﻿ Boundedness of Calderón-Zygmund Operator and Their Commutator on Herz Spaces with Variable Exponent

Applied Mathematics
Vol.08 No.04(2017), Article ID:75530,16 pages
10.4236/am.2017.84035

Boundedness of Calderón-Zygmund Operator and Their Commutator on Herz Spaces with Variable Exponent

Omer Abdalrhman1,2*, Afif Abdalmonem1,3, Shuangping Tao1

1College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China

2College of Education, Shendi University, Shendi, River Nile State, Sudan

3Faculty of Science, University of Dalanj, Dalanj, South kordofan, Sudan    Received: March 13, 2017; Accepted: April 17, 2017; Published: April 20, 2017

ABSTRACT

The aim of this paper is to study the boundedness of Calderón-Zygmund operator and their commutator on Herz Spaces with two variable exponents $p\left(.\right),q\left(.\right)$ . By applying the properties of the Lebesgue spaces with variable exponent, the boundedness of the Calderón-Zygmund operator and the commutator generated by BMO function and Calderón-Zygmund operator is obtained on Herz space.

Keywords:

Calderón-Zygmund Operator, Commutator, Herz Spaces with Variable Exponent, BMO Spaces 1. Introduction

Definition 1.1. Let $T$ be a bounded linear operator from $S\left({ℝ}^{n}\right)$ to ${S}^{\prime }\left({ℝ}^{n}\right)$ (see  ,  ). $T$ is called a standard operator if $T$ satisfies the following conditions:

1) $T$ extends to a bounded linear operator on ${L}^{2}\left({ℝ}^{n}\right)$ .

2) There exists a function $K\left(x,y\right)$ defined by $\left\{\left(x,y\right)\in \left({ℝ}^{n}\right)×\left({ℝ}^{n}\right);x\ne y\right\}$ satisfies

$|K\left(x,y\right)|\le C/{|x-y|}^{n},$ (1.1)

where $C>0$ .

3) $〈Tf,g〉={\int }_{\left({ℝ}^{n}\right)}{\int }_{\left({ℝ}^{n}\right)}K\left(x,y\right)f\left(y\right)g\left(x\right)\text{d}x\text{d}y,$ for $f,g\in S\left({ℝ}^{n}\right)$ with $\text{supp}\left(f\right)\cap \text{supp}\left(g\right)=\varnothing$

A standard operator $T$ is called a $\gamma$ -Calderón $\text{-}$ Zygmund operator if $K$ is a standard kernel satisfies:

$|K\left(x,y\right)-K\left(z,y\right)|\le C{|x-z|}^{\gamma }/{|x-y|}^{n+\gamma };$ (1.2)

$|K\left(y,x\right)-K\left(y,z\right)|\le C{|x-z|}^{\gamma }/{|x-y|}^{n+\gamma },$ (1.3)

if $|x-z|<\frac{1}{2}|x-y|$ for some $0<\gamma \le 1$ .

The bounded mean oscillation BMO space and BMO norm are defined, respectively, by

$BMO\left({ℝ}^{n}\right)=\left\{b\in {L}_{loc}^{1}\left({ℝ}^{n}\right):{‖b‖}_{BMO\left({ℝ}^{n}\right)}<\infty \right\},$ (1.4)

${‖b‖}_{BMO\left({ℝ}^{n}\right)}=\underset{B:\text{ball}}{\mathrm{sup}}1/|B|{\int }_{B}|b\left(x\right)-{b}_{B}|\text{d}x.$ (1.5)

The commutator of the Calderón-Zygmund operator is defined by

$\left[b,T\right]f\left(x\right)=b\left(x\right)Tf\left(x\right)-T\left(bf\right)\left(x\right).$ (1.6)

In 1983, J.-L. Jouné proved $\gamma$ -Calderón $\text{-}$ Zygmund operator is bounded on ${L}^{p}\left({ℝ}^{n}\right)$ in  . Coifman, Rochberg and Weiss proved that commutator [b,T] is bounded on ${L}^{p}\left({ℝ}^{n}\right)\left(1 (see  ).

Kovácik and Rákosník introduced Lebesgue spaces and Sobolev spaces with variable exponents (see  ). The function spaces with variable exponent has been recently obtained an increasing interest by a number of authors since many applications are found in many different fields, for example, in fluid dynamics (see  ), image restoration (see    ) and differential equations.

Herz spaces play an important role in harmonic analysis. After they were introduced in  , the boundedness of some operators and some characteriza- tions of Herz spaces with variable exponents were studied extensively (see  -  ). In 2015, Wang and Tao introduced the Herz spaces with two variable exponents $p\left(.\right),q\left(.\right)$ , and studied the parameterized Littlewood-Paley operators and their commutators on Herz spaces with variable exponents in  .

In this paper, we will discuss the boundedness of the Calderón-Zygmund operator $T$ and their commutator $\left[b,T\right]$ are bounded on Herz spaces with two variable exponents $p\left(.\right),q\left(.\right)$ .

2. Definitions of Function Spaces with Variable Exponent

In this section we recall some definitions. Let $\Omega$ be a measurable set in ${ℝ}^{n}$ with $|\Omega |>0$ . We firstly recall the definition of the Lebesgue spaces with variable exponent.

