The commutator generated by a linear operator $T$ and a locally integrable function $b$ is defined by

$T_b\left(f\right):=bT\left(f\right)-T\left(bf\right).$ (2.1)

The investigation into the operator $T_b$ commenced with the groundbreaking studied by Coifman-Rochberg-Weiss in [13] . Two primary motivations underlie the consideration of commutators in this context. First, the boundedness of commutators provides valuable insights into the characterization of function spaces [14] - [20] . Second, commutator theory is of great importance in analyzing the regularity of solutions to second-order elliptic and parabolic partial differential equations [21] - [23] . The well-posedness of solutions to many PDEs depends on the boundedness of commutators for integral operators [24] - [26] . Fractional integrals play an important role in harmonic analysis and PDEs.

Let $0<\alpha <n$. Then the fractional integral is defined by

$I_{\alpha }f\left(x\right)={\displaystyle {\int }_{{ℝ}^n}\frac{f\left(y\right)}{{\left|y-x\right|}^{n-\alpha }}\text{d}y}$ (2.2)

Let $b$ be a locally integrable function. $I_{\alpha ,b}^m$ is the m-th commutator generated by $I_{\alpha }$ and $b$, which is defined by

$I_{\alpha ,b}^m\left(f\right)={\displaystyle {\int }_{{ℝ}^n}{\left(b\left(x\right)-b\left(y\right)\right)}^m\frac{f\left(y\right)}{{\left|x-y\right|}^{n-\alpha }}\text{d}y}$ (2.3)

In [27] , Chanillo studied the characterization of BMO space via boundedness of $I_{\alpha ,b}$. Paluszynski derived the characterization of the ${\text{Lip}}_{\beta }\left({ℝ}^n\right)$ space, which involves the boundedness of $I_{\alpha ,b}$ [19] . In contrast to the extensive and profound results regarding commutators with symbol functions in BMO spaces and lipschite spaces, research on Morrey spaces has been relatively scarce. However, recent developments have emerged, providing innovative characterizations of Morrey spaces through the analysis of the boundedness of commutators associated with Calderón-Zygmund singular integral operators. These developments utilize novel methodologies, moving away from the traditional dependence on the sharp maximal function theorem [28] . This previous breakthrough marks the initial exploration of commutator studies where the symbol function belongs to Morrey spaces.

Let $f$ be a locally integrable function on ${ℝ}^n$. Hardy-Littlewood maximal operator $M$ is defined by

$Mf\left(x\right):=\underset{x\in B}{\text{sup}}\frac{1}{\left|B\right|}{\displaystyle {\int }_B\left|f\left(y\right)\right|\text{d}y}.$ (2.4)

given $\alpha$, $0<\alpha <n$ , the fractional maximal operator $M_{\alpha }$ is defined by

$M_{\alpha }f\left(x\right):=\underset{x\in B}{\text{sup}}\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}{\displaystyle {\int }_B\left|f\left(y\right)\right|\text{d}y}.$ (2.5)

where the supremum is again taken over all balls B which contain $x$. In the limiting case $\alpha =0$, the fractional maximal operator reduces to the Hardy-Littlewood maximal operator.

Let $p(\cdot )$ be a measurable function on ${ℝ}^n$ taking values in $\left[1,\infty \right)$, the Lebesgue space with variable exponent $L^{p(\cdot )}\left({ℝ}^n\right)$ is defined by

$L^{p(\cdot )}\left({ℝ}^n\right):=\left\{f\text{is}\text{measurable}\text{on}{ℝ}^n:{\displaystyle {\int }_{{ℝ}^n}{\left(\frac{\left|f\left(x\right)\right|}{\lambda }\right)}^{p\left(x\right)}\text{d}x}<\infty \text{for}\text{some}\lambda >0\right\}.$ (2.6)

Then $L^{p(\cdot )}\left({ℝ}^n\right)$ is a Banach function space equipped with the norm

${\Vert f\Vert }_{L^{p(\cdot )}}:=\text{inf}\left\{\lambda >0:{\displaystyle {\int }_{{ℝ}^n}{\left(\frac{\left|f\left(x\right)\right|}{\lambda }\right)}^{p\left(x\right)}\text{d}x}\le 1\right\}.$ (2.7)

The space $L_{\text{loc}}^{p(\cdot )}\left({ℝ}^n\right)$ is defined by $L_{\text{loc}}^{p(\cdot )}\left({ℝ}^n\right):=\{f:f{\chi }_K\in L^{p(\cdot )}\left({ℝ}^n\right)$ for all compact subsets $K\subset {ℝ}^n\}$, where and what follows, ${\chi }_S$ denotes the characteristic function of a measurable set $S\subset {ℝ}^n$. Let $p(\cdot ):{ℝ}^n\to \left(0,\infty \right)$, we denote $p^{-}:=\text{ess}\underset{x\in {ℝ}^n}{\text{inf}}p\left(x\right)$, $p^{+}:=\text{ess}\underset{x\in {ℝ}^n}{\text{sup}}p\left(x\right)$. The set consists of all $p(\cdot )$ satisfying $p^{-}>1$ and $p^{+}<\infty$; consists of all $p(\cdot )$ satisfying $p^{-}>0$ and $p^{+}<\infty$. $L^{p(\cdot )}$ can be similarly defined as above for . Let $p^{\prime }(\cdot )$ be the conjugate exponent of $p(\cdot )$, that means $1/p(\cdot )+1/p^{\prime }(\cdot )=1$.

