Yitao Gu, Yanqi Yang^*
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China.
DOI: 10.4236/jamp.2025.1310191   PDF    HTML   XML   49 Downloads   249 Views   Citations

Abstract

In this paper, by applying the extension of Rubio de Francia’s extrapolation theorem in Lebesgue spaces with variable exponent, the boundedness of the θ-type Calderón-Zygmund operator $T_{\theta }$ is bounded in variable exponent Lebesgue spaces. In addition, the commutators generated by $T_{\theta }$ and Besov, BMO and Lipschitz functions are respectively obtained in $L^{p(\cdot )}\left({ℝ}^n\right)$ .

Keywords

θ-Type Calderón-Zygmund Operator, Commutator, Lebesgue Space with Variable Exponent

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1. Introduction and Main Results

Given a measurable function $p:{ℝ}^n\to \left[1,\infty \right)$ , $L^{p(\cdot )}\left({ℝ}^n\right)$ denotes the set of measurable functions $f$ on ${ℝ}^n$ such that for some $\lambda >0$ ,

${\displaystyle {\int }_{{ℝ}^n}{\left(\frac{\left|f\left(x\right)\right|}{\lambda }\right)}^{p\left(x\right)}\text{d}x}<\infty .$

Equipped with the Luxemburg-Nakano norm

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

then $L^{p(\cdot )}\left({ℝ}^n\right)$ becomes a Banach function space. If $p\left(x\right)=p_0$ is constant, then $L^{p(\cdot )}\left({ℝ}^n\right)$ equals the usual Lebesgue spaces $L^{p_0}\left({ℝ}^n\right)$ .

Lebesgue spaces with variable exponent $L^{p(\cdot )}\left({ℝ}^n\right)$ were defined originally by Orlicz (see [1]). As the theory of the spaces with variable exponent was applied in some fields such as fluid dynamics, elasticity dynamics, Calculus of variations and differential equations with non-standard growth conditions (see [2]-[5]), the boundedness of some typical operators is being studied with keen interest (see [6]-[8]). The aim of this paper is to study the boundedness of the θ-type Calderón-Zygmund operators and their commutators in $L^{p(\cdot )}\left({ℝ}^n\right)$ .

In 1985, Yabuta introduced certain θ-type Calderón-Zygmund operators to facilitate his study of certain classes of pseudodifferential operators (see [9]). Since then, such type of operators is extensively applied in PDE with non-smooth area. Singular integral operators and variable exponent spaces play a fundamental role in the modern theory of partial differential equations (PDEs). These operators provide essential tools for establishing regularity and well-posedness results in non-standard function spaces, while variable exponent spaces offer a natural framework for problems with non-uniform ellipticity or growth conditions. Their interplay has led to significant advances in the analysis of nonlinear PDEs, especially in materials science and fluid dynamics, where physical properties exhibit sharp variations. Further applications have been found for such type of operators (see [10]-[13]).

With the further research, in [10], Quek and Yang introduced certain Calderón-Zygmund operator boundedness on space such as weighted Lebesgue spaces, weighted weak Lebesgue spaces, weighted Hardy spaces and weighted weak Hardy spaces. After that, Ri and Zhang obtained the boundedness of θ-type Calderón-Zygmund operator on Hardy spaces with non-doubling measures and non-homogeneous metric measure spaces (see [11] [12]), and Wang proved the boundedness of θ-type Calderón-Zygmund and commutators in the generalized weighted Morrey spaces (see [13]).

Definition 1.1 [9] Let $\theta$ be a non-negative, non-decreasing function on ${ℝ}^{+}=\left(0,\infty \right)$ satisfying

${\displaystyle {\int }_0^1\frac{\theta \left(t\right)}{t}\text{d}t}<\infty .$ (1.1)

A measurable function $K\left(\cdot ,\cdot \right)$ on $\left({ℝ}^n\times {ℝ}^n\left\{\left(x,x\right):x\in {ℝ}^n\right\}\right)$ is said to be a θ-type Calderón-Zygmund kernel if it satisfies

$\left|K\left(x,y\right)\right|\le C{\left|x-y\right|}^{-n}$ (1.2)

and

$\left|K\left(x,y\right)-K\left(x^{\prime },y\right)\right|+\left|K\left(y,x\right)-K\left(y,x^{\prime }\right)\right|\le C\theta \left(\frac{\left|x-x^{\prime }\right|}{\left|x-y\right|}\right){\left|x-y\right|}^{-n},$ (1.3)

when $\left|x-y\right|\ge 2\left|x-x^{\prime }\right|$ .

Definition 1.2 [9] Let $T_{\theta }$ be a linear operator from $\mathcal{S}$ into its dual ${\mathcal{S}}^{\prime }$ . One can say that $T_{\theta }$ is a θ-type Calderón-Zygmund operator if:

1) $T_{\theta }$ can be extended to be a bounded linear operator on $L^2\left({ℝ}^n\right)$ ;

2) There is a $T_{\theta }$ -type kernel $K\left(x,y\right)$ such that

$T_{\theta }f\left(x\right):={\displaystyle {\int }_{{ℝ}^n}K\left(x,y\right)f\left(y\right)\text{d}y}$ (1.4)

for all $f\in C_c^{\infty }\left({ℝ}^n\right)$ and for all $x\notin \text{supp}f$ , where $C_c^{\infty }\left({ℝ}^n\right)$ is the space consisting of all infinitely differentiable functions on ${ℝ}^n$ with compact supports.