Definition 2.1.  Let $p\left(\cdot \right):\Omega \to \left[1,\infty \right)$ be a measurable function. The Lebesgue space with variable exponent ${L}^{p\left(\cdot \right)}\left(\Omega \right)$ is defined by

${L}^{p\left(\cdot \right)}\left(\Omega \right)=\left\{f\text{ismeasurable}:{\int }_{\Omega }{\left(\frac{|f\left(x\right)|}{\eta }\right)}^{p\left(x\right)}\text{d}x<\infty \text{forsomeconstant}\eta >0\right\}.$ (2.1)

For all compact $K\subset \Omega$ , the space ${L}_{loc}^{p\left(\cdot \right)}\left(\Omega \right)$ is defined by

${L}_{loc}^{p\left(\cdot \right)}\left(\Omega \right)=\left\{\text{ }f\text{ismeasurable}:f\in {L}^{p\left(\cdot \right)}\left(K\right)\right\}.$ (2.2)

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

${‖f‖}_{{L}^{p\left(\cdot \right)}\left(\Omega \right)}=\mathrm{inf}\left\{\eta >0:{\int }_{\Omega }{\left(\frac{|f\left(x\right)|}{\eta }\right)}^{p\left(x\right)}\text{d}x\le 1\right\}.$ (2.3)

We denote ${p}_{-}=ess\text{inf}\left\{p\left(x\right):x\in \Omega \right\},\text{}{p}_{+}=ess\mathrm{sup}\left\{p\left(x\right):x\in \Omega \right\}$ . Then $\mathcal{P}\left(\Omega \right)$ consists of all $p\left(\cdot \right)$ satisfying ${p}_{-}>1$ and ${p}_{+}<\infty$ . Let $M$ be the Hardy-Littlewood maximal operator. We denote $\mathcal{B}\left(\Omega \right)$ to be the set of all function $p\left(\cdot \right)\in \mathcal{P}\left(\Omega \right)$ satisfying the $M$ is bounded on ${L}^{p\left(\cdot \right)}\left(\Omega \right)$ .

Definition 2.2.  Let $p\left(\cdot \right),q\left(\cdot \right)\in \mathcal{P}\left(\Omega \right)$ . The mixed Lebesgue sequence space with variable exponent ${\mathcal{l}}^{q\left(\cdot \right)}\left({L}^{p\left(\cdot \right)}\right)$ is the collection of all sequences ${\left\{{f}_{j}\right\}}_{j=0}^{\infty }$ of the measurable functions on ${ℝ}^{n}$ such that

$\begin{array}{l}{‖{\left\{{f}_{j}\right\}}_{j=0}^{\infty }‖}_{{\mathcal{l}}^{q\left(\cdot \right)}\left({L}^{p\left(\cdot \right)}\right)}=\mathrm{inf}\left\{\eta >0:{Q}_{{\mathcal{l}}^{q\left(\cdot \right)}\left({L}^{p\left(\cdot \right)}\right)}\left(\begin{array}{c}{\left\{\frac{{f}_{j}}{\zeta }\right\}}_{j=0}^{\infty }\end{array}\right)\le 1\right\}<\infty ,\\ {Q}_{{\mathcal{l}}^{q\left(\cdot \right)}\left({L}^{p\left(\cdot \right)}\right)}\left({\left\{{f}_{j}\right\}}_{j=0}^{\infty }\right)=\underset{j=0}{\overset{\infty }{\sum }}\mathrm{inf}\left\{{\zeta }_{j}>0;{\int }_{{R}^{n}}{\left(\frac{|{f}_{j}\left(x\right)|}{{\zeta }_{j}^{\frac{1}{q\left(x\right)}}}\right)}^{p\left(x\right)}\text{d}x\le 1\right\}.\end{array}$ (2.4)

Let ${B}_{k}=\left\{x\in {ℝ}^{n}:|x|\le {2}^{k}\right\},{C}_{k}={B}_{k}\{B}_{k-1},{\chi }_{k}={\chi }_{{C}_{k}}$ , $k\in ℤ.$ , for ${q}_{+}<\infty$ , we have that

${Q}_{{\mathcal{l}}^{q\left(\cdot \right)\left({L}^{p\left(\cdot \right)}\right)}}\left({\left\{{f}_{j}\right\}}_{j=0}^{\infty }\right)=\underset{j=0}{\overset{\infty }{\sum }}{‖{|{f}_{j}|}^{q\left(\cdot \right)}‖}_{{L}^{\frac{p\left(\cdot \right)}{q\left(\cdot \right)}}}.$ (2.5)

Let ${B}_{k}=\left\{x\in {ℝ}^{n}:|x|\le {2}^{k}\right\},{C}_{k}={B}_{k}\{B}_{k-1},{\chi }_{k}={\chi }_{{C}_{k}}$ , $k\in ℤ.$

Definition 2.3.  Let $\alpha \in {ℝ}^{n},q\left(\cdot \right),p\left(\cdot \right)\in \mathcal{P}\left({ℝ}^{n}\right)$ . The homogeneous Herz space with variable exponent ${\stackrel{˙}{K}}_{p\left(\cdot \right)}^{\alpha ,q\left(\cdot \right)}\left({ℝ}^{n}\right)$ is defined by

${\stackrel{˙}{K}}_{p\left(\cdot \right)}^{\alpha ,q\left(\cdot \right)}\left({ℝ}^{n}\right)=\left\{f\in {L}_{loc}^{p\left(\cdot \right)}\left({ℝ}^{n}\\left\{0\right\}\right):{‖f‖}_{{\stackrel{˙}{K}}_{p\left(\cdot \right)}^{\alpha ,q\left(\cdot \right)}\left({ℝ}^{n}\right)}<\infty \right\}.$

Equipped the norm

$\begin{array}{l}{‖f‖}_{{\stackrel{˙}{K}}_{{p}_{\left(\cdot \right)}}^{\alpha ,q\left(\cdot \right)}\left({ℝ}^{n}\right)}={‖{\left\{{2}^{k\alpha }|f{\chi }_{k}|\right\}}_{k=0}^{\infty }‖}_{{l}^{q\left(\cdot \right)}\left({L}^{p\left(\cdot \right)}\right)}\\ =\mathrm{inf}\left\{\eta >0:\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|f{\chi }_{k}|}{\eta }\right)}^{q\left(\cdot \right)}‖}_{{L}^{\frac{p\left(\cdot \right)}{q\left(\cdot \right)}}}\le 1\right\}.\end{array}$

Remark 2.1.  Let ${q}_{1}\left(\cdot \right),{q}_{2}\left(\cdot \right)\in \mathcal{P}\left({ℝ}^{n}\right)$ satisfying ${\left({q}_{1}\right)}_{+}\le {\left({q}_{2}\right)}_{+}$ and satisfy the following results:

1) ${\stackrel{˙}{K}}_{p\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)\subset {\stackrel{˙}{K}}_{p\left(\cdot \right)}^{\alpha ,{q}_{2}\left(\cdot \right)}\left({ℝ}^{n}\right).$