Basic facts concerning Lebesgue spaces with variable exponent can be found in [29] [30]

Lemma 2.1 see [31] Let $f$ belong to the Lebesgue space $L^{p(\cdot )}\left({ℝ}^n\right)$ and $g$ belong to $L^{p^{\prime }(\cdot )}\left({ℝ}^n\right)$, where . Then, the product function $fg$ is integrable over ${ℝ}^n$, and the following inequality holds.

${\displaystyle {\int }_{{ℝ}^n}\left|f\left(x\right)g\left(x\right)\right|\text{d}x}\le r_p{\Vert f\Vert }_{p(\cdot )}{\Vert g\Vert }_{p^{\prime }(\cdot )},$ (2.8)

where $r_p$ is defined as $r_p=1+\frac{1}{p_{-}}-\frac{1}{p_{+}}$.

Theorem 2.2 see [32] Let Ω be an open set in ${ℝ}^n$, and $\alpha$, $0<\alpha <n$, Let $p(\cdot ):\Omega \to \left[1,\infty \right)$, be such that $1<p_{-}\le p_{+}<n/\alpha$. Suppose further that $p(\cdot )$ satisfies

(i) For any $x,y\in \Omega$, $\left|x-y\right|<1/2$, if

$\left|p\left(x\right)-p\left(y\right)\right|\le \frac{C}{-\log \left|x-y\right|},$ (2.9)

(ii) For any $x,y\in \Omega$, $\left|y\right|\ge \left|x\right|$, if

$\left|p\left(x\right)-p\left(y\right)\right|\le \frac{C}{\log \left(e+\left|x\right|\right)},$ (2.10)

Define $q(\cdot ):\Omega \to \left[1,\infty \right)$, $x\in \Omega$, by

$\frac{1}{p\left(x\right)}-\frac{1}{q\left(x\right)}=\frac{\alpha }{n}$ (2.11)

Then the fractional maximal operator $M_{\alpha }$ is bounded from $L^{p(\cdot )}\left(\Omega \right)$ to $L^{q(\cdot )}\left(\Omega \right)$.

Theorem 2.3 see [32] Let $\Omega ,\alpha ,p(\cdot )$ and $q(\cdot )$ are as in Theorem 2.2, then the fractional integral operator $I_{\alpha }$ is bounded from $L^{p(\cdot )}\left(\Omega \right)$ to $L^{q(\cdot )}\left(\Omega \right)$.

The Morrey spaces ${ℳ}^{p(\cdot ),\lambda (\cdot )}\left(\Omega \right)$ with variable exponents $p(\cdot )$ and $\lambda (\cdot )$ over an open set $\Omega \subset {ℝ}^n$ have recently been introduced almost simultaneously by different authors. Relevant works can be found in [33] - [37] .

Let $\lambda (\cdot )$ be a measurable function on Ω with the values in $\left[0,n\right]$. The variable exponents Morrey space ${ℳ}^{p(\cdot ),\lambda (\cdot )}\left(\Omega \right)$ is defined [33] as the set of measurable functions f on Ω with the finite norm

${\Vert f\Vert }_{{ℳ}^{p(\cdot ),\lambda (\cdot )}\left(\Omega \right)}=\underset{x\in \Omega ,r>0}{\sup }{r}^{-\frac{\lambda \left(x\right)}{p\left(x\right)}}{\Vert f{\chi }_{B^{\prime }\left(x,r\right)}\Vert }_{L^{p(\cdot )}\left(\Omega \right)}.$ (3.1)

The Campanato space ${\mathcal{C}}^{p(\cdot ),\lambda (\cdot )}\left(\Omega \right)$ of variable order, in the Euclidean case, are defined [38] via the norm

${\Vert f\Vert }_{{\mathcal{c}}^{p(\cdot ),\lambda (\cdot )}\left(\Omega \right)}:={\Vert f\Vert }_{L^1\left(\Omega \right)}+\underset{x\in \Omega ,r>0}{\sup }{r}^{-\frac{\lambda \left(x\right)}{p\left(x\right)}}{\Vert f-f_{B^{\prime }\left(x,r\right)}\Vert }_{L^{p(\cdot )}\left(\Omega \right)}.$ (3.2)

Let $p(\cdot ):{ℝ}^n\to \left[1,\infty \right)<\infty$ and $\omega \left(x,\cdot \right):\left(0,\infty \right)\to \left(0,\infty \right)$. $\omega \left(x,r\right)$ satisfies the doubling condition

$\omega \left(x,2r\right)\le C\omega \left(x,r\right),\text{for}\text{any}x\in \Omega ,r>0,$

where $C>0$ is a constant independent of $x$ and $r$.