Note that the classical Calderón-Zygmund operator with standard kernel (see [4] [14]) is a special case of θ-type operator $T_{\theta }$ when $\theta \left(t\right)=t^{\delta }$ with $0<\delta \le 1$ .

Definition 1.3 [15] Let $0<\alpha <1$ , $1<r,s<\infty$ . Then, the Besov spaces ${\dot{\wedge }}_{\alpha }^{r,s}\left({ℝ}^n\right)$ consists of all functions $f$ in $L^r\left({ℝ}^n\right)$ for which the norm

${\Vert f\Vert }_{{\dot{\wedge }}_{\alpha }^{r,s}\left({ℝ}^n\right)}={\Vert f\Vert }_r+{\left({\displaystyle {\int }_{{ℝ}^n}\frac{{\left({\Vert f\left(x+t\right)-f\left(x\right)\Vert }_r\right)}^s}{{\left|t\right|}^{n+\alpha s}}\text{d}t}\right)}^{\frac{1}{s}}$

is finite.

Definition 1.4 [16] Let . If there exist $C>0$ such that for any $x,y\in {ℝ}^n$ ,

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

$\left|p\left(x\right)-p\left(0\right)\right|\le \frac{C}{\text{log}\left(e+1/\left|x\right|\right)},\left|y\right|\ge \left|x\right|.$

then $p(\cdot )$ is said to satisfy the log-Hölder condition.

We denote

$p_{-}=\text{essinf}\left\{p\left(x\right):x\in {ℝ}^n\right\},p_{+}=\text{esssup}\left\{p\left(x\right):x\in {ℝ}^n\right\}.$

Then, consists all $p(\cdot )$ satisfying $p_{-}>1$ and $p_{+}<\infty$ .

Let $M$ be the Hardy-Littlewood maximal operator. We denote $ℬ\left({ℝ}^n\right)$ to be the set of all functions satisfying the condition that $M$ is bounded on $L^{p(\cdot )}\left({ℝ}^n\right)$ .

Given a measurable function $f\in L_{\text{loc}}\left({ℝ}^n\right)$ , the sharp maximal operators $M_{r,\rho }$ and $M^{�}$ are respectively defined by

$M_{r,\rho }f\left(x\right)=\underset{x\in Q}{\text{sup}}{\left(\frac{1}{{\left|Q\right|}^{1-\frac{r\rho }{n}}}{\displaystyle {\int }_Q{\left|f\left(y\right)\right|}^r\text{d}y}\right)}^{\frac{1}{r}},$

and

$M^{�}f\left(x\right)=\underset{x\in Q}{\text{sup}}{\displaystyle {\int }_Q\left|f\left(y\right)-f_Q\right|\text{d}y},$

where $f_Q=\frac{1}{\left|Q\right|}{\displaystyle {\int }_{Q}f\left(y\right)\text{d}y}$ , $r\ge 1$ and $0<\rho <\frac{n}{r}$ .

When $0<\beta \le 1$ , the Homogeneous Lipschitz spaces ${\text{Lip}}_{\beta }\left({ℝ}^n\right)$ is the space of functions such that

${\Vert f\Vert }_{{\text{Lip}}_{\beta }}=\underset{x,h\in {ℝ}^n;h\ne 0}{\text{sup}}\frac{\left|f\left(x+h\right)-f\left(x\right)\right|}{{\left|h\right|}^{\beta }}<\infty .$ (1.5)

Theorem 1.5 Suppose that $p(\cdot )\in ℬ\left({ℝ}^n\right)$ , $\theta$ satisfies (1.1). Then, there exists a constant $C$ independent of $f$ such that

${\Vert T_{\theta }\left(f\right)\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\le C{\Vert f\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}.$

Theorem 1.6 Suppose that $q\left(x\right)$ satisfies the log-Hölder’s inequality and $\theta$ satisfies

${\displaystyle {\int }_0^1\frac{\theta \left(t\right)\cdot \left|\log t\right|}{t}\text{d}t}<\infty .$ (1.6)

If $b\in {\dot{\wedge }}_{\alpha }^{r,s}\left({ℝ}^n\right),0<\alpha <1$ , then for any $f\left(x\right)\in L^{q(\cdot )}\left({ℝ}^n\right)$ , there exists a constant $C$ independent of $f$ such that

${\Vert \left[b,T_{\theta }\right]f\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\le C{\Vert b\Vert }_{{\dot{\wedge }}_{\alpha }^{r,s}}{\Vert f\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}.$

Theorem 1.7 Let $b\in {\text{Lip}}_{\beta }\left({ℝ}^n\right)$ . Suppose that $0<\beta <1$ , $\theta$ satisfies (1.6) and is such that $p_{+}<\frac{n}{\beta }$ . Define $q(\cdot )$ by