2) If $\frac{{q}_{2}\left(\cdot \right)}{{q}_{1}\left(\cdot \right)}\in \mathcal{P}\left({ℝ}^{n}\right)$ and $\frac{{q}_{2}\left(\cdot \right)}{{q}_{1}\left(\cdot \right)}\ge 1$ . For any $f\in {\stackrel{˙}{K}}_{p\left(\cdot \right)}^{\alpha ,q\left(\cdot \right)}\left({ℝ}^{n}\right)$ , by using Lemma 3.7 and Remark 2.2, we have

$\begin{array}{c}\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|f{\chi }_{k}|}{\eta }\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{p\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\le \underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|f{\chi }_{k}|}{\eta }\right)}^{{q}_{1}\left(\cdot \right)}‖}_{{L}^{\frac{p\left(\cdot \right)}{{q}_{1}\left(\cdot \right)}}}^{{p}_{v}}\\ \le {\left\{\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|f{\chi }_{k}|}{\eta }\right)}^{{q}_{1}\left(\cdot \right)}‖}_{{L}^{\frac{p\left(\cdot \right)}{{q}_{1}\left(\cdot \right)}}}^{{p}_{h}}\right\}}^{{p}_{*}}\le 1.\end{array}$

where

${p}_{v}=\left\{\begin{array}{l}{\left(\frac{{q}_{2}\left(\cdot \right)}{{q}_{1}\left(\cdot \right)}\right)}_{-},\frac{{2}^{k\alpha }|f{\chi }_{k}|}{\eta }\le 1,\hfill \\ {\left(\frac{{q}_{2}\left(\cdot \right)}{{q}_{1}\left(\cdot \right)}\right)}_{+},\frac{{2}^{k\alpha }|f{\chi }_{k}|}{\eta }>1.\hfill \end{array}$

${p}_{*}=\left\{\begin{array}{l}\underset{v\in ℕ}{\mathrm{min}}{p}_{v},\underset{v=0}{\overset{\infty }{\sum }}{a}_{v}\le 1,\hfill \\ \underset{v\in ℕ}{\mathrm{max}}{p}_{v},\underset{v=0}{\overset{\infty }{\sum }}{a}_{v}>1.\hfill \end{array}$

This implies that ${\stackrel{˙}{K}}_{p\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)\subset {\stackrel{˙}{K}}_{p\left(\cdot \right)}^{\alpha ,{q}_{2}\left(\cdot \right)}\left({ℝ}^{n}\right)$ .

Remark 2.2. Let $v\in ℕ,{a}_{v}\ge 0,1\le {p}_{v}<\infty$ . Then we have

$\underset{v=0}{\overset{\infty }{\sum }}{a}_{v}\le {\left(\underset{v=0}{\overset{\infty }{\sum }}{a}_{h}\right)}^{{p}_{*}},$

where

${p}_{*}=\left\{\begin{array}{l}\underset{v\in ℕ}{\mathrm{min}}{p}_{v},\underset{v=0}{\overset{\infty }{\sum }}{a}_{v}\le 1,\hfill \\ \underset{v\in ℕ}{\mathrm{max}}{p}_{v},\underset{v=0}{\overset{\infty }{\sum }}{a}_{v}>1.\hfill \end{array}$

3. Properties and Lemmas of Variable Exponent

In this section, we recall some properties and some lemmas of variable exponent belonging to the class $\mathcal{B}\left({ℝ}^{n}\right)$ .

Proposition 3.1.  If $p\left(\cdot \right)\in \mathcal{P}\left({ℝ}^{n}\right)$ satisfies

$|p\left(x\right)-p\left(y\right)|\le \frac{-C}{\text{Log}\left(|x-y|\right)},|x-y|\le 1/2;$ (3.1)

$|p\left(x\right)-p\left(y\right)|\le \frac{C}{\text{Log}\left(e+|x|\right)},|y|\ge |x|.$ (3.2)

Hence we have $p\left(\cdot \right)\in \mathcal{B}\left({ℝ}^{n}\right)$ .

Lemma 3.1.  Given $p\left(\cdot \right):{ℝ}^{n}\to \left[1,\infty \right)$ have that for all functions $f$ and $g$ ,

${\int }_{{ℝ}^{n}}|f\left(x\right)g\left(x\right)|\text{d}x\le C{‖f‖}_{{L}^{p\left(\cdot \right)}\left({ℝ}^{n}\right)}{‖g‖}_{{L}^{{p}^{\prime }\left(\cdot \right)}\left({ℝ}^{n}\right)}.$ (3.3)

where ${C}_{p}=1+\frac{1}{{p}_{-}}-\frac{1}{{p}_{+}}$ .

Lemma 3.2.  Suppose that $p\left(\cdot \right),{p}_{1}\left(\cdot \right),{p}_{2}\left(\cdot \right)\in \mathcal{P}\left({ℝ}^{n}\right)$ , for any $f\in {L}^{{p}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right),g\in {L}^{{p}_{2}\left(\cdot \right)}\left({ℝ}^{n}\right)$ , when $\frac{1}{p\left(\cdot \right)}=\frac{1}{{p}_{2}\left(\cdot \right)}+\frac{1}{{p}_{1}\left(\cdot \right)}$ , we get

${‖f\left(x\right)g\left(x\right)‖}_{{L}^{p\left(\cdot \right)}\left({ℝ}^{n}\right)}\le C{‖g\left(x\right)‖}_{{L}^{{p}_{2}}\left({ℝ}^{n}\right)}{‖f\left(x\right)‖}_{{L}^{{p}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)},$ (3.4)

where ${C}_{{p}_{1},{p}_{2}}={\left[1+\frac{1}{{p}_{1-}}-\frac{1}{{p}_{1+}}\right]}^{\frac{1}{{p}_{-}}}$ .