Let $B$ be the ball of centre at $x$ and radius $r$. The generalized variable Morrey space ${ℳ}^{p(\cdot ),\omega }$ was introduced in [39] as the space of functions equipped with the norm

${\Vert f\Vert }_{{ℳ}^{p(\cdot ),\omega }}=\underset{B}{\text{sup}}\frac{1}{\omega \left(x,r\right)}{r}^{-\frac{n}{p\left(x\right)}}{\Vert f{\chi }_{B\left(x,r\right)}\Vert }_{L^{p(\cdot )}}.$ (3.3)

The generalized variable Campanato space ${\mathcal{C}}^{p(\cdot ),\omega }$ is defined by the norm

${\Vert f\Vert }_{{\mathcal{C}}^{p(\cdot ),\omega }}=\underset{B}{\sup }\frac{1}{\omega \left(x,r\right)}{r}^{-\frac{n}{p\left(x\right)}}{\Vert \left(f-f_B\right){\chi }_{B\left(x,r\right)}\Vert }_{L^{p(\cdot )}}.$ (3.4)

The generalized Campanato spaces with variable exponents are a key link in the “space-operator-equation” chain in functional analysis. They not only generalize the theory of classical spaces but also provide a powerful functional analysis framework for dealing with problems of non-uniform regularity, such as analysis under variable coefficients and inhomogeneous media.

In [40] , a characterization of Campanato spaces was provided via the boundedness of $I_{\alpha ,b}$ under the assumption that $b$ satisfies a mean value inequality. In [41] , authors respectively characterized the strong and weak type boundedness of $I_{\alpha ,b}$ on the generalized Morrey spaces. Inspired by these results, in this paper, we obtain a characterization of ${\mathcal{C}}^{p(\cdot ),\omega }$ via the boundedness of $I_{\alpha ,b}$ under a suitable assumption that $b$ satisfies the mean value inequality. A function $f$ is said to satisfy the well-known mean value inequality if there exists a constant $C>0$ such that for any ball $B$ in ${ℝ}^n$,

$\underset{B}{\text{sup}}\left|f\left(x\right)-f_B\right|\le \frac{C}{\left|B\right|}{\displaystyle {\int }_B\left|f\left(x\right)-f_B\right|\text{d}x}.$ (3.5)

(3.5) is also called the reverse Hölder class which contains many kinds of functions. For example, if $P\left(x\right)$ is a polynomial and $\xi >0$, then $f\left(x\right)={\left|P\left(x\right)\right|}^{\xi }$ satisfies (3.5) [42] . Besides polynomial functions, the mean value equalities also characterize harmonic functions [43] . For more theories about the reverse Hölder classes see [44] and [45] for example.

Our main results can be stated as follows.

Theorem 3.1 Let $0<\alpha <n$, $p(\cdot ):\Omega \to \left[1,\infty \right)<\infty$, $\frac{1}{p\left(x\right)}-\frac{1}{q\left(x\right)}=\frac{\alpha }{n}$, $\omega$ is a non-negative measurable function on $\Omega \times \left(0,\ell \right)$, $\ell =diam\Omega$, and $\omega$ satisfies the doubling condition. Then the fractional integral $I_{\alpha }$ is a bounded operator from ${ℳ}^{p(\cdot ),\omega }$ to ${ℳ}^{q(\cdot ),\omega }$.

Theorem 3.2 Let $p(\cdot ):{ℝ}^n\to \left[1,\infty \right)<\infty$. $\omega \left(x,r\right)$ satisfies the doubling condition, $\omega ={\omega }_1{\omega }_2$, $0<\alpha <n$, $b$ satisfy (3.5). Then the following statements are equivalent.

(a) $b\in {\mathcal{C}}^{p_1(\cdot ),{\omega }_1}$;

(b) $I_{\alpha ,b}$ is a bounded operator from ${ℳ}^{q(\cdot ),{\omega }_2}$ to ${\mathcal{C}}^{p(\cdot ),\omega }$;

(c) For any ball $B\subset {ℝ}^n$ and $m\in {\mathbb{N}}^{+}$, there exists a constant $C>0$ such that

${\Vert I_{\alpha ,b}^{m}f\Vert }_{{\mathcal{C}}^{p(\cdot ),\omega }}\le C{\Vert f\Vert }_{{ℳ}^{q(\cdot ),{\omega }_2}}$

As mentioned in a prior study, solutions to a substantial category of elliptic second-order PDEs fulfill the mean value inequality (3.5). In this manner, our primary findings can provide characterizations for the spaces of solutions to some second-order elliptic PDEs. Let $b$ be a solution to the following Laplace equation

