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Multilinear fractional integral operators on central Morrey spaces with variable exponent

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Abstract

In this paper, we obtain the boundedness of the multilinear fractional integral operators and their commutators on central Morrey spaces with variable exponent.

Introduction

The study of multilinear integral operators was motivated not only as the generalization of the theory of linear ones but also their natural appearance in analysis. It has increasing attention and much development in recent years, such as the study of the bilinear Hilbert transform by Lacey and Thiele [21, 22] and the systematic treatment of multilinear Calderón–Zygmund operators by Grafokas and Torres [13, 14], Grafokas and Kalton [12]. The importance of fractional integral operators is owing to the fact that they are smooth operators and have been extensively used in various areas such as potential analysis, harmonic analysis and partial differential equations. As one of the most important operators, the multilinear fractional integral operator (also known as the multilinear Riesz potential) has also attracted more attention, see for example [3, 11, 18, 24].

It is well known that function spaces with variable exponent arouse strong interest not only in harmonic analysis but also in applied mathematics. The theory of function spaces with variable exponent has made great progress since some elementary properties were given by Kováčik and Rákosník [19] in 1991. Lebesgue and Sobolev spaces with integrability exponent have been widely studied, see [5, 8] and the references therein. Many applications of these spaces were given, for example, in the modeling of electrorheological fluids [26], in the study of image processing [4], and in differential equations with nonstandard growth [15]. On the other hand, the λ-central bounded mean oscillation spaces, Morrey type spaces and related function spaces have interesting applications in studying boundedness of operators including singular integral operators; see for example [1, 9, 19, 27,28,29]. In 2015, Mizuta, Ohno and Shimomura introduced the non-homogeneous central Morrey spaces of variable exponent in [23]. Recently, Fu et al. introduced the λ-central BMO spaces and the central Morrey spaces with variable exponent and gave the boundedness of some operators in [10]. In [2, 6, 7, 17] and [31,32,33,34], the authors proved the boundedness of some integral operators on variable function spaces, respectively. Meanwhile, some authors gave the boundedness of multilinear integral operators and their commutators on variable exponent function spaces, such as [16, 30, 35].

Motivated by [9, 10, 29], we will study the boundedness of the multilinear fractional integral operators and their commutators on the central Morrey spaces with variable exponent.

Let us explain the outline of this article. In Sect. 2, we first briefly recall some standard notations and lemmas in variable Lebesgue spaces. Then we will recall the definiton of the λ-central BMO spaces and central Morrey spaces with variable exponent. In Sect. 3, we will establish the boundedness for a class of multi-sublinear fractional integral operators on central Morrey spaces with variable exponent. Subsequently the boundedness of multilinear fractional integral commutators on central Morrey spaces with variable exponent will be obtained in Sect. 4. In Sect. 5, we will also consider the boundedness of another multilinear fractional integral commutators.

In addition, we denote the Lebesgue measure and the characteristic function of a measurable set \(A\subset\mathbb{R}^{n}\) by \(|A|\) and \(\chi_{A}\), respectively. The notation \(f\approx g\) means that there exist constants \(C_{1},C_{2}>0\) such that \(C_{1}g\leq f\leq C_{2}g\).

Variable exponent function spaces

Firstly we give some notation and basic definitions on variable exponent Lebesgue spaces.

Given an open set \(E\subset\mathbb{R}^{n}\), and a measurable function \(p(\cdot):E\rightarrow[1,\infty)\). \(p'(\cdot)\) is the conjugate exponent defined by \(p'(\cdot)=p(\cdot)/(p(\cdot)-1)\).