$\frac{1}{p\left(x\right)}-\frac{1}{q\left(x\right)}=\frac{\beta }{n}.$

If $q(\cdot )\left(1-\frac{\beta }{n}\right)\in ℬ\left({ℝ}^n\right)$ , then there exists a constant $C$ independent of $f$ such that

${\Vert \left[b,T_{\theta }\right]\left(f\right)\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\le C{\Vert b\Vert }_{{\text{Lip}}_{\beta }\left({ℝ}^n\right)}{\Vert f\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}.$

Theorem 1.8 Let $b\in \text{BMO}\left({ℝ}^n\right)$ . Suppose that $p(\cdot )\in ℬ\left({ℝ}^n\right)$ and $\theta$ satisfies (1.6). Then, there exists a constant $C$ independent of $f$ such that

${\Vert \left[b,T_{\theta }\right]\left(f\right)\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\le C{\Vert b\Vert }_{\ast }{\Vert f\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}.$

2. Preliminary Lemmas

Lemmas 2.1 [17] Let $p(\cdot )\in ℬ\left({ℝ}^n\right)$ , then for all $f\left(x\right)\in L^{p(\cdot )}\left({ℝ}^n\right)$ , we have

${\Vert f\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\lesssim {\Vert M^{�}f\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}.$

Lemmas 2.2 [15] Let $b\in {\dot{\wedge }}_{\alpha }^{r,s}\left({ℝ}^n\right)$ , then for $0<\alpha <1$ , $1<r<s<\infty$ , we have

$\underset{y\in B}{\text{sup}}\frac{1}{{\left|B\right|}^{1+\frac{r}{n}-\frac{1}{r}}}{\displaystyle {\int }_B\left|b-b_B\right|\text{d}y}\le \underset{y\in B}{\text{sup}}\frac{1}{{\left|B\right|}^{\frac{\alpha }{n}+\frac{1}{s}-\frac{1}{r}}}{\left({\displaystyle {\int }_B{\left|b-b_B\right|}^s\text{d}y}\right)}^{\frac{1}{s}}\le C{\Vert b\Vert }_{{\dot{\wedge }}_{\alpha }^{r,s}\left({ℝ}^n\right)}.$

Lemmas 2.3 [18] Let . Then, $C_c^{\infty }\left({ℝ}^n\right)$ is dense in $L^{p(\cdot )}\left({ℝ}^n\right)$ , where $C_c^{\infty }\left({ℝ}^n\right)$ denotes the infinity times differentiable functions on ${ℝ}^n$ with compact support set.

Lemmas 2.4 [14] Let . If endowing the spaces $L^{p(\cdot )}\left({ℝ}^n\right)$ with the following Orlicz-type norm:

${\Vert f\Vert }_{L^{p(\cdot )}}^0=\text{sup}\left\{{\displaystyle {\int }_{{ℝ}^n}\left|f\left(x\right)g\left(x\right)\text{d}x:{\Vert g\Vert }_{L^{p(\cdot )}}\le 1\right|}\right\},$

then the norm ${\Vert \cdot \Vert }_{L^{p(\cdot )}}^0$ above is equivalent to the Luxemburg-Nakano norm ${\Vert \cdot \Vert }_{L^{p(\cdot )}}$ .

Lemmas 2.5 [19] Let . Then, the following conditions are equivalent:

1) $p(\cdot )\in ℬ\left({ℝ}^n\right)$ .

2) $p^{\prime }(\cdot )\in ℬ\left({ℝ}^n\right)$ .

3) $p(\cdot )/q\in ℬ\left({ℝ}^n\right)$ for some $1<q<p_{-}$ .

4) ${\left(p(\cdot )/q\right)}^{\prime }\in ℬ\left({ℝ}^n\right)$ for some $1<q<p_{-}$ .

Lemmas 2.6 [20] Let . Then, for $f\in L^{p(\cdot )}\left({ℝ}^n\right)$ and $g\in L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)$ ,

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

where $C_p=1+1/p^{-}-1/p^{+}$ .

Given $0<\alpha <n$ , define the fractional integral operator $I_{\alpha }$ by

$I_{\alpha }\left(f\right)\left(x\right)={\displaystyle {\int }_{{ℝ}^n}\frac{f\left(y\right)}{{\left|x-y\right|}^{n-\alpha }}\text{d}y}.$

Lemmas 2.7 [20] Let be such that $p_{+}<n/\alpha$ and

$\frac{1}{p\left(x\right)}-\frac{1}{q\left(x\right)}=\frac{\alpha }{n}.$

If $q(\cdot )\left(n-\alpha \right)/n\in ℬ\left({ℝ}^n\right)$ , then

${\Vert I_{\alpha }\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\le C{\Vert f\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}.$

For $\delta >0$ , $f\in L_{\text{loc}}^{\delta }\left({ℝ}^n\right)$ , let

$M_{\delta }\left(f\right)\left(x\right)=\underset{x\in Q}{\text{sup}}{\left(\left|Q\right|{\displaystyle {\int }_Q{\left|f\left(y\right)\right|}^{\delta }\text{d}y}\right)}^{\frac{1}{\delta }},$

and

$f_{\delta }^{�}=\underset{x\in Q}{\text{sup}}\underset{c\in ℝ}{\text{inf}}{\left(\left|Q\right|{\displaystyle {\int }_Q{\left|f\left(y\right)-c\right|}^{\delta }\text{d}y}\right)}^{\frac{1}{\delta }},$

