }{\overset{L}{\sum }}}{\left|{\lambda }_j\right|}^q+\underset{L\le 0,L\in \mathbb{Z}}{\sup }{2}^{-L\lambda q}{\displaystyle \underset{j=L}{\overset{-1}{\sum }}}{\left|{\lambda }_j\right|}^q{\displaystyle \underset{j=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q/2}\\ +\Delta \underset{L\le 0,L\in \mathbb{Z}}{\sup }{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{2}^{\left(\lambda -{\alpha }_{\infty }/2\right)j/2q}{\displaystyle \underset{k=-\infty }{\overset{L}{\sum }}}{2}^{kq\alpha \left(0\right)-L\lambda q}\end{array}$

$\begin{array}{l}\le \Delta +\underset{L\le 0,L\in \mathbb{Z}}{\sup }{\displaystyle \underset{j=L}{\overset{-1}{\sum }}}{2}^{-j\lambda q}{\left|{\lambda }_j\right|}^q{2}^{\left(j-L\right)\lambda q}{\displaystyle \underset{j=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q/2}+\Delta \\ \le \Delta +\Delta \underset{L\le 0,L\in \mathbb{Z}}{\sup }{\displaystyle \underset{j=L}{\overset{-1}{\sum }}}{2}^{\left(j-L\right)\lambda q}{\displaystyle \underset{j=-\infty }{\overset{j}{\sum }}}{2}^{\alpha \left(0\right)\left(k-j\right)q/2}\\ �\end{array}$

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Marcinkiewicz, J. (1938) Sur quelques integrales du type de Dini. Annales de la Societe Polonaise de Mathematique, 17, 42-50.
[2]	Stein, E.M. (1958) On the Function of Little Wood-Paley, Lusin and Marcinkiewicz. Transactions of the American Mathematical Society, 88, 430-466. https://doi.org/10.1090/S0002-9947-1958-0112932-2
[3]	Ding, Y., Fan, D. and Pan, Y. (1999) Weighted Boundedness for a Class of Rough Marcinkiewicz Integrals. Indiana University Mathematics Journal, 48, 1037-1056. https://doi.org/10.1512/iumj.1999.48.1696
[4]	Torchinsky, A. and Wang, S. (1990) A Note on the Marcinkiewicz Integral. Colloquium Mathematicum, 60, 235-243. https://doi.org/10.4064/cm-60-61-1-235-243
[5]	Xu, J. and Yang, X. (2016) The Molecular Characterization of Herz-Morrey-Haredy Space. Journal of Mathematical Inequalities, 10, 977-1008. https://doi.org/10.7153/jmi-10-79
[6]	Xu, J. and Yang, X. (2015) Herz-Morrey-Hardy Spaces with Variable Exponents and Their Applications. Journal of Function Spaces, 2015, Article ID: 160635, 19 p. https://doi.org/10.1155/2015/160635
[7]	Omer, A. and Tao S. (2019) Boundedness of Commutators on Herz-Morry-Hardy Spaces with Variable Exponents. Open Journal of Mathematical Science, 3, 94-108. https://doi.org/10.30538/oms2019.0052
[8]	Yang, Y. and Tao, S. (2018) Singular Integrals with Variable Kernel and Fractional Differentiation in Homogeneous Morrey-Herz-Type Hardy Spaces with Variable Exponents. Open Mathematics, 16, 326-345. https://doi.org/10.1515/math-2018-0036
[9]	Afiff, A., Omer, A. and Tao, S.P. (2018) Boundedness of Fractional Marcinkiewicz Integral with Variable Kernel on Variable Morrey-Herz Spaces. Open Journal of Mathematical Science, 2, 93-114. https://doi.org/10.30538/oms2018.0020
[10]	Nantomah, K. (2017) Generalized Hölder’s and Minkowski’s Inequalities for Jackson’s q-Integral and Some Applications to the Incomplete q-Gamma Function. Abstract and Applied Analysis, 2017, Article ID: 9796873. https://doi.org/10.1155/2017/9796873
[11]	Wu, J. and Wen, J. (2017) Boundedness for Higher Order Commutators of Fractional Integrals on Variable Exponent Herz-Morrey Spaces. Mediterranean Journal of Mathematics, 14, Article No. 198. https://doi.org/10.1007/s00009-017-1002-y
[12]	Tan, J. and Liu, Z. (2015) Some Boundedness of Homogeneous Fractional Integrals on Variable Exponent Function Spaces. Acta Mathematica Sinica (Chinese Series), 58, 309-320.
[13]	Izuki, M. (2010) Boundedness of Commutators on Herz Spaces with Variable Exponent. Rendiconti del Circolo Matematico di Palermo, 59, 199-213. https://doi.org/10.1007/s12215-010-0015-1
[14]	Wang, H.B. and Liao, F. (2020) Boundedness of Singular Integral Operators on Herz-Morrey Spaces with Variable Exponent. Chinese Annals of Mathematics, Series B, 41, 99-116. https://doi.org/10.1007/s11401-019-0188-7
[15]	Wang, H.B. (2020) The Continuity for Commutator of Maricnkiewicz Integrals on Herz-Type Space with Variable Exponent. Journal of University of Chinese Academy of Sciences, 37, 155-161.
[16]	Wang, L.W. (2014) Marcinkiewicz Integral Operators and Commutators on Herz Spaces with Variable Exponents. Journal of Function Spaces, 2014, Article ID: 430365. https://doi.org/10.1155/2014/430365
[17]	Xuan, Z.-H. and Shu, L.-S. (2013) Boundedness of Higher order Commutators on Herz-Morrey Spaces with Variable Exponent. Journal of Nanjing University Mathematical Biquarterly, 30, 188-196.
[18]	Yong, D., Shan, Z. and Qing, X. (2002) Maricnkiewicz Integral on Hardy Spaces. Integral Equations and Operator Theory, 42, 174-182. https://doi.org/10.1007/BF01275514
[19]	Wang, H. (2017) Commutators of Singular Integuler Operator on Herz-Type Hardy Spaces with Variable Exponents. Journal of the Korean Mathematical Society, 54, 713-732. https://doi.org/10.4134/JKMS.j150771
[20]	Tao, Sh.P. and Li, L.L. (2016) Boundednesss of Marcikiewicz Integrals and Commutators on Morrey Spaces with Varible Exponent. Chinese Journal of Contemporary Mathematics, 37, 53-68.