The set \(\mathcal{P}(E)\) consists of all \(p(\cdot):E\rightarrow[1,\infty)\) satisfying

$$\begin{gathered} p^{-}=\operatorname{ess} \inf\bigl\{ p(x):x\in E\bigr\} >1, \\ p^{+}=\operatorname{ess} \sup\bigl\{ p(x):x\in E\bigr\} < \infty.\end{gathered} $$

By \(L^{p(\cdot)}(E)\) we denote the space of all measurable functions f on E such that, for some \(\lambda>0\),

$$\int_{E} \biggl(\frac{ \vert f(x) \vert }{\lambda} \biggr)^{p(x)}\,dx < \infty. $$

This is a Banach function space with respect to the Luxemburg–Nakano norm,

$$\Vert f \Vert _{L^{p(\cdot)}(E)}=\inf \biggl\{ \lambda>0: \int_{E} \biggl(\frac{ \vert f(x) \vert }{\lambda} \biggr)^{p(x)}\,dx \leq1 \biggr\} . $$

The space \(L_{\mathrm{loc}}^{p(\cdot)}(\varOmega)\) is defined by \(L_{\mathrm{loc}}^{p(\cdot)}(\varOmega):=\{f: f\in L^{p(\cdot)}(E)\) for all compact subsets \({E\subset\varOmega\}}\).

Let \(f\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n})\), the Hardy–Littlewood maximal operator is defined by

$$Mf(x)=\sup_{r>0}\frac{1}{ \vert B_{r}(x) \vert } \int_{B_{r}(x)} \bigl\vert f(y) \bigr\vert \,dy, $$

where \(B_{r}(x)=\{y\in\mathbb{R}^{n}:|x-y|< r\}\). The set \(\mathcal{B}(\mathbb{R}^{n})\) consists of \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) satisfying the condition that M is bounded on \(L^{p(\cdot)}(\mathbb{R}^{n})\).

In variable \(L^{p}\) spaces there are some important lemmas as follows.

Lemma 2.1

([7])

If \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\)and satisfies

$$ \bigl\vert p(x)-p(y) \bigr\vert \leq\frac{C}{-\log( \vert x-y \vert )}, \quad \vert x-y \vert \leq1/2, $$
(2.1)

and

$$ \bigl\vert p(x)-p(y) \bigr\vert \leq\frac{C}{\log( \vert x \vert +e)}, \quad \vert y \vert \geq \vert x \vert , $$
(2.2)

then \(p(\cdot)\in\mathcal{B}(\mathbb{R}^{n})\), that is, the Hardy–Littlewood maximal operatorMis bounded on \(L^{p(\cdot)}(\mathbb{R}^{n})\).

Lemma 2.2

([20] (Generalized Hölder inequality))

Let \(p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\). If \(f\in L^{p(\cdot)}(\mathbb {R}^{n})\)and \(g\in L^{p'(\cdot)}(\mathbb{R}^{n})\), thenfgis integrable on \(\mathbb {R}^{n}\)and

$$\int_{\mathbb{R}^{n}} \bigl\vert f(x)g(x) \bigr\vert \,dx \leq r_{p} \Vert f \Vert _{L^{p(\cdot )}(\mathbb{R}^{n})} \Vert g \Vert _{L^{p'(\cdot)}(\mathbb{R}^{n})}, $$

where

$$r_{p}=1+1/p^{-} -1/p^{+}. $$

Lemma 2.3

([17])

Suppose \(p(\cdot)\in \mathcal{B}(\mathbb{R}^{n})\). Then there exists a positive constantCsuch that, for all ballsBin \(\mathbb{R}^{n}\),

$$\frac{1}{ \vert B \vert } \Vert \chi_{B} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{p'(\cdot)}(\mathbb{R}^{n})}\leq C. $$

Lemma 2.4

([17])