Proposition 3.2.  Let $p\left(\cdot \right)\in \mathcal{B}\left({ℝ}^{n}\right)$ and $T$ be a Calderón $\text{-}$ Zygmund operator. Then we have

${‖Tf‖}_{{L}^{p\left(\cdot \right)}\left({ℝ}^{n}\right)}\le C{‖f‖}_{{L}^{p\left(\cdot \right)}\left({ℝ}^{n}\right)}.$ (3.5)

Lemma 3.3.  Let $p\left(\cdot \right)\in \mathcal{B}\left({ℝ}^{n}\right),b\in \text{BMO}$ function and $T$ be a Calderón $\text{-}$ Zygmund operator.Then

${‖\left[b,T\right]f‖}_{{L}^{p\left(\cdot \right)}\left({ℝ}^{n}\right)}\le C{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}{‖f‖}_{{L}^{p\left(\cdot \right)}\left({ℝ}^{n}\right)}$ (3.6)

Lemma 3.4.  Let $b\in \text{BMO}\left({ℝ}^{n}\right)$ . If $i,j\in ℤ$ with $i , then we have

1. ${C}^{-1}{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}\le \underset{B}{\mathrm{sup}}\frac{1}{{‖{\chi }_{B}‖}_{{L}^{p\left(\cdot \right)}\left({ℝ}^{n}\right)}}{‖\left(b-{b}_{B}\right){\chi }_{B}‖}_{{L}^{p\left(\cdot \right)}\left({ℝ}^{n}\right)}\le C{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}.$

2. ${‖\left(b-{b}_{{B}_{i}}\right){\chi }_{{B}_{j}}‖}_{{L}^{q\left(\cdot \right)}\left({ℝ}^{n}\right)}\le C\left(j-i\right){‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}{‖{\chi }_{{B}_{j}}‖}_{{L}^{q\left(\cdot \right)}\left({ℝ}^{n}\right)}.$

Lemma 3.5.  Let ${p}_{u}\left(\cdot \right)\in \mathcal{B}\left({ℝ}^{n}\right)\left(u=1,2\right)$ , then there exist constants $0<{\iota }_{u1},{\iota }_{u2}<1$ , and $C>0$ such that for all balls $B\subset {ℝ}^{n}$ and all measurable subset $R\subset B$ ,

$\frac{{‖{\chi }_{R}‖}_{{L}^{{p}_{u}\left(\cdot \right)}\left({ℝ}^{n}\right)}}{{‖{\chi }_{B}‖}_{{L}^{{p}_{u}\left(\cdot \right)}\left({ℝ}^{n}\right)}}\le C{\left(\frac{|R|}{|B|}\right)}^{{\iota }_{u1}},\frac{{‖{\chi }_{R}‖}_{{L}^{{{p}^{\prime }}_{u}\left(\cdot \right)}\left({ℝ}^{n}\right)}}{{‖{\chi }_{B}‖}_{{L}^{{{p}^{\prime }}_{u}\left(\cdot \right)}\left({ℝ}^{n}\right)}}\le C{\left(\frac{|R|}{|B|}\right)}^{{\iota }_{u2}}.$ (3.7)

Lemma 3.6.  If $p\left(\cdot \right)\in \mathcal{B}\left({ℝ}^{n}\right)$ , there exist a constant $C>0$ such that for any balls B in ${ℝ}^{n}$ , we have

$\frac{1}{|B|}{‖{\chi }_{B}‖}_{{L}^{p\left(\cdot \right)}\left({ℝ}^{n}\right)}{‖{\chi }_{B}‖}_{{L}^{{p}^{\prime }\left(\cdot \right)}\left({ℝ}^{n}\right)}\le C.$ (3.8)

Lemma 3.7.  Suppose that $p\left(\cdot \right),q\left(\cdot \right)\in \mathcal{P}\left({\mathcal{B}}^{n}\right)$ . If $f\in {L}^{p\left(\cdot \right)q\left(\cdot \right)}$ , then

$\mathrm{min}\left({‖f‖}_{{L}^{p\left(\cdot \right)q\left(\cdot \right)}}^{{q}_{+}},{‖f‖}_{{L}^{p\left(\cdot \right)q\left(\cdot \right)}}^{{q}_{-}}\right)\le {‖{|f|}^{q\left(\cdot \right)}‖}_{{L}^{p\left(\cdot \right)}}\le \mathrm{max}\left({‖f‖}_{{L}^{p\left(\cdot \right)q\left(\cdot \right)}}^{{q}_{+}},{‖f‖}_{{L}^{p\left(\cdot \right)q\left(\cdot \right)}}^{{q}_{-}}\right).$ (3.9)

4. The Main Theorems and Their Proofs

Theorem 4.1. Suppose that ${p}_{1}\left(\cdot \right)\in \mathcal{B}\left({ℝ}^{n}\right),{q}_{1}\left(\cdot \right),{q}_{2}\left(\cdot \right)\in \mathcal{P}\left({ℝ}^{n}\right)$ with ${\left({q}_{2}\right)}_{-}\ge {\left({q}_{1}\right)}_{+}$ . If $-n{\iota }_{12}<\alpha with ${\iota }_{11},{\iota }_{12}$ as defined in Lemma 3.5, then the operator $T$ is bounded from ${\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{2}\left(\cdot \right)}\left({ℝ}^{n}\right)$ to ${\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)$ .

Proof Let $h\left(x\right)\in {\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)$ . We write

$h\left(x\right)=\underset{j=-\infty }{\overset{\infty }{\sum }}h\left(x\right){\chi }_{j}=\underset{j=-\infty }{\overset{\infty }{\sum }}{h}_{j}\left(x\right).$

By Definition 2.3, we have

#Math_135# (4.1)

Since

$\begin{array}{c}{‖{\left(\frac{{2}^{k\alpha }|T\left(h\right){\chi }_{k}|}{\eta }\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\le {‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{\infty }T\left({h}_{j}\right){\chi }_{k}|}{{\sum }_{i=1}^{3}{\eta }_{1i}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\\ \le {‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{k-2}T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{11}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\\ \text{}+{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=k-2}^{k+2}T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{12}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\\ \text{}+{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=k+2}^{\infty }T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{13}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}},\end{array}$ (4.2)

where

${\eta }_{11}={‖{\left\{{2}^{k\alpha }|\underset{j=-\infty }{\overset{k-2}{\sum }}T\left({h}_{j}\right){\chi }_{k}|\right\}}_{k=-\infty }^{\infty }‖}_{{\mathcal{l}}^{{q}_{2}\left(\cdot \right)}\left({L}^{{p}_{1}\left(\cdot \right)}\right)},$ (4.3)