The non-increasing rearrangement of a measurable function $f$ on ${ℝ}^n$ is defined by

$f^{*}\left(t\right)=\text{inf}\left\{\lambda >0:|x\in {ℝ}^n:|f\left(x\right)>\lambda \left\}\right|\le t\right\},\left(0<t<\infty \right).$

Furthermore, for $\tau \in \left(0,1\right)$ and a measurable function $f$ on ${ℝ}^n$ , the local sharp maximal operator $M_{\tau }^{�}$ is defined by

$M_{\tau }^{�}\left(f\right)\left(x\right)=\underset{x\in Q}{\text{sup}}\underset{c\in ℝ}{\text{inf}}{\left(\left(f-c\right){\chi }_Q\right)}^{*}\left(\tau \left|Q\right|\right).$

Lemmas 2.8 [6] Let $\delta >0$ , $\tau \in \left(0,1\right)$ and $f\in L_{\text{loc}}^1\left({ℝ}^n\right)$ . Then, for any $x\in {ℝ}^n$

$M_{\tau }^{�}\left(f\right)\left(x\right)\le {\left(1/\tau \right)}^{1/\delta }{f}_{\delta }^{�}\left(x\right).$

Lemmas 2.9 [2] Let $g\in L_{\text{loc}}^1$ , $\tau \in \left(0,1\right)$ and a measurable function $f$ satisfying

$\left|\left\{x:\left|f\left(x\right)>\alpha \right|\right\}\right|<\infty \text{for}\text{all}\alpha >0.$ (2.1)

Then

${\displaystyle {\int }_{{ℝ}^n}\left|f\left(x\right)g\left(x\right)\right|\text{d}x}\le C{\displaystyle {\int }_{{ℝ}^n}{M}_{\tau }^{�}\left(f\right)\left(x\right)M\left(g\right)\left(x\right)\text{d}x}.$

Lemmas 2.10 [21] Suppose that $\theta$ satisfies (1.1) and ${\displaystyle {\int }_0^1\frac{\theta \left(t\right)\cdot \left|\log t\right|}{t}\text{d}t}<\infty$ respectively in 1) and 2). Then, for all $f\in C_c^{\infty }$ and $x\in {ℝ}^n$ , there exists a constant $C=C_{\delta >0}$ such that:

1) If $0<\delta <1$ , then

$M_{\delta }^{�}\left(T_{\theta }f\right)\left(x\right)\le CM\left(f\right)\left(x\right).$

2) If $0<\delta <l<1$ , $m\in \mathbb{N}$ and $b\in \text{BMO}\left({ℝ}^n\right)$ , then

${\left(\left(\left[b,T_{\theta }\right]f\right)\left(x\right)\right)}_{\delta }^{�}\le C\left({\Vert b\Vert }_{\ast }{M}_l\left(\left[b,T_{\theta }\right]\left(f\right)\right)\left(x\right)+{\Vert b\Vert }_{\ast }{M}^2\left(f\right)\left(x\right)\right).$

Lemmas 2.11 Let $1<r,s<\infty$ , $b\in {\dot{\wedge }}_{\alpha }^{r,s}\left({ℝ}^n\right)$ , $\frac{1}{t}+\frac{1}{s}=1$ . Then, for any $f\in C_c^{\infty }\left({ℝ}^n\right)$ , there is a constant $C$ , we have

$M^{�}\left(\left[b,T_{\theta }\right]f\right)\left(x\right)\lesssim m^{\frac{n}{2}}{\Vert b\Vert }_{{\dot{\wedge }}_{\alpha }^{r,s}\left({ℝ}^n\right)}\left[M_{t,\frac{\alpha r-n}{r}}\left(T_{\theta }f\right)\left(x\right)+M_{t,\frac{\alpha r-n}{r}}f\left(x\right)+M_{\frac{s}{\sqrt{s}-1}}f\left(x\right)\right],x\in {ℝ}^n.$

Proof Let $f\in C_c^{\infty }\left({ℝ}^n\right)$ . In order to prove our theorem, we only need to show that for any $x_0$ , there is a cube $Q$ center at $x_0$ , and a constant $C_Q$ , such that

$\begin{array}{l}\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q\left|\left[b,T_{\theta }\right]f\left(y\right)-C_Q\right|\text{d}y}\\ \lesssim m^{\frac{n}{2}}{\Vert b\Vert }_{{\dot{\wedge }}_{\alpha }^{r,s}}\left[M_{t,\frac{\alpha r-n}{r}}\left(T_{\theta }f\right)\left(x\right)+M_{t,\frac{\alpha r-n}{r}}f\left(x\right)+\underset{x\in Q^{*}}{\text{sup}}Mf\left(x\right)\right],x\in {ℝ}^n\end{array}$

Decompose $f$ as

$f=f_1+f_2=f{\chi }_{Q^{*}}+f{\chi }_{{ℝ}^n\backslash Q^{*}},$

where $Q^{*}$ denotes to be a cube, which center is the same with $Q$ and $l\left(Q^{*}\right)=2\sqrt{n}l\left(Q\right)$ .