$\text{Δ}u=0$ (3.6)

where $\text{Δ}$ is the Laplace operator and $u$ is a function defined on a bounded domain $\Omega \subset {ℝ}^n$. As shown in a previous study, $b$ satisfies (3.5) and is bounded [38] . Thus, the corresponding commutator is bounded from ${ℳ}^{q(\cdot ),{\omega }_2}$ to ${\mathcal{C}}^{p(\cdot ),\omega }$. Then the space of solutions to (3.8) is ${\mathcal{C}}^{p(\cdot ),\omega }$. The regularity of solutions to some elliptic PDEs with smooth boundaries can be credited to the boundedness of corresponding commutators with smooth kernels. The question of what occurs when the boundary conditions are relaxed can be addressed by studying the boundedness of commutators with rough kernels. For this purpose, the results are extended to the case of rough kernels.

This can be justified by simply stating that the proof follows that of Theorem 3.2 with minor adjustments to the estimates to account for the kernel $\Omega \left(x-y\right)$. we can claim that Theorem 3.2 also holds true for fractional integrals with homogeneous kernels, which were defined [40] by

$T_{\Omega ,\alpha }f\left(x\right)={\displaystyle {\int }_{{ℝ}^n}\frac{\Omega \left(x-y\right)}{{\left|y-x\right|}^{n-\alpha }}\text{d}y},0<\alpha <n.$ (3.7)

where $\Omega \left(x\right)\in {\text{C}}^{\infty }\left({\text{S}}^{n-1}\right)$ is homogeneous of degree 0 and

${\displaystyle {\int }_{{\text{S}}^{n-1}}\Omega \left(x^{\prime }\right)\text{d}\sigma \left(x^{\prime }\right)}=0.$ (3.8)

Here $\text{d}\sigma$ is the normalized Lebesgue measure and $x^{\prime }=x/\left|x\right|$. When $\Omega =1$, $T_{\Omega ,\alpha }$ is the same as the fractional integral $I_{\alpha }$. If $\alpha =0$, then $T_{\Omega ,\alpha }$ becomes a Calderόn-Zygmund singular integral operator in the sense of the principal value Cauchy integral.

Corollary 3.3 Let $p(\cdot ),\omega ,{\omega }_1,{\omega }_2,\alpha$ and $b$ be as Theorem 3.2. Then the following conditions are equivalent.

(1) $b\in {\mathcal{C}}^{p_1(\cdot ),{\omega }_1}$;

(2) $T_{\Omega ,\alpha ,b}$ is a bounded operator from ${ℳ}^{q(\cdot ),{\omega }_2}$ to ${\mathcal{C}}^{p(\cdot ),\omega }$;

(3) For any ball $B\subset {ℝ}^n$ and $m\in {\mathbb{N}}^{+}$, there exists a constant $C>0$ such that

${\Vert T_{\Omega ,\alpha ,b}^{m}f\Vert }_{{\mathcal{C}}^{p(\cdot ),\omega }}\le C{\Vert f\Vert }_{{ℳ}^{q(\cdot ),{\omega }_2}}$

This section begins with some lemmas which will be used in the proof of our results.

Lemma 3.4 Let $p(\cdot ),\omega ,{\omega }_1,{\omega }_2,\alpha$ and $b$ be as Theorem 3.2, and $\frac{1}{s\left(x\right)}=\frac{1}{p_1\left(x\right)}+\frac{1}{q\left(x\right)}$. Then

${\Vert \left(b-b_B\right)f{\chi }_B\Vert }_{L^{s(\cdot )}}\le C\omega \left(x,r\right)r^{\frac{n}{p\left(x\right)}}{\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}{\Vert f\Vert }_{{ℳ}^{q(\cdot ),{\omega }_2}}.$ (3.9)

Proof. By applying the Hölder inequality we get

$\begin{array}{c}{\Vert \left(b-b_B\right)f{\chi }_B\Vert }_{L^{s(\cdot )}}\le C{\Vert \left(b-b_B\right){\chi }_B\Vert }_{L^{p_1(\cdot )}}{\Vert f{\chi }_B\Vert }_{L^{q(\cdot )}}\\ \le C{\omega }_1\left(x,r\right)r^{\frac{n}{p_1\left(x\right)}}{\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot )},{\omega }_1}{\omega }_2\left(x,r\right)r^{\frac{n}{q\left(x\right)}}{\Vert f\Vert }_{{ℳ}^{q(\cdot ),{\omega }_2}}\\ \le C\omega \left(x,r\right)r^{\frac{n}{p\left(x\right)}}{\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}{\Vert f\Vert }_{{ℳ}^{q(\cdot ),{\omega }_2}}\end{array}$

where $\frac{1}{q\left(x\right)}=\frac{1}{p_2\left(x\right)}+\frac{\alpha }{n}$.

Lemma 3.5 Let $B_{*}\subset B$, $b\in {\mathcal{C}}^{p_1(\cdot ),{\omega }_1}$.Then the following inequality holds.