${\eta }_{12}={‖{\left\{{2}^{k\alpha }|\underset{j=k-2}{\overset{k+2}{\sum }}T\left({h}_{j}\right){\chi }_{k}|\right\}}_{k=-\infty }^{\infty }‖}_{{\mathcal{l}}^{{q}_{2}\left(\cdot \right)}\left({L}^{{p}_{1}\left(\cdot \right)}\right)},$ (4.4)

${\eta }_{13}={‖{\left\{{2}^{k\alpha }|\underset{j=k+2}{\overset{\infty }{\sum }}T\left({h}_{j}\right){\chi }_{k}|\right\}}_{k=-\infty }^{\infty }‖}_{{\mathcal{l}}^{{q}_{2}\left(\cdot \right)}\left({L}^{{p}_{1}\left(\cdot \right)}\right)},$

and

$\eta =\underset{i=1}{\overset{3}{\sum }}{\eta }_{1i}.$

Thus,

$\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|T\left(h\right){\chi }_{k}|}{\eta }\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\le C.$

We easily see that

${‖T\left(h\right)‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{2}\left(\cdot \right)}\left({ℝ}^{n}\right)}\le C\eta =C\underset{i=1}{\overset{3}{\sum }}{\eta }_{1i}.$ (4.6)

This implies that we only need to prove ${\eta }_{11},{\eta }_{12},{\eta }_{13}\le C{‖h‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)}$ . Denote ${\eta }_{10}={‖h‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)}.$

First, we consider ${\eta }_{12}$ . By virtue of Lemma 3.7, we get

$\begin{array}{l}\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=k-2}^{k+2}T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\\ \le \underset{k=-\infty }{\overset{\infty }{\sum }}{‖\frac{{2}^{k\alpha }|{\sum }_{j=k-2}^{k+2}T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}‖}_{{L}^{{p}_{1}\left(\cdot \right)}}^{{\left({q}_{2}^{1}\right)}_{k}}\\ \le \underset{k=-\infty }{\overset{\infty }{\sum }}{\left({‖\frac{{2}^{k\alpha }|{\sum }_{j=k-2}^{k+2}T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}‖}_{{L}^{{p}_{1}\left(\cdot \right)}}\right)}^{{\left({q}_{2}^{1}\right)}_{k}},\end{array}$ (4.7)

where,

${\left({q}_{2}^{1}\right)}_{k}=\left\{\begin{array}{l}{\left({q}_{2}\right)}_{-},{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=k-2}^{k+2}T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\le 1,\hfill \\ {\left({q}_{2}\right)}_{+},{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=k-2}^{k+2}T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}>1.\hfill \end{array}$

In the above, we use the Proposition 3.2 and Remark 2.2. Since $h\left(x\right)\in {\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)$ , we have ${‖\frac{{2}^{k\alpha }|h{\chi }_{k}|}{{\eta }_{10}}‖}_{{L}^{{p}_{1}\left(\cdot \right)}}\le 1$ and ${\sum }_{k=-\infty }^{\infty }{‖{\left(\frac{{2}^{k\alpha }|h{\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{1}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{1}\left(\cdot \right)}}}\le 1$ , we get

$\begin{array}{l}\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=k-2}^{k+2}T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\\ \le C\underset{k=-\infty }{\overset{\infty }{\sum }}{\left(\underset{j=k-2}{\overset{k+2}{\sum }}{‖\frac{{2}^{k\alpha }|{h}_{j}|}{{\eta }_{10}}‖}_{{L}^{{p}_{1}\left(\cdot \right)}}\right)}^{{\left({q}_{2}^{1}\right)}_{k}}\\ \le C\underset{k=-\infty }{\overset{\infty }{\sum }}{‖\frac{{2}^{k\alpha }|h{\chi }_{k}|}{{\eta }_{10}}‖}_{{L}^{{p}_{1}\left(\cdot \right)}}^{{\left({q}_{2}^{1}\right)}_{k}}\\ \le C\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|h{\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{1}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{1}\left(\cdot \right)}}}^{\frac{{\left({q}_{2}^{1}\right)}_{k}}{{\left({q}_{1}\right)}_{+}}}\\ \le C{\left\{\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|h{\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{1}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{1}\left(\cdot \right)}}}\right\}}^{{q}_{*}}\\ \le C.\end{array}$

Here ${\left({p}_{1}\right)}_{+}\le {\left({p}_{2}\right)}_{-}\le {\left({q}_{2}^{1}\right)}_{k}$ and ${q}_{*}=\underset{k\in N}{\mathrm{min}}\frac{{\left({q}_{2}^{1}\right)}_{k}}{{\left({q}_{1}\right)}_{+}}$ . That is

${\eta }_{12}\le C{\eta }_{10}\le C{‖h‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)}.$ (4.8)

Let us now turn to estimate ${\eta }_{11}$ . Noting that $x\in {A}_{j}$ and $j\le k-2$ , by the generalized Hölder's inequality and the Minkowski’s inequality, we get

$\begin{array}{c}|T{h}_{j}\left(x\right)|\le {\int }_{{A}_{j}}|K\left(x,y\right){h}_{j}\left(y\right)|\text{d}y\\ \le C{\int }_{{A}_{j}}|{h}_{j}\left(y\right)|/{|x-y|}^{n}\text{d}y\\ \le C{2}^{-kn}{\int }_{{A}_{j}}|{h}_{j}\left(y\right)|\text{d}y\\ \le C{2}^{-kn}{‖{h}_{j}‖}_{{L}^{1}\left({ℝ}^{n}\right)}.\end{array}$ (4.9)