Set $C_Q=T_{\theta }\left(b-b_{Q^{*}}{f}_2\right)\left(x_0\right)$ . Noting that $\left[b,T_{\theta }\right]f=\left[\left(b-b_{Q^{*}}\right),T_{\theta }\right]f$ , then it follows

$\begin{array}{l}\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q\left|\left[b,T_{\theta }\right]f\left(y\right)-C_Q\right|\text{d}y}\\ \le \frac{1}{\left|Q\right|}{\displaystyle {\int }_Q\left|b\left(y\right)-b_{Q^{*}}\right|\left|T_{\theta }f\left(y\right)\right|\text{d}y}+\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q\left|T_{\theta }\left(\left(b-b_{Q^{*}}\right)f_1\right)\left(y\right)\right|\text{d}y}\\ +\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q\left|T_{\theta }\left(\left(b-b_{Q^{*}}\right)f_2\right)\left(y\right)-T_{\theta }\left(\left(b-b_{Q^{*}}\right)f_2\right)\left(x_0\right)\right|\text{d}y}\\ =I_1+I_2+I_3.\end{array}$

We first estimate the term $I_1$ . By using the Hölder’s inequality and Lemma 2.2, we have

$\begin{array}{c}{I}_1=\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q\left|b\left(y\right)-b_{Q^{*}}\right|\left|T_{\theta }f\left(y\right)\right|\text{d}y}\\ \le {\left(\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q{\left|b\left(y\right)-b_{Q^{*}}\right|}^s\text{d}y}\right)}^{\frac{1}{s}}{\left(\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q{\left|T_{\theta }f\left(y\right)\right|}^t\text{d}y}\right)}^{\frac{1}{t}}\\ \le \frac{1}{{\left|Q\right|}^{\frac{1}{r}-\frac{\alpha }{n}}}{\Vert b\Vert }_{{\dot{\wedge }}_{\alpha }^{r,s}}{\left(\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q{\left|T_{\theta }f\left(y\right)\right|}^t\text{d}y}\right)}^{\frac{1}{t}}\\ \le C{\Vert b\Vert }_{{\dot{\wedge }}_{\alpha }^{r,s}}{M}_{t,\frac{\alpha r-n}{r}}\left(T_{\theta }f\right)\left(x\right).\end{array}$

Secondly, we consider $I_2$ . Taking $p:=\sqrt{s}$ , by using the Hölder’s inequality and the fact that ${\Vert T_{\theta }f\Vert }_{L^p\left({ℝ}^n\right)}\le C{\Vert f\Vert }_{L^p\left({ℝ}^n\right)}$ [13], then we have

$\begin{array}{c}{I}_2\le \frac{1}{\left|Q\right|}{\displaystyle {\int }_Q\left|T_{\theta }\left(\left(b-b_{Q^{*}}\right)f_1\right)\left(y\right)\right|\text{d}y}\\ \le C\frac{1}{\left|Q\right|}{\left({\displaystyle {\int }_Q{\left|T_{\theta }\left(\left(b-b_{Q^{*}}\right)f_1\right)\left(y\right)\right|}^p\text{d}y}\right)}^{\frac{1}{p}}{\left|Q\right|}^{1-\frac{1}{p}}\\ \le C\frac{1}{\left|Q\right|}{\left({\displaystyle {\int }_{Q^{*}}{\left|b\left(y\right)-b_{Q^{*}}\right|}^p{\left|f\left(y\right)\right|}^p\text{d}x}\right)}^{\frac{1}{p}}{\left|Q\right|}^{1-\frac{1}{p}}\\ \le C{\left({\displaystyle {\int }_{Q^{*}}{\left|b\left(y\right)-b_{Q^{*}}\right|}^{pp}}\right)}^{\frac{1}{pp}}{\left({\displaystyle {\int }_{Q^{*}}{\left|f\left(y\right)\right|}^{p{p}^{\prime }}\text{d}x}\right)}^{\frac{1}{p{p}^{\prime }}}{\left|Q\right|}^{1-\frac{1}{p}}\\ \le C{\left({\displaystyle {\int }_{Q^{*}}{\left|b\left(y\right)-b_{Q^{*}}\right|}^s}\right)}^{\frac{1}{s}}\frac{1}{{\left|Q\right|}^{\frac{1}{p}}}{\left({\displaystyle {\int }_{Q^{*}}{\left|f\left(y\right)\right|}^{p{p}^{\prime }}\text{d}x}\right)}^{\frac{1}{p{p}^{\prime }}}\\ \le C{\left(\frac{1}{\left|Q^{*}\right|}{\displaystyle {\int }_{Q^{*}}{\left|b\left(y\right)-b_{Q^{*}}\right|}^s}\right)}^{\frac{1}{s}}\frac{1}{{\left|Q\right|}^{\frac{1}{p}}}{\left(\frac{1}{\left|Q^{*}\right|}{\displaystyle {\int }_{Q^{*}}{\left|f\left(y\right)\right|}^{p{p}^{\prime }}\text{d}x}\right)}^{\frac{1}{p{p}^{\prime }}}{\left|Q^{*}\right|}^{\frac{1}{p{p}^{\prime }}}{\left|Q^{*}\right|}^{\frac{1}{s}}\\ \le C{\Vert b\Vert }_{{\dot{\wedge }}_{\alpha }^{r,s}\left({ℝ}^n\right)}{M}_{\frac{s}{\sqrt{s}-1}}\left(f\right)\left(x\right).\end{array}$