$\left|b_{B_{*}}-b_B\right|\le C{\omega }_1\left(x,r\right){\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}.$ (3.10)

Proof. We suppose two cases.

Case 1. When $B_{*}\subset B\subseteq 2B_{*}$. By using Hölder inequality

$\begin{array}{c}\left|b_{B_{*}}-b_B\right|\le \frac{1}{\left|B_{*}\right|}{\displaystyle {\int }_{B_{*}}\left|b\left(x\right)-b_B\right|\text{d}x}+\frac{1}{\left|B\right|}{\displaystyle {\int }_B\left|b\left(x\right)-b_B\right|\text{d}x}\\ \le \frac{C}{\left|B\right|}{\displaystyle {\int }_B\left|b\left(x\right)-b_B\right|\text{d}x}\\ \le C{\left|B\right|}^{-\frac{1}{p_1\left(x\right)}}{\Vert b-b_B\Vert }_{L^{p_1\left(x\right)}}\\ \le C{\omega }_1\left(x,r\right){\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}.\end{array}$

Case 2. When $2B_{*}\subset B$. Choose a sequence of nested cubes

$B_{*}:=B_1\subset B_2\subset \cdot \cdot \cdot \subset B=:B_{m+1}$

Similarly to case 1, we have

$\begin{array}{c}\left|b_{B_{*}}-b_B\right|=\left|b_{B_1}-b_{B_2}+b_{B_2}-b_{B_3}+\cdot \cdot \cdot +b_{B_m}-b_{B_{m+1}}\right|\\ \le \underset{i=1}{\overset{\infty }{{\displaystyle \sum }}}\left|b_{B_i}-b_{B_{i+1}}\right|,\\ \le C{\omega }_1\left(x,r\right){\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}.\end{array}$

Proof of Theorem 3.1. Let $f$ be a function in ${ℳ}^{p(\cdot ),\omega }\left(\Omega \right)$. For any fixed $r>0$, denote $B\left(0,r\right)$ by $B$. $B^{\prime }=B\cap \Omega$. For any ball $B$, let $2B=B\left(0,2r\right)$. We write $f$ as $f=f_1+f_2$, where $f_1=f{\chi }_{2B}$ and $f_2=f-f_1$. By the Minkowski inequality, we have

$\begin{array}{c}{\Vert I_{\alpha }f\Vert }_{{ℳ}^{q(\cdot ),\omega }}\le {\Vert I_{\alpha }{f}_1\Vert }_{{ℳ}^{q(\cdot ),\omega }}+{\Vert I_{\alpha }{f}_2\Vert }_{{ℳ}^{q(\cdot ),\omega }}:\\ =I_1+I_2\end{array}$

We first estimate $I_1$.

Note that $f_1\in L^{p(\cdot )}$ and $I_{\alpha }$ is bounded from $L^{p(\cdot )}$ to $L^{q(\cdot )}$, we get

$\begin{array}{c}{\Vert I_{\alpha }{f}_1{\chi }_{B^{\prime }\left(x,r\right)}\Vert }_{L^{q(\cdot )}}\le {\Vert I_{\alpha }{f}_1\Vert }_{L^{q(\cdot )}}\lesssim {\Vert f_1\Vert }_{L^{p(\cdot )}}\\ ={\Vert f{\chi }_{2B}\Vert }_{L^{p(\cdot )}}.\end{array}$

Thus,

$\begin{array}{c}{\Vert I_{\alpha }{f}_1\Vert }_{{ℳ}^{q(\cdot ),\omega }}=\underset{B}{\text{sup}}\frac{1}{\omega \left(x,r\right)}{r}^{-\frac{n}{q\left(x\right)}}{\Vert I_{\alpha }{f}_1{\chi }_{B^{\prime }\left(x,r\right)}\Vert }_{L^{q(\cdot )}}\\ \lesssim \frac{1}{\omega \left(x,r\right)}{r}^{-\frac{n}{q\left(x\right)}}{\Vert f{\chi }_{2B}\Vert }_{L^{p(\cdot )}}\\ \lesssim \frac{1}{\omega \left(x,r\right)}{r}^{-\frac{n}{q\left(x\right)}}\omega \left(x,2r\right){\left(2r\right)}_{}^{\frac{n}{p\left(x\right)}}{\Vert f\Vert }_{{ℳ}^{p(\cdot ),\omega }}\\ \lesssim r^{\alpha }{\Vert f\Vert }_{{ℳ}^{p(\cdot ),\omega }}.\end{array}$

Now we turn to prove $I_2$. For $x\in B$ and $y\in 2^{k+1}B\backslash 2^{k}B$, we have $\left|x-y\right|\approx {\left|2^{k}B\right|}^{1/n}$, together with the Hölder inequality, we obtain