By Lemmas 3.5-3.7 and the fact that ${‖\frac{{2}^{j\alpha }|h{\chi }_{j}|}{{\eta }_{10}}‖}_{{L}^{{p}_{1}\left(\cdot \right){q}_{1}}}\le 1$ , we easily see that (4.10)

where

${\left({q}_{2}^{2}\right)}_{k}=\left\{\begin{array}{l}{\left({q}_{2}\right)}_{-},{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{k-2}T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\le 1,\hfill \\ {\left({q}_{2}\right)}_{+},{‖{\left(\frac{{2}^{k\alpha }|{{\sum }_{j=-\infty }^{k-2}T\left({h}_{j}\right)\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}>1.\hfill \end{array}$

Therefore, if ${\left({q}_{1}\right)}_{+}<1$ and ${\left({p}_{1}\right)}_{+}\le {\left({p}_{2}\right)}_{-}\le {\left({q}_{2}^{2}\right)}_{k}$ , we can get

$\begin{array}{l}\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{k-2}T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\\ \le C{\left\{\underset{j=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{|{2}^{j\alpha }h{\chi }_{j}|}{{\eta }_{10}}\right)}^{{q}_{1}\left(\cdot \right)}‖}_{{L}^{{p}_{1}\left(\cdot \right){q}_{1}\left(\cdot \right)}}\underset{k=j+2}{\overset{\infty }{\sum }}{2}^{\left(k-j\right)\left(\alpha -n{\iota }_{11}\right)}\right\}}^{{q}_{*}}\\ \le C,\end{array}$

where ${q}_{*}=\underset{k\in ℕ}{\mathrm{min}}\frac{{\left({q}_{2}^{1}\right)}_{k}}{{\left({q}_{1}\right)}_{+}}$ .

If ${\left({q}_{1}\right)}_{+}\ge 1$ and ${\left({q}_{2}^{2}\right)}_{k}\ge {\left({q}_{2}\right)}_{-}\ge {\left({q}_{2}\right)}_{+}\ge 1$ . By Remark 2.2 and applying the generalized Hölder’s inequality, we obtain

$\begin{array}{l}\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{k-2}T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\\ \le C\underset{k=-\infty }{\overset{\infty }{\sum }}{\left\{\underset{j=-\infty }{\overset{k-2}{\sum }}\left(k-j\right){2}^{\left(k-j\right)\left(\alpha -n{\iota }_{11}\right){\left({q}_{1}\right)}_{+}/2}{‖{\left(\frac{|{2}^{j\alpha }h{\chi }_{j}|}{{\eta }_{10}}\right)}^{{q}_{1}\left(\cdot \right)}‖}_{{L}^{{p}_{1}\left(\cdot \right){q}_{1}\left(\cdot \right)}}\right\}}^{\frac{{\left({q}_{2}^{2}\right)}_{k}}{\left({q}_{1}\right)+}}\\ \text{}×{\left(\underset{j=-\infty }{\overset{k-2}{\sum }}{2}^{\left(k-j\right)\left(\alpha -n{\iota }_{11}\right){\left({\left({q}_{1}\right)}_{+}\right)}^{\prime }/2}\right)}^{\frac{{\left({q}_{2}^{2}\right)}_{k}}{{\left(\left({q}_{1}\right)+\right)}^{\prime }}}\\ \le C{\left\{\underset{j=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{|{2}^{j\alpha }h{\chi }_{j}|}{{\eta }_{10}}\right)}^{{q}_{1}\left(\cdot \right)}‖}_{{L}^{{p}_{1}\left(\cdot \right){q}_{1}\left(\cdot \right)}}\underset{k=j+2}{\overset{\infty }{\sum }}{2}^{\left(k-j\right)\left(\alpha -n{\iota }_{11}\right){\left({q}_{1}\right)}_{+}/2}\right\}}^{{q}_{*}}\\ \le C,\end{array}$

where ${q}_{*}=\underset{k\in ℕ}{\mathrm{min}}\frac{{\left({q}_{2}^{2}\right)}_{k}}{{\left({q}_{1}\right)}_{+}}$ .

Hence, we see that

${\eta }_{11}\le C{\eta }_{10}\le C{‖h‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)}.$ (4.11)

Finally, we estimate ${\eta }_{13}$ . Noting that for each $x\in {A}_{j}$ and $j\ge k+2$ , we have

$|T{h}_{j}\left(x\right)|\le {\int }_{{A}_{j}}|K\left(x,y\right){h}_{j}\left(y\right)|\text{d}y\le C{\int }_{{A}_{j}}|{h}_{j}\left(y\right)|/{|x-y|}^{n}\text{d}y\le C{2}^{-jn}{‖{h}_{j}‖}_{{L}^{1}\left({ℝ}^{n}\right)}.$ (4.12)

By Lemma 3.7 and ${‖\frac{{2}^{j\alpha }|h{\chi }_{j}|}{{\eta }_{10}}‖}_{{L}^{{p}_{1}\left(\cdot \right){q}_{1}\left(\cdot \right)}}\le 1$ , we get (4.13)

where

${\left({q}_{2}^{3}\right)}_{k}=\left\{\begin{array}{l}{\left({q}_{2}\right)}_{-},{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=k+2}^{\infty }T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\le 1,\hfill \\ {\left({q}_{2}\right)}_{+},{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=k+2}^{\infty }T\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}>1.\hfill \end{array}$

Then we have ${\eta }_{13}\le C{\eta }_{10}\le C{‖h‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)}$ , by using the same argument in ${\eta }_{11}$ . Thus, we prove Theorem 4.1. $�$

Theorem 4.2. Let $b\in \text{BMO}\left({ℝ}^{n}\right)$ . Suppose that ${p}_{1}\left(\cdot \right)\in \mathcal{B}\left({ℝ}^{n}\right),{q}_{1}\left(\cdot \right),{q}_{2}\left(\cdot \right)\in \mathcal{P}\left({ℝ}^{n}\right)$ with ${\left({q}_{2}\right)}_{-}\ge {\left({q}_{1}\right)}_{+}$ . If $-n{\iota }_{12}<\alpha with ${\iota }_{11},{\iota }_{12}$ as defined in lemma 3.5, then the commutator $\left[b,T\right]$ is bounded from ${\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{2}\left(\cdot \right)}\left({ℝ}^{n}\right)$ to ${\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)$ .