Now, we turn to estimate of $I_3$ . Noting the fact that

${\displaystyle \sum _{k=0}^{\infty }\theta }\left(2^{-k}\right)\le C{\displaystyle \sum _{k=0}^{\infty }{\displaystyle {\int }_{2^{-k+1}}^{2^{-k}}\frac{\theta \left(2^{-k}\right)}{2^{1-k}}\text{d}t}}\le C{\displaystyle {\int }_0^1\frac{\theta \left(t\right)}{t}\text{d}t}<\infty .$

For $I_3$ , $y\in Q$ , $z\in \left(Q^{*}\right)$ , so we have $2\left|y-x_0\right|\le \left|y-z\right|$ , one has

$\begin{array}{c}{I}_3=\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q\left|T_{\theta }\left(\left(b-b_{Q^{*}}\right)f_2\right)\left(y\right)-T_{\theta }\left(\left(b-b_{Q^{*}}\right)f_2\right)\left(x_0\right)\right|\text{d}y}\\ \le C\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q{\displaystyle {\int }_{{ℝ}^n\backslash Q^{*}}\left|\left(b\left(z\right)-b_{Q^{*}}\right)f\left(z\right)\right|\cdot \left|K\left(y,z\right)-K\left(x_0,z\right)\right|\text{d}z\text{d}y}}\\ \le C\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q{\displaystyle \sum _{k=0}^{\infty }{\displaystyle {\int }_{2^{k+1}{Q}^{*}\backslash 2^k{Q}^{*}}\left|\left(b\left(z\right)-b_{Q^{*}}\right)f\left(z\right)\right|\cdot \frac{1}{{\left|y-z\right|}^n}\theta \left(\frac{\left|x_0-y\right|}{\left|y-z\right|}\right)\text{d}z\text{d}y}}}\\ \le C\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q{\displaystyle \sum _{k=0}^{\infty }\theta }}\left(2^{-k}\right){\displaystyle {\int }_{2^{k+1}{Q}^{*}\backslash 2^k{Q}^{*}}\left|\left(b\left(z\right)-b_{Q^{*}}\right)f\left(z\right)\right|\frac{1}{\left|2^{k+1}{Q}^{*}\right|}\theta \left(2^{-k}\right)\text{d}z\text{d}y}\\ \le C{\displaystyle \sum _{k=0}^{\infty }\theta }\left(2^{-k}\right){\left({\displaystyle {\int }_{2^{k+1}{Q}^{*}}\frac{1}{\left|2^{k+1}{Q}^{*}\right|}{\left|b\left(z\right)-b_{Q^{*}}\right|}^s\text{d}z}\right)}^{\frac{1}{s}}{\left(\frac{1}{\left|2^{k+1}{Q}^{*}\right|}\cdot {\displaystyle {\int }_{2^{k+1}{Q}^{*}}{\left|f\left(z\right)\right|}^t\text{d}z}\right)}^{\frac{1}{t}}\\ \le C{\Vert b\Vert }_{{\dot{\wedge }}_{\alpha }^{r,s}}{M}_{t,\frac{\alpha r-n}{r}}f\left(x\right).\end{array}$

Combining the estimates for $I_1,I_2$ and $I_3$ , the proof of Lemma 2.11 is finished.

Then, we have the following conclusion.

3. Proof of Main Results

Proof of Theorem 1.5. By applying the extension of Rubio de Francia’s extrapolation theorem in the scale of the variable Lebesgue spaces [22], together with $T_{\theta }$ is bounded in $L^p\left({ℝ}^n\right)$ [10], and combining with the results of Cruz-Uribe and Wang [23], we can know, let and $\omega$ be a weight. If the Hardy-Littlewood maximal operator $M$ is bounded on $L^{p(\cdot )}\left(\omega \right)$ and on $L^{p^{\prime }(\cdot )}\left({\omega }^{-\frac{1}{p(\cdot )-1}}\right)$ , then holds for all $f\in L^{p(\cdot )}\left(\omega \right)$ and all measurable functions $g$ , we have

${\Vert T_{\theta }\left(f\right)\Vert }_{{\Vert f\Vert }_{L^{p(\cdot )}\left(\omega \right)}}\le C{\Vert f\Vert }_{{\Vert f\Vert }_{L^{p(\cdot )}\left(\omega \right)}}.$

Due to the boundedness of the $T_{\theta }$ in the aforementioned space $L^{p(\cdot )}\left(\omega \right)$ , weakening the conditions of $p\left(x\right)$ , then boundedness of $T_{\theta }$ in the $L^{p(\cdot )}\left({ℝ}^n\right)$ must hold, namely

${\Vert T_{\theta }\left(f\right)\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\le C{\Vert f\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}.$

This finishes the proof of Theorem 1.5.