$\begin{array}{c}\left|I_{\alpha }{f}_2\left(x\right)\right|=\left|I_{\alpha }\left(f{\chi }_{{(2B)}^c}\right)\left(x\right)\right|\\ \le \underset{k=1}{\overset{\infty }{{\displaystyle \sum }}}{\displaystyle {\int }_{2^{k+1}B\backslash 2^{k}B}\frac{\left|f\left(y\right)\right|}{{\left|y-x\right|}^{n-\alpha }}\text{d}y}\\ \lesssim \underset{k=1}{\overset{\infty }{{\displaystyle \sum }}}{\left|2^{k}B\right|}^{-1+\frac{\alpha }{n}}{\displaystyle {\int }_{2^{k+1}B}\left|f\left(y\right)\right|\text{d}y}\\ \lesssim \underset{k=1}{\overset{\infty }{{\displaystyle \sum }}}{\left|2^{k}B\right|}^{-1+\frac{\alpha }{n}}{\Vert {\chi }_{2^{k+1}B}\Vert }_{L^{p^{\prime }(\cdot )}}{\Vert f{\chi }_{2^{k+1}B}\Vert }_{L^{p(\cdot )}}.\end{array}$

Therefore,

$\begin{array}{c}{\Vert I_{\alpha }{f}_2\Vert }_{{ℳ}^{q(\cdot ),\omega }}=\underset{B}{\text{sup}}\frac{1}{\omega \left(x,r\right)}{r}^{-\frac{n}{q\left(x\right)}}{\Vert I_{\alpha }{f}_2{\chi }_{B^{\prime }\left(x,r\right)}\Vert }_{L^{q(\cdot )}}\\ \lesssim \frac{1}{\omega \left(x,r\right)}{r}^{-\frac{n}{q\left(x\right)}}\underset{k=1}{\overset{\infty }{{\displaystyle \sum }}}{\left|2^{k}B\right|}^{-1+\frac{\alpha }{n}}{\Vert {\chi }_{2^{k+1}B}\Vert }_{L^{p^{\prime }(\cdot )}}{\Vert f{\chi }_{2^{k+1}B}\Vert }_{L^{p(\cdot )}}{\Vert {\chi }_B\Vert }_{L^{q(\cdot )}}\\ \lesssim \frac{1}{\omega \left(x,r\right)}{r}^{-\frac{n}{q\left(x\right)}}\underset{k=1}{\overset{\infty }{{\displaystyle \sum }}}{\left|2^{k}B\right|}^{-1+\frac{\alpha }{n}}{\Vert {\chi }_{2^{k+1}B}\Vert }_{L^{p^{\prime }(\cdot )}}{\Vert {\chi }_B\Vert }_{L^{q(\cdot )}}\omega \left(x,2^{k+1}r\right)\\ \times {\left(2^{k+1}r\right)}^{\frac{n}{p\left(x\right)}}{\Vert f\Vert }_{{ℳ}^{p(\cdot ),\omega }}\\ \lesssim \frac{1}{\omega \left(x,r\right)}{r}^{-\frac{n}{q\left(x\right)}}\underset{k=1}{\overset{\infty }{{\displaystyle \sum }}}{\left|2^{k}B\right|}^{-1+\frac{\alpha }{n}}{\left|2^{k+1}B\right|}^{1-\frac{1}{p_{-}}}{\left|B\right|}^{\frac{1}{q_{-}}}\omega \left(x,2^{k+1}r\right)\\ \times {\left(2^{k+1}r\right)}^{\frac{n}{p\left(x\right)}}{\Vert f\Vert }_{{ℳ}^{p(\cdot ),\omega }}\\ \lesssim {\Vert f\Vert }_{{ℳ}^{p(\cdot ),\omega }}.\end{array}$

This completes the proof.

Proof of Theorem 3.2. Let $f$ be a function in ${ℳ}^{q(\cdot ),{\omega }_2}$. For any fixed $r>0$, denote $B\left(0,r\right)$ by $B$. It suffices to show that the fact

${\Vert \left(I_{\alpha ,b}f-{\left(I_{\alpha ,b}f\right)}_B\right){\chi }_B\Vert }_{L^{p(\cdot )}}\le C\omega \left(x,r\right)r^{\frac{n}{p\left(x\right)}}{\Vert f\Vert }_{{ℳ}^{q(\cdot ),{\omega }_2}}$

For any ball $B$, let $2B=B\left(0,2r\right)$. We write $f$ as $f=f_1+f_2$, where $f_1=f{\chi }_{2B}$ and $f_2=f-f_1$. By using the facts $I_{\alpha ,b}f=I_{\alpha ,\left(b-b_B\right)}f$, and $\left|I_{\alpha ,\left(b-b_B\right)}f\left(y\right)-{\left(I_{\alpha ,\left(b-b_B\right)}f\right)}_B\right|\le 2\left|I_{\alpha ,\left(b-b_B\right)}f\left(y\right)-\left(I_{\alpha }\left(b-b_B\right)f_2\right)\left(x_B\right)\right|$ ( [19] , p. 5, [40] , p. 6) imply the following.