Let \(p(\cdot)\in \mathcal{B}(\mathbb{R}^{n})\). Then there exists a positive constantCsuch that, for all ballsBin \(\mathbb{R}^{n}\)and all measurable subsets \(S\subset B\),

$$\frac{ \Vert \chi_{B} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})}}{ \Vert \chi_{S} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})}}\leq C \frac{ \vert B \vert }{ \vert S \vert },\qquad\frac{ \Vert \chi_{S} \Vert _{L^{p(\cdot)}(\mathbb {R}^{n})}}{ \Vert \chi_{B} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})}}\leq C \biggl( \frac{ \vert S \vert }{ \vert B \vert } \biggr)^{\delta_{1}} $$

and

$$\frac{ \Vert \chi_{S} \Vert _{L^{p'(\cdot)}(\mathbb{R}^{n})}}{ \Vert \chi_{B} \Vert _{L^{p'(\cdot)}(\mathbb{R}^{n})}}\leq C \biggl(\frac{ \vert S \vert }{ \vert B \vert } \biggr)^{\delta_{2}}, $$

where \(\delta_{1}\), \(\delta_{2}\)are constants with \(0<\delta_{1}, \delta_{2}<1\).

Lemma 2.5

([8])

Let \(p(\cdot)\in\mathcal {P}(\mathbb{R}^{n})\)satisfies conditions (2.1) and (2.2) in Lemma 2.1. Then

$$\Vert \chi_{Q} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})}\approx \textstyle\begin{cases} \vert Q \vert ^{\frac{1}{p(x)}}& \textit{if } \vert Q \vert \leq2^{n} \textit{ and } x\in Q,\\ \vert Q \vert ^{\frac{1}{p(\infty)}}& \textit{if } \vert Q \vert \geq1 \end{cases} $$

for every cube (or ball) \(Q\subset\mathbb{R}^{n}\), where \(p(\infty)=\lim_{x\rightarrow\infty}p(x)\).

Lemma 2.6

([8])

Let \(p(\cdot), q(\cdot), s(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\)be such that

$$\frac{1}{s(x)}=\frac{1}{p(x)}+\frac{1}{q(x)} $$

for almost every \(x\in\mathbb{R}^{n}\). Then

$$\Vert fg \Vert _{L^{s(\cdot)}(\mathbb{R}^{n})}\leq2 \Vert f \Vert _{L^{p(\cdot )}(\mathbb{R}^{n})} \Vert g \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} $$

for all \(f\in L^{p(\cdot)}(\mathbb{R}^{n})\)and \(g\in L^{q(\cdot )}(\mathbb{R}^{n})\).

Now we recall that the central Morrey space with variable exponent and the λ-central bounded mean oscillation space with variable exponent in [10] are defined as follows.

Definition 2.1

([10])

Let \(q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) and \(\lambda\in \mathbb{R}\). The central Morrey space with variable exponent \(\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb {R}^{n})\) is defined by

$$\dot{\mathcal{B}}^{q(\cdot),\lambda}\bigl(\mathbb{R}^{n}\bigr)= \bigl\{ f \in L_{\mathrm{loc}}^{q(\cdot)}\bigl(\mathbb{R}^{n}\bigr): \Vert f \Vert _{\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb {R}^{n})}< \infty \bigr\} , $$

where

$$\Vert f \Vert _{\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb{R}^{n})}=\sup_{R>0} \frac{ \Vert f\chi_{B(0,R)} \Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}}{ \vert B(0,R) \vert ^{\lambda} \Vert \chi_{B(0,R)} \Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}}. $$

Definition 2.2

([10])

Let \(q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) and \(\lambda<1/n\). The λ-central BMO space with variable exponent \(\mathrm {CBMO}^{q(\cdot),\lambda}(\mathbb{R}^{n})\) is defined by

$$\mathrm{CBMO}^{q(\cdot),\lambda}\bigl(\mathbb{R}^{n}\bigr)= \bigl\{ f\in L_{\mathrm{loc}}^{q(\cdot)}\bigl(\mathbb{R}^{n}\bigr): \Vert f \Vert _{\mathrm{CBMO}^{q(\cdot),\lambda}(\mathbb{R}^{n})}< \infty \bigr\} , $$

where

$$\Vert f \Vert _{\mathrm{CBMO}^{q(\cdot),\lambda}(\mathbb{R}^{n})}=\sup_{R>0} \frac{ \Vert (f-f_{B(0,R)})\chi_{B(0,R)} \Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}}{ \vert B(0,R) \vert ^{\lambda} \Vert \chi_{B(0,R)} \Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}}. $$

Remark 2.1

Denote by \(\mathcal{B}^{q(\cdot),\lambda}(\mathbb{R}^{n})\) and \(\mathrm{CMO}^{q(\cdot),\lambda}(\mathbb{R}^{n})\) the inhomogeneous versions of the central Morrey space and the λ-central BMO space with variable exponent, which are defined, respectively, by taking the supremum over \(R\geq1\) in Definition 2.1 and Definition 2.2 instead of \(R>0\) there.