Proof of Theorem 1.6. From Lemma 2.1 and Theorem 2.11, we easily see

$\begin{array}{c}{\Vert \left[b,T_{\theta }\right]f\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\le C{\Vert M^{�}\left[b,T_{\theta }\right]f\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\\ \le C{\Vert b\Vert }_{{\dot{\wedge }}_{\alpha }^{r,s}\left({ℝ}^n\right)}[{\Vert M_{t,\frac{\alpha r-n}{r}}\left(Tf\right)\left(x\right)\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\\ +{\Vert M_{t,\frac{\alpha r-n}{r}}f\left(x\right)\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}+{\Vert M_{\frac{s}{\sqrt{s}-1}}f\left(x\right)\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}].\end{array}$

It is easy to verify that ${\Vert M_{\frac{s}{\sqrt{s}-1}}f\left(x\right)\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\le C{\Vert f\left(x\right)\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}$ , and

$\begin{array}{c}{\Vert M_{t,\frac{\alpha r-n}{r}}\left(Tf\right)\left(x\right)\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}={\Vert {\left(M_{1,\frac{\left(\alpha r-n\right)t}{r}}{\left|T_{\theta }f\right|}^t\right)}^{\frac{1}{t}}\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\\ ={\Vert M_{1,\frac{\left(\alpha r-n\right)t}{r}}{\left|T_{\theta }f\right|}^t\Vert }_{L^{\frac{q(\cdot )}{t}}\left({ℝ}^n\right)}^{\frac{1}{t}}\\ \le C{\Vert {\left|T_{\theta }f\right|}^t\Vert }_{L^{\frac{q(\cdot )}{t}}\left({ℝ}^n\right)}^{\frac{1}{t}}\\ \le C{\Vert f\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}.\end{array}$

Also, we can get

${\Vert M_{t,\frac{\alpha r-n}{r}}f\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\le C{\Vert f^t\Vert }_{\frac{L^{q(\cdot )}\left({ℝ}^n\right)}{t}}^{\frac{1}{t}}={\Vert f\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}.$

Thus, we have

${\Vert \left[b,T_{\theta }\right]f\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\le C{m}^{\frac{n}{2}}{\Vert b\Vert }_{{\dot{\wedge }}_{\alpha }^{r,s}}{\Vert f\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}.$

This finishes the proof of Theorem 1.6.

Proof of Theorem 1.7. Let $b\in {\text{Lip}}_{\beta }\left({ℝ}^n\right)$ , $0<\beta <1$ . Then, by (1.5), we shall get

$\left|b\left(x\right)-b\left(y\right)\right|\le {\left|x-y\right|}^{\beta }{\Vert b\Vert }_{{\text{Lip}}_{\beta }}.$

Thus, for any $f\in L^{p(\cdot )}\left({ℝ}^n\right)$ ,

$\left[b,T_{\theta }\right]\left(f\right)={\displaystyle {\int }_{{ℝ}^n}\left[b\left(x\right)-b\left(y\right)\right]K\left(x,y\right)f\left(y\right)\text{d}y}$

$\begin{array}{c}\left|\left[b,T_{\theta }\right]\left(f\right)\right|=\left|{\displaystyle {\int }_{{ℝ}^n}\left[b\left(x\right)-b\left(y\right)\right]K\left(x,y\right)f\left(y\right)\text{d}y}\right|\\ \le C{\displaystyle {\int }_{{ℝ}^n}{\left|x-y\right|}^{\beta }{\Vert b\Vert }_{{\text{Lip}}_{\beta }}K\left(x,y\right)f\left(y\right)\text{d}y}\\ \le C{\displaystyle {\int }_{{ℝ}^n}{\Vert b\Vert }_{{\text{Lip}}_{\beta }}\frac{\left|f\left(y\right)\right|}{{\left|x-y\right|}^{n-\beta }}f\left(y\right)\text{d}y}\\ \le C{\Vert b\Vert }_{{\text{Lip}}_{\beta }}{I}_{\beta }\left(\left|f\right|\right).\end{array}$

Applying Lemma 2.7, we get

${\Vert \left[b,T_{\theta }\right]\left(f\right)\Vert }_{L^{q(\cdot )}\left({ℝ}^n\right)}\le C{\Vert b\Vert }_{{\text{Lip}}_{\beta }}{I}_{\beta }\left(\left|f\right|\right)\le C{\Vert b\Vert }_{{\text{Lip}}_{\beta }}{\Vert f\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}.$

Hence, the proof of Theorem 1.7 is finished.

Proof of Theorem 1.8. Let $b\in \text{BMO}\left({ℝ}^n\right)$ , $f\in C_c^{\infty }\left({ℝ}^n\right)$ . Then, by Lemma 2.3, we have $f\in L^{p(\cdot )}\left({ℝ}^n\right)$ . For any $g\in L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)\in L_{\text{loc}}^1\left({ℝ}^n\right)$ to be ${\Vert g\Vert }_{L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)}\le 1$ , noting the $\left[b,T_{\theta }\right]$ is bounded on usual Lebesgue spaces $L^p\left({ℝ}^n\right)$ (see [13]), so it satisfies (2.1) in Lemma 2.9.