Proof Let $h\left(x\right)\in {\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right),b\in \text{BMO}\left({ℝ}^{n}\right)$ .We write

$h\left(x\right)=\underset{j=-\infty }{\overset{\infty }{\sum }}h\left(x\right){\chi }_{j}=\underset{j=-\infty }{\overset{\infty }{\sum }}{h}_{j}\left(x\right)$

By virtue of the definition of ${\stackrel{˙}{K}}_{p\left(\cdot \right)}^{\alpha ,q\left(\cdot \right)}\left({ℝ}^{n}\right)$ , we have

${‖\left[b,T\right]\left(h\right)‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{2}\left(\cdot \right)}\left({ℝ}^{n}\right)}=\mathrm{inf}\left\{\eta >0:\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|\left[b,T\right]\left(h\right){\chi }_{k}|}{\eta }\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\le 1\right\}.$ (4.14)

Since

$\begin{array}{l}{‖{\left(\frac{{2}^{k\alpha }|\left[b,T\right]\left(h\right){\chi }_{k}|}{\eta }\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\le {‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{k-2}\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|}{{\sum }_{i=1}^{3}{\eta }_{2i}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\\ \le {‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{\infty }\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{21}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}+{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=k-2}^{k+2}\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{22}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\\ \text{}+{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=k+2}^{\infty }\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{23}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}.\end{array}$ (4.15)

Let

${\eta }_{21}={‖{\left\{{2}^{k\alpha }|\underset{j=-\infty }{\overset{k-2}{\sum }}\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|\right\}}_{k=-\infty }^{\infty }‖}_{{\mathcal{l}}^{{q}_{2}\left(\cdot \right)}\left({L}^{{p}_{1}\left(\cdot \right)}\right)},$ (4.16)

${\eta }_{22}={‖{\left\{{2}^{k\alpha }|\underset{j=k-2}{\overset{k+2}{\sum }}\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|\right\}}_{k=-\infty }^{\infty }‖}_{{\mathcal{l}}^{{q}_{2}\left(\cdot \right)}\left({L}^{{p}_{1}\left(\cdot \right)}\right)},$ (4.17)

${\eta }_{23}={‖{\left\{{2}^{k\alpha }|\underset{j=k+2}{\overset{\infty }{\sum }}\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|\right\}}_{k=-\infty }^{\infty }‖}_{{\mathcal{l}}^{{q}_{2}\left(\cdot \right)}\left({L}^{{p}_{1}\left(\cdot \right)}\right)},$ (4.18)

and

$\eta =\underset{i=1}{\overset{3}{\sum }}{\eta }_{2i}.$

Therefore, we can obtain

$\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|\left[b,T\right]\left(h\right){\chi }_{k}|}{\eta }\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\le C.$

Thus it follows that,

${‖\left[b,T\right]\left(h\right)‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{2}\left(\cdot \right)}\left({ℝ}^{n}\right)}\le C\eta =C\underset{i=1}{\overset{3}{\sum }}{\eta }_{1i}.$ (4.20)

Hence ${\eta }_{21},{\eta }_{22},{\eta }_{23}\le C{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}{‖h‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)}$ . Denoting ${\eta }_{10}=C{‖h‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)}$ , firstly we estimate ${\eta }_{22}$ as in Theorem 4.1. Applying Lemma 3.3, we imme- diately arrive at

$\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=k-2}^{k+2}\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\le C.$

So we can get that

${\eta }_{21}\le C{\eta }_{10}{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}\le C{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}{‖h‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)}.$ (4.21)

Next we estimate ${\eta }_{21}$ , Let $x\in {A}_{j},j\le k-2$ .

$\begin{array}{l}|\left[b,T\right]{h}_{j}|\le {\int }_{{A}_{j}}|K\left(x,y\right)\left(b\left(x\right)-b\left(y\right)\right){h}_{j}\left(y\right)|\text{d}y\\ \le C{\int }_{{A}_{j}}|\left(b\left(x\right)-b\left(y\right)\right){h}_{j}\left(y\right)|/{|x-y|}^{n}\text{d}y\\ \le C{2}^{-nk}|b\left(x\right)-{b}_{{B}_{j}}|{\int }_{{A}_{j}}|{h}_{j}\left(y\right)|\text{d}y+{\int }_{{A}_{j}}|{b}_{{B}_{j}}-b\left(y\right)||{h}_{j}\left(y\right)|\text{d}y\\ \le C{2}^{-nk}|b\left(x\right)-{b}_{{B}_{j}}|{‖{h}_{j}‖}_{{L}^{1}\left({ℝ}^{n}\right)}+{‖b\left(\cdot \right)-\left({b}_{{B}_{j}}\right){h}_{j}‖}_{{L}^{1}\left({ℝ}^{n}\right)}.\end{array}$ (4.22)

Thus, from Lemmas 3.4-3.7, We obtain that Therefore, we get

$\begin{array}{l}\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{\infty }\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\\ \le C\underset{k=-\infty }{\overset{\infty }{\sum }}{\left\{\underset{j=-\infty }{\overset{k-2}{\sum }}\left(k-j\right){2}^{\left(k-j\right)\left(\alpha -n{\iota }_{11}\right)}{‖{\left(\frac{|{2}^{j\alpha }h{\chi }_{j}|}{{\eta }_{10}}\right)}^{{q}_{1}\left(\cdot \right)}‖}_{{L}^{{p}_{1}\left(\cdot \right){q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)}^{\frac{1}{\left({q}_{1}\right)+}}\right\}}^{{\left({q}_{2}^{2}\right)}_{k}},\end{array}$ (4.23)

where

${\left({q}_{2}^{2}\right)}_{k}=\left\{\begin{array}{l}{\left({q}_{2}\right)}_{-},{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{k-2}\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\le 1,\hfill \\ {\left({q}_{2}\right)}_{+},{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{k-2}\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}>1.\hfill \end{array}$

This, for ${\left({q}_{1}\right)}_{+}<1$ , ${\left({p}_{1}\right)}_{+}\le {\left({p}_{2}\right)}_{-}\le {\left({q}_{2}^{2}\right)}_{k}$ , along with Remark 2.2, tells us that