Remark 2.2

Our results in this paper remain true for the inhomogeneous versions of λ-central BMO spaces and central Morrey spaces with variable exponent.

Multilinear fractional integral operators

Let \(m\in\mathbb{N}\) and \(K(y_{0},y_{1},\ldots,y_{m})\) be a function defined away from the diagonal \(y_{0}=y_{1}=\cdots=y_{m}\) in \((\mathbb {R}^{n})^{m+1}\). We denote by f⃗ the m-tuple \((f_{1},\ldots ,f_{m})\). Now we consider that T is an m-linear operator defined on the product of test functions such that, for K, the integral representation below is valid:

$$T(\vec{f}) (x)= \int_{\mathbb{R}^{n}}\cdots \int_{\mathbb {R}^{n}}K(x,y_{1},\ldots,y_{m})\prod _{j=1}^{m}f_{j}(y_{j})\,dy_{1} \cdots dy_{m}, $$

whenever \(f_{j}\), \(j=1,\ldots,m\), are smooth functions with compact support and \(x\notin \bigcap_{j=1}^{m}\operatorname{supp}f_{j}\).

Particularly, there is a kind of multilinear operator \(T_{\alpha,m}\), which is called multilinear fractional integral operator, whose kernel is

$$ K(x,y_{1},\ldots,y_{m})= \bigl\vert (x-y_{1}, \ldots,x-y_{m}) \bigr\vert ^{\alpha-mn},\quad 0< \alpha< mn. $$
(3.1)

In 1999, Kenig and Stein [18] gave the boundedness of the above multilinear fractional integral operator \(T_{\alpha,m}\) on the product of Lebesgue spaces.

Theorem A

([18])

Let \(0< \alpha<mn\)and \(T_{\alpha,m}\)be anm-linear fractional integral operator with kernelKsatisfying (3.1). Suppose \(1\leq p_{1},p_{2},\ldots,p_{m}\leq \infty\), \(1/q=1/p_{1}+\cdots+1/p_{m}- \alpha/n>0\).

  1. (1)

    If each \(p_{j}>1\), \(j=1,\ldots,m\), then

    $$\bigl\Vert T_{\alpha,m}(\vec{f}) \bigr\Vert _{L^{q}(\mathbb{R}^{n})}\leq C\prod _{i=1}^{m} \Vert f_{i} \Vert _{L^{p_{i}}(\mathbb{R}^{n})}. $$
  2. (2)

    If each \(p_{j}=1\)for somej, then

    $$\bigl\Vert T_{\alpha,m}(\vec{f}) \bigr\Vert _{L^{q,\infty}(\mathbb{R}^{n})}\leq C\prod _{i=1}^{m} \Vert f_{i} \Vert _{L^{p_{i}}(\mathbb{R}^{n})}. $$

In the variable exponent case, Tan, Liu and Zhao [30] gave the following result.

Theorem B

([30])

Let \(m\in\mathbb{N}\), \(0< \alpha<mn\), \(q(\cdot),p_{1}(\cdot),\ldots,p_{m}(\cdot)\in\mathcal {P}(\mathbb{R}^{n})\)satisfy conditions (2.1) and (2.2) in Lemma 2.1and \(1/q(\cdot)=1/p_{1}(\cdot)+\cdots+1/p_{m}(\cdot)-\alpha/n\). Then

$$\bigl\Vert T_{\alpha,m}(\vec{f}) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C\prod _{i=1}^{m} \Vert f_{i} \Vert _{L^{p_{i}(\cdot)}(\mathbb{R}^{n})}. $$

Next we will give the boundedness of a class of multi-sublinear fractional integral operators T on the product of central Morrey spaces with variable exponent.