$\begin{array}{l}{\Vert \left(I_{\alpha ,b}f-{\left(I_{\alpha ,b}f\right)}_B\right){\chi }_B\Vert }_{L^{p(\cdot )}}\\ ={\Vert \left(I_{\alpha ,\left(b-b_B\right)}f-{\left(I_{\alpha ,\left(b-b_B\right)}f\right)}_B\right){\chi }_B\Vert }_{L^{p(\cdot )}}\\ \le 2{\Vert \left(I_{\alpha ,\left(b-b_B\right)}f-\left(I_{\alpha }\left(b-b_B\right)f_2\right)\left(x_B\right)\right){\chi }_B\Vert }_{L^{p(\cdot )}}\\ \le 2{\Vert \left(\left(b-b_B\right)I_{\alpha }f\right){\chi }_B\Vert }_{L^{p(\cdot )}}+2{\Vert \left(I_{\alpha }\left(b-b_B\right)f_1\right){\chi }_B\Vert }_{L^{p(\cdot )}}\\ +2{\Vert I_{\alpha }\left(b-b_B\right)f_2-\left(I_{\alpha }\left(\left(b-b_B\right)f_2\right)\left(x_B\right)\right){\chi }_B\Vert }_{L^{p(\cdot )}}\\ =:E_1+E_2+E_3\end{array}$

Now we first estimate $E_1$,

$\begin{array}{c}{\Vert \left(\left(b-b_B\right)I_{\alpha }f\right){\chi }_B\Vert }_{L^{p(\cdot )}}\lesssim {\Vert \left(b-b_B\right){\chi }_B\Vert }_{L^{p_1(\cdot )}}{\Vert \left(I_{\alpha }f\right){\chi }_B\Vert }_{L^{p_2(\cdot )}}\\ \lesssim {\omega }_1\left(x,r\right)r^{\frac{n}{p_1\left(x\right)}}{\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}{\Vert f{\chi }_B\Vert }_{L^{q(\cdot )}}\\ \lesssim {\omega }_1\left(x,r\right)r^{\frac{n}{p_1\left(x\right)}}{\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}{\omega }_2\left(x,r\right)r^{\frac{n}{q\left(x\right)}}{\Vert f\Vert }_{{ℳ}^{q(\cdot ),{\omega }_2}}\\ \lesssim \omega \left(x,r\right)r^{\frac{n}{p\left(x\right)}}{\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}{\Vert f\Vert }_{{ℳ}^{q(\cdot ),{\omega }_2}}.\end{array}$

where $\frac{1}{q\left(x\right)}=\frac{1}{p_2\left(x\right)}+\frac{\alpha }{n}$, $\frac{1}{p\left(x\right)}=\frac{1}{p_1\left(x\right)}+\frac{1}{p_2\left(x\right)}$.

Next we will find the estimate for $E_2$

$\begin{array}{c}\left(I_{\alpha }\left(b-b_B\right)f_1\right){\chi }_B{}_{L^{p(\cdot )}}\lesssim {\Vert \left(\left(b-b_B\right)f_1\right){\chi }_B\Vert }_{L^{s(\cdot )}}\\ \lesssim {\Vert \left(b-b_B\right){\chi }_B\Vert }_{L^{p_1(\cdot )}}{\Vert f{\chi }_{2B}\Vert }_{L^{q(\cdot )}}\\ \lesssim {\omega }_1\left(x,r\right)r^{\frac{n}{p_1\left(x\right)}}{\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}{\omega }_2\left(x,2r\right){\left(2r\right)}^{\frac{n}{q\left(x\right)}}{\Vert f\Vert }_{{ℳ}^{q(\cdot ),{\omega }_2}}\\ \lesssim \omega \left(x,r\right)r^{\frac{n}{p\left(x\right)}}{\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}{\Vert f\Vert }_{{ℳ}^{q(\cdot ),{\omega }_2}}.\end{array}$

where $\frac{1}{s\left(x\right)}=\frac{1}{p\left(x\right)}+\frac{\alpha }{n}$, $\frac{1}{s\left(x\right)}=\frac{1}{p_1\left(x\right)}+\frac{1}{q\left(x\right)}$.

Now we will find the estimate for the term $E_3$,

$\begin{array}{l}\left|I_{\alpha }\left(b-b_B\right)f_2\left(y\right)-\left(I_{\alpha }\left(\left(b-b_B\right)f_2\right)\right)\left(x_B\right)\right|\\ \le {\displaystyle {\int }_{{\left(2B\right)}^c}\frac{\left|y-x_B\right|}{{\left|x_B-z\right|}^{n+1-\alpha }}\left|b\left(z\right)-b_B\right|\left|f\left(z\right)\right|\text{d}z}\\ \le \underset{k=2}{\overset{\infty }{{\displaystyle \sum }}}{\displaystyle {\int }_{2^{k}B\backslash 2^{k-1}B}{2}^{-k}{\left|2^{k}B\right|}^{-1+\frac{\alpha }{n}}\left|b\left(z\right)-b_B\right|\left|f\left(z\right)\right|\text{d}z}\\ \lesssim \underset{k=2}{\overset{\infty }{{\displaystyle \sum }}}\frac{1}{2^k{\left|2^{k}B\right|}^{1-\frac{\alpha }{n}}}{\displaystyle {\int }_{2^{k}B}\left|b\left(z\right)-b_{2^{k}B}\right|\left|f\left(z\right)\right|\text{d}z}\\ +\underset{k=2}{\overset{\infty }{{\displaystyle \sum }}}\frac{1}{2^k{\left|2^{k}B\right|}^{1-\frac{\alpha }{n}}}{\displaystyle {\int }_{2^{k}B}\left|b_B-b_{2^{k}B}\right|\left|f\left(z\right)\right|\text{d}z}\end{array}$