$\begin{array}{l}\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{k-2}\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}{‖b‖}_{BMO\left({ℝ}^{n}\right)}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\\ \le C{\left\{\underset{j=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{|{2}^{j\alpha }h{\chi }_{j}|}{{\eta }_{10}}\right)}^{{q}_{1}\left(\cdot \right)}‖}_{{L}^{{p}_{1}\left(\cdot \right){q}_{1}\left(\cdot \right)}}\underset{k=j+2}{\overset{\infty }{\sum }}\left(k-j\right){2}^{\left(k-j\right)\left(\alpha -n{\iota }_{11}\right)}\right\}}^{{q}_{*}}\le C,\end{array}$

where ${q}_{*}=\underset{k\in N}{\mathrm{min}}\frac{{\left({q}_{2}^{2}\right)}_{k}}{{\left({q}_{1}\right)}_{+}}.$

If ${\left({q}_{1}\right)}_{+}\le 1$ , it is follows from Remark 2.2 and Hölder’s inequality that

$\begin{array}{l}\underset{k=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{k-2}\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\\ \le C\underset{k=-\infty }{\overset{\infty }{\sum }}{\left\{\underset{j=-\infty }{\overset{k-2}{\sum }}\left(k-j\right){2}^{\left(k-j\right)\left(\alpha -n{\iota }_{11}\right){\left({q}_{1}\right)}_{+}/2}{‖{\left(\frac{|{2}^{j\alpha }h{\chi }_{j}|}{{\eta }_{10}}\right)}^{{q}_{1}\left(\cdot \right)}‖}_{{L}^{{p}_{1}\left(\cdot \right){q}_{1}\left(\cdot \right)}}\right\}}^{\frac{{\left({q}_{2}^{2}\right)}_{k}}{\left({q}_{1}\right)+}}\\ \text{}×{\left(\underset{j=-\infty }{\overset{k-2}{\sum }}\left(k-j\right){2}^{\left(k-j\right)\left(\alpha -n{\iota }_{11}\right){\left({\left({q}_{1}\right)}_{+}\right)}^{\prime }/2}\right)}^{\frac{{\left({q}_{2}^{2}\right)}_{k}}{{\left(\left({q}_{1}\right)+\right)}^{\prime }}}\\ \le C{\left\{\underset{j=-\infty }{\overset{\infty }{\sum }}{‖{\left(\frac{|{2}^{j\alpha }h{\chi }_{j}|}{{\eta }_{10}}\right)}^{{q}_{1}\left(\cdot \right)}‖}_{{L}^{{p}_{1}\left(\cdot \right){q}_{1}\left(\cdot \right)}}\underset{k=j+2}{\overset{\infty }{\sum }}\left(k-j\right){2}^{\left(k-j\right)\left(\alpha -n{\iota }_{11}\right){\left({q}_{1}\right)}_{+}/2}\right\}}^{{q}_{*}}\\ \le C,\end{array}$

where ${q}_{*}=\underset{k\in N}{\mathrm{min}}\frac{{\left({q}_{2}^{2}\right)}_{k}}{{\left({q}_{1}\right)}_{+}}$ .

This implies that

${\eta }_{21}\le C{\eta }_{10}{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}\le C{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}{‖h‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)}.$ (4.24)

Finally we estimate ${\eta }_{23}$ , for any $x\in {A}_{j},j\ge k+2$ , by the same way to argument in ${\eta }_{21}$ , we obtain that

$\begin{array}{c}|\left[b,T\right]{h}_{j}|\le {\int }_{{A}_{j}}|K\left(x,y\right)\left(b\left(x\right)-b\left(y\right)\right){h}_{j}\left(y\right)|\text{d}y\\ \le C{\int }_{{A}_{j}}|\left(b\left(x\right)-b\left(y\right)\right){h}_{j}\left(y\right)|/{|x-y|}^{n}\text{d}y\\ \le C{2}^{-nj}|b\left(x\right)-{b}_{{B}_{k}}|{\int }_{{A}_{j}}|{h}_{j}\left(y\right)|\text{d}y+{\int }_{{A}_{j}}|{b}_{{B}_{k}}-b\left(y\right)||{h}_{j}\left(y\right)|\text{d}y\\ \le C{2}^{-nj}|b\left(x\right)-{b}_{{B}_{j}}|{‖{h}_{j}‖}_{{L}^{1}\left({ℝ}^{n}\right)}+{‖b\left(\cdot \right)-\left({b}_{{B}_{j}}\right){h}_{j}‖}_{{L}^{1}\left({ℝ}^{n}\right)},\end{array}$ (4.25)

and (4.26)

where

${\left({q}_{2}^{3}\right)}_{k}=\left\{\begin{array}{l}{\left({q}_{2}\right)}_{-},{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{k-2}\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}\le 1,\hfill \\ {\left({q}_{2}\right)}_{+},{‖{\left(\frac{{2}^{k\alpha }|{\sum }_{j=-\infty }^{k-2}\left[b,T\right]\left({h}_{j}\right){\chi }_{k}|}{{\eta }_{10}}\right)}^{{q}_{2}\left(\cdot \right)}‖}_{{L}^{\frac{{p}_{1}\left(\cdot \right)}{{q}_{2}\left(\cdot \right)}}}>1.\hfill \end{array}$

Hence, we arrive at that ${\eta }_{23}\le C{\eta }_{10}{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}\le C{‖b‖}_{\text{BMO}\left({ℝ}^{n}\right)}{‖h‖}_{{\stackrel{˙}{K}}_{{p}_{1}\left(\cdot \right)}^{\alpha ,{q}_{1}\left(\cdot \right)}\left({ℝ}^{n}\right)}$ by the similar argument in the proof Theorem 4.1.

This completes the proof of Theorem 4.2. $�$

Acknowledgements

This paper is supported by National Natural Foundation of China (Grant No. 11561062).

Cite this paper

Abdalrhman, O., Abdalmonem, A. and Tao, S.P. (2017) Boundedness of Calderón-Zygmund Operator and Their Commutator on Herz Spaces with Variable Exponent. Applied Mathematics, 8, 428-443. https://doi.org/10.4236/am.2017.84035

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