Theorem 3.1

Let \(m\in\mathbb{N}\), \(0< \alpha<mn\)andTbe a multi-sublinear fractional integral operator such that

$$ \bigl\vert T(\vec{f}) (x) \bigr\vert \leq C \int_{(\mathbb{R}^{n})^{m}}\frac{ \vert f_{1}(y_{1}) \vert \cdots \vert f_{m}(y_{m}) \vert }{ \vert (x-y_{1},\ldots,x-y_{m}) \vert ^{mn-\alpha}}\,dy_{1}\cdots dy_{m} $$
(3.2)

for any integrable functions \(f_{1},\ldots,f_{m}\)with compact support and \(x\notin\bigcap_{j=1}^{m}\operatorname{supp}f_{j}\). Suppose \(\lambda _{j}<-\frac{\alpha}{mn}\), \(\lambda=\sum_{j=1}^{m}\lambda_{j}+\alpha/n\), \(p_{j}(\cdot)\) (\(j=1,\ldots,m\)), \(q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\)satisfy conditions (2.1) and (2.2) in Lemma 2.1and \(1/q(\cdot)=\sum_{j=1}^{m}1/p_{j}(\cdot)-\alpha/n>0\). IfTis bounded from \(L^{p_{1}(\cdot)}(\mathbb{R}^{n})\times\cdots \times L^{p_{m}(\cdot)}(\mathbb{R}^{n})\)into \(L^{q(\cdot)}(\mathbb {R}^{n})\), thenTis also bounded from \(\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})\times \cdots\times\dot{\mathcal{B}}^{p_{m}(\cdot),\lambda_{m}}(\mathbb {R}^{n})\)into \(\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb{R}^{n})\).

If \(0< \alpha<mn\) and \(T_{\alpha,m}\) is an m-linear fractional integral operator, then the condition (3.2) is obviously satisfied by (3.1). By Theorem B we can get the following corollary of Theorem 3.1.

Corollary 3.1

Let \(m\in\mathbb{N}\), \(0< \alpha<mn\)and \(T_{\alpha,m}\)be anm-linear fractional integral operator with kernelKsatisfying (3.1). Suppose \(\lambda_{j}<-\frac{\alpha}{mn}\), \(\lambda=\sum_{j=1}^{m}\lambda_{j}+\alpha/n\), \(p_{j}(\cdot)\) (\(j=1,\ldots,m\)), \(q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\)satisfy conditions (2.1) and (2.2) in Lemma 2.1and \(1/q(\cdot)=\sum_{j=1}^{m}1/p_{j}(\cdot)-\alpha/n>0\). Then \(T_{\alpha,m}\)is bounded from \(\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})\times \cdots\times\dot{\mathcal{B}}^{p_{m}(\cdot),\lambda_{m}}(\mathbb {R}^{n})\)into \(\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb{R}^{n})\).

Proof of Theorem 3.1

In order to simplify the proof, we consider only the situation when \(m=2\). Actually, a similar procedure works for all \(m\in\mathbb{N}\). Let \(f_{1}\), \(f_{2}\) be functions in \(\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda _{1}}(\mathbb{R}^{n})\) and \(\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda _{2}}(\mathbb{R}^{n})\), respectively. For fixed \(R>0\), denote \(B(0,R)\) by B. We need to prove

$$\bigl\Vert T(f_{1},f_{2})\chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C \vert B \vert ^{\lambda} \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb{R}^{n})} , $$

where C is a constant independent of R.