$\begin{array}{c}{E}_3\lesssim {\Vert \underset{k=2}{\overset{\infty }{{\displaystyle \sum }}}\frac{1}{2^k{\left|2^{k}B\right|}^{1-\frac{\alpha }{n}}}{\displaystyle {\int }_{2^{k}B}\left|b\left(z\right)-b_{2^{k}B}\right|\left|f\left(z\right)\right|\text{d}z}\Vert }_{L^{p(\cdot )}}\\ +{\Vert \underset{k=2}{\overset{\infty }{{\displaystyle \sum }}}\frac{1}{2^k{\left|2^{k}B\right|}^{1-\frac{\alpha }{n}}}{\displaystyle {\int }_{2^{k}B}\left|b_B-b_{2^{k}B}\right|\left|f\left(z\right)\right|\text{d}z}\Vert }_{L^{p(\cdot )}}\\ =:I_1+I_2\end{array}$

$\begin{array}{c}{I}_1\lesssim {\Vert \underset{k=2}{\overset{\infty }{{\displaystyle \sum }}}\frac{1}{2^k}{\displaystyle {\int }_{2^{k}B}\frac{1}{{\left|2^{k}B\right|}^{1-\frac{\alpha }{n}}}\left|b\left(z\right)-b_{2^{k}B}\right|\left|f\left(z\right)\right|\text{d}z}\Vert }_{L^{p(\cdot )}}\\ \lesssim \underset{k=2}{\overset{\infty }{{\displaystyle \sum }}}\frac{1}{2^k}{\Vert M_{\alpha }\left|b-b_{2^{k}B}\right|\left|f\right|\Vert }_{L^{p(\cdot )}}\\ \lesssim \underset{k=2}{\overset{\infty }{{\displaystyle \sum }}}\frac{1}{2^k}{\Vert \left|b-b_{2^{k}B}\right|\left|f\right|\Vert }_{L^{s(\cdot )}}\\ \lesssim \omega \left(x,r\right)r^{\frac{n}{p\left(x\right)}}{\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}{\Vert f\Vert }_{{ℳ}^{q(\cdot ),{\omega }_2}}.\end{array}$

Now we find the estimate for $I_1$, by using the boundedness of $M_{\alpha }$ (Theorem 2.2) and Lemma 3.4, we have.

Now we find the estimate for $I_2$. By using the boundedness of $M_{\alpha }$ (Theorem 2.2) and Lemma 3.5, we have.

$\begin{array}{c}{I}_2\lesssim {\Vert \underset{k=2}{\overset{\infty }{{\displaystyle \sum }}}\frac{1}{2^k}{\displaystyle {\int }_{2^{k}B}\frac{1}{{\left|2^{k}B\right|}^{1-\frac{\alpha }{n}}}\left|b_B-b_{2^{k}B}\right|\left|f\left(z\right)\right|\text{d}z}\Vert }_{L^{p(\cdot )}}\\ \lesssim \underset{k=2}{\overset{\infty }{{\displaystyle \sum }}}\frac{1}{2^k}\left|b_B-b_{2^{k}B}\right|{\Vert M_{\alpha }f{\chi }_{2^{k}B}\Vert }_{L^{p(\cdot )}}\\ \lesssim \underset{k=2}{\overset{\infty }{{\displaystyle \sum }}}\frac{1}{2^k}{\omega }_1\left(x,r\right){\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}{\Vert f{\chi }_{2^{k}B}\Vert }_{L^{s(\cdot )}}\\ \lesssim \underset{k=2}{\overset{\infty }{{\displaystyle \sum }}}\frac{1}{2^k}{\omega }_1\left(x,r\right){\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}{\Vert {\chi }_{2^{k}B}\Vert }_{L^{p_1(\cdot )}}{\Vert f{\chi }_{2^{k}B}\Vert }_{L^{q(\cdot )}}\\ \lesssim \omega \left(x,r\right)r^{\frac{n}{p\left(x\right)}}{\Vert b\Vert }_{{\mathcal{C}}^{p_1(\cdot ),{\omega }_1}}{\Vert f\Vert }_{{ℳ}^{q(\cdot ),{\omega }_2}}.\end{array}$