Thus, applying Lemma 2.9 and 2.10, we obtain

$\begin{array}{c}{\displaystyle {\int }_{{ℝ}^n}\left[b,T_{\theta }\right]\left(f\right)\left(x\right)g\left(x\right)\text{d}x}\le C{\displaystyle {\int }_{{ℝ}^n}{M}_{\tau }^{�}\left(\left[b,T_{\theta }\right]\right)\left(f\right)\left(x\right)M\left(g\right)\left(x\right)\text{d}x}\\ \le C{\displaystyle {\int }_{{ℝ}^n}{\left(1/\tau \right)}^{1/\delta }{\left(\left[b,T_{\theta }\right]\left(f\right)\right)}_{\delta }^{�}\left(x\right)M\left(g\right)\left(x\right)\text{d}x},\end{array}$

where $\delta ,\tau \in \left(0,1\right)$ .

By 2) in Lemma 2.10, for $0<\delta <l<1$

$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^n}\left[b,T_{\theta }\right]\left(f\right)\left(x\right)g\left(x\right)\text{d}x}\\ \le C{\Vert b\Vert }_{\ast }{\displaystyle {\int }_{{ℝ}^n}{M}_l\left(\left[b,T_{\theta }\right]\left(f\right)\right)\left(x\right)M\left(g\right)\left(x\right)\text{d}x}\\ +C{\Vert b\Vert }_{\ast }{\displaystyle {\int }_{{ℝ}^n}{M}^2\left(f\right)\left(x\right)M\left(g\right)\left(x\right)\text{d}x}\\ =I_1+I_2.\end{array}$

Observing that for $0<l<1$ , we have

$M_l\left[b,T_{\theta }\right]\left(f\right))\left(x\right)\le M\left(\left[b,T_{\theta }\right]\left(f\right)\right)\left(x\right)\text{a}.\text{e}.x\in {ℝ}^n.$

By Lemma 2.5 and the generalized Hölder’s inequality (Lemma 2.6), for $p(\cdot )\in ℬ\left({ℝ}^n\right)$ , we have

$\begin{array}{c}{I}_1=C{\Vert b\Vert }_{\ast }{\displaystyle {\int }_{{ℝ}^n}{M}_l\left(\left[b,T_{\theta }\right]\left(f\right)\right)\left(x\right)M\left(g\right)\left(x\right)\text{d}x}\\ \le C{\Vert b\Vert }_{\ast }{\displaystyle {\int }_{{ℝ}^n}M\left(\left[b,T_{\theta }\right]\left(f\right)\right)\left(x\right)M\left(g\right)\left(x\right)\text{d}x}\\ \le C{\Vert b\Vert }_{\ast }{\Vert M\left(T_{\theta }\left(f\right)\right)\Vert }_{L^{p(\cdot )}}{\Vert Mg\Vert }_{L^{p^{\prime }(\cdot )}}\\ \le C{\Vert b\Vert }_{\ast }{\Vert T_{\theta }\left(f\right)\Vert }_{L^{p(\cdot )}}{\Vert Mg\Vert }_{L^{p^{\prime }(\cdot )}}\\ \le C{\Vert b\Vert }_{\ast }{\Vert f\Vert }_{L^{p(\cdot )}}{\Vert Mg\Vert }_{L^{p^{\prime }(\cdot )}}.\end{array}$

On the other hand, also using Lemma 2.5 and the generalized Hölder’s inequality (Lemma 2.6), for $p(\cdot )\in ℬ\left({ℝ}^n\right)$ ,

$\begin{array}{c}{I}_2\le C{\Vert b\Vert }_{\ast }{\Vert M^2\left(f\right)\Vert }_{L^{p(\cdot )}}{\Vert Mg\Vert }_{L^{p^{\prime }(\cdot )}}\\ \le C{\Vert b\Vert }_{\ast }{\Vert M\left(f\right)\Vert }_{L^{p(\cdot )}}{\Vert g\Vert }_{L^{p^{\prime }(\cdot )}}\\ \le C{\Vert b\Vert }_{\ast }{\Vert f\Vert }_{L^{p(\cdot )}}.\end{array}$

According to the estimates of $I_1$ and $I_2$ above and Lemma 2.5, we can obtain

${\displaystyle {\int }_{{ℝ}^n}\left|\left[b,T_{\theta }\right]\left(f\right)\left(x\right)g\left(x\right)\right|\text{d}x}\le I_1+I_2\le C{\Vert b\Vert }_{\ast }{\Vert f\Vert }_{L^{p(\cdot )}},$

and

${\Vert \left[b,T_{\theta }\right]\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\le C{\Vert \left[b{T}_{\theta }\right]\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}^0\le C{\Vert b\Vert }_{\ast }{\Vert f\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}.$

Hence, by Lemma 2.3, for any $f\in L^{p(\cdot )}\left({ℝ}^n\right)$ , we have

${\Vert \left[b,T_{\theta }\right]\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}\le C{\Vert b\Vert }_{\ast }{\Vert f\Vert }_{L^{p(\cdot )}\left({ℝ}^n\right)}.$

This finishes the proof of Theorem 1.8.

Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant No. 12361018) and Key Laboratory of Computational Science and Application of Hainan Province (Grant No. JSKX 202304).

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.