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

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(.),q(.). 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.

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).

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

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

  Calderón, A. and Zygmund, A. (1956) On Singular Integrals. American Journal of Mathematics, 78, 289-309. https://doi.org/10.2307/2372517  Calderón, A. and Zygmund, A. (1978) On Singular Integral with Variable Kernels. Applied Analysis, 7, 221-238. https://doi.org/10.1080/00036817808839193  Jouné, J.-L. (1983) Calderón-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón. In: Lecture Notes in Math, Vol. 994, Springer-Verlag, Berlin, Heidelberg.  Coifman, R., Rochberg, R. and Weiss, G. (1976) Factorization Theorems for Hardy Spaces in Several Variables. Annals of Mathematics, 103, 611-635. https://doi.org/10.2307/1970954  Kovácik, O. and Rákosník, J. (1991) On Spaces Lp(x) and Wk,p(x). Czechoslovak Mathematical Journal, 41, 592-618.  Diening, L. and Ruicka, M. (2003) Calderón-Zygmund Operators on Generalized Lebesgue Spaces Lp(.) and Problems Related to Fluid Dynamics. Journal für die Reine und Angewandte Mathematik, 563, 197-220. https://doi.org/10.1515/crll.2003.081  Chen, Y., Levine, S. and Rao, M. (2006) Variable Exponent, Linear Growth Functionals in Image Restoration. SIAM Journal on Applied Mathematics, 66, 1383-1406. https://doi.org/10.1137/050624522  Li, F., Li, Z. and Pi, L. (2010) Variable Exponent Functionals in Image Restoration. Applied Mathematics and Computation, 216, 870-882.  Harjulehto, P., Hasto, P., Latvala, V. and Toivanen, O. (2013) Critical Variable Exponent Functionals in Image Restoration. Applied Mathematics Letters, 26, 56-60.  Izuki, M. (2009) Herz and Amalgam Spaces with Variable Exponent, the Haar Wavelets and Greediness of the Wavelet System. East Journal on Approximations, 15, 87-109.  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  Izuki, M. (2010) Fractional Integrals on Herz-Morrey Spaces with Variable Exponent. Hiroshima Mathematical Journal, 40, 343-355.  Wang, L. and Tao, S. (2014) Boundedness of Littlewood-Paley Operators and Their Commutators on Herz-Morrey Spaces with Variable Exponent. Journal of Inequalities and Applications, 2014, 227. https://doi.org/10.1186/1029-242x-2014-227  Wang, L. and Tao, S. (2015) Parameterized Littlewood-Paley Operators and Their Commutators on Lebegue Spaces with Variable Exponent. Analysis in Theory and Applications, 31, 13-24.  Tan, J. and Liu, Z. (2015) Some Boundedness of Homogeneous Fractional Integrals on Variable Exponent Function Spaces. ACTA Mathematics Science (Chinese Series), 58, 310-320.  Omer, A., Afif, A. and Tao, S. (2016) The Boundedness for Commutators of Calderón-Zygmund Operator on Herz-Tybe Hardy Spaces with Variable Exponent. Journal of Applied Mathematics and Physics, 4, 1157-1167. https://doi.org/10.4236/jamp.2016.46120  Wang, L. and Tao, S. (2016) Parameterized Littlewood-Paley Operators and Their Commutators on Herz Spaces with Variable Exponents. Turkish Journal of Mathematics, 40, 122-145. https://doi.org/10.3906/mat-1412-52  Cuz-Uribe, D., Fiorenza, A., Martell, J.M. and Perez, C. (2006) The Boundedness of Classical Operators on Variable Lp Spaces. Annales Academiae Scientiarum Fennicae-Mathematica, 31, 239-264.  Almeida, A., Hasanov, J. and Samko, S. (2008) Maximal and Potential Operators in Variable Exponent Morrey Spaces. Georgian Mathematical Journal, 15, 195-208.  Cruz-Uribe, D. and Fiorenza, A. (2013) Variable Lebesgue Spaces: Foundations and Harmonic Analysis. In: Applied and Numerical Harmonic Analysis, Springer, New York.  Diening, L. (2005) Maximal Function on Musielak-Orlicz Spaces and Generalized Lebesgue Spaces. Bulletin des Sciences Mathématiques, 129, 657-700. https://doi.org/10.1016/j.bulsci.2003.10.003 