By the Minkowski inequality we write

$$ \begin{aligned}[b] \bigl\Vert T(f_{1},f_{2}) \chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}& \leq \bigl\Vert T(f_{1}\chi_{2B},f_{2}\chi_{2B}) \chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})} \\ & \quad+ \bigl\Vert T(f_{1}\chi_{(2B)^{c}},f_{2} \chi_{2B})\chi_{B} \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \\ & \quad+ \bigl\Vert T(f_{1}\chi_{2B},f_{2} \chi_{(2B)^{c}})\chi_{B} \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \\ & \quad+ \bigl\Vert T(f_{1}\chi_{(2B)^{c}},f_{2} \chi_{(2B)^{c}})\chi_{B} \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \\ & =:I_{1}+I_{2}+I_{3}+I_{4}. \end{aligned} $$
(3.3)

We first estimate \(I_{1}\). Using Lemma 2.4 and the boundedness of T from \(L^{p_{1}(\cdot)}(\mathbb{R}^{n})\times L^{p_{2}(\cdot)}(\mathbb {R}^{n})\) into \(L^{q(\cdot)}(\mathbb{R}^{n})\), we have

$$ \begin{aligned}[b] I_{1}& \leq C \Vert f_{1} \chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert f_{2} \chi_{2B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \vert 2B \vert ^{\lambda_{1}} \Vert \chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert 2B \vert ^{\lambda_{2}} \Vert \chi_{2B} \Vert _{L^{p_{2}(\cdot)}(\mathbb {R}^{n})} \\ & \leq C \vert 2B \vert ^{\lambda_{1}+\lambda_{2}+\alpha/n} \vert 2B \vert ^{-\alpha/n} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \Vert \chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{2B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\hspace{-12pt} \\ & \leq C \vert 2B \vert ^{\lambda_{1}+\lambda_{2}+\alpha/n} \Vert f_{1} \Vert _{\dot{\mathcal {B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \Vert \chi_{2B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \vert B \vert ^{\lambda} \Vert \chi_{2B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb{R}^{n})}, \end{aligned} $$
(3.4)

where

$$\Vert \chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{2B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\approx \vert 2B \vert ^{1/p_{1}(\cdot )+1/p_{2}(\cdot)}= \vert 2B \vert ^{1/q(\cdot)+\alpha/n}\approx \vert 2B \vert ^{\alpha/n} \Vert \chi_{2B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}. $$

Next we estimate \(I_{2}\). Noting that \(|(x-y_{1}, x-y_{2})|^{2n-\alpha}\geq |x-y_{1}|^{2n-\alpha}\), by using (3.2), \({\lambda_{1}<-\frac{\alpha }{2n}}\), Lemma 2.3, the Minkowski inequality and the generalized Hölder inequality, we have

$$\begin{aligned} I_{2}& = \bigl\Vert T(f_{1} \chi_{(2B)^{c}},f_{2}\chi_{2B})\chi_{B} \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \\ & \leq\sum_{k=1}^{\infty}\bigl\Vert T(f_{1}\chi_{2^{k+1}B\setminus2^{k}B},f_{2}\chi _{2B}) \chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ & \leq C\sum_{k=1}^{\infty}\biggl\Vert \int_{2B} \int_{2^{k+1}B\setminus 2^{k}B}\frac{ \vert f_{1}(y_{1}) \vert \vert f_{2}(y_{2}) \vert }{ \vert (\cdot-y_{1}, \cdot-y_{2}) \vert ^{2n-\alpha }}\,dy_{1}\,dy_{2} \chi_{B}(\cdot) \biggr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \Vert f_{2}\chi_{2B} \Vert _{L^{1}(\mathbb{R}^{n})} \sum_{k=1}^{\infty}\biggl\Vert \int_{2^{k+1}B\setminus2^{k}B}\frac{ \vert f_{1}(y_{1}) \vert }{ \vert \cdot -y_{1} \vert ^{2n-\alpha}}\,dy_{1} \chi_{B}(\cdot) \biggr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})} \\ & \leq C \Vert f_{2}\chi_{2B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2B} \Vert _{L^{p'_{2}(\cdot)}(\mathbb{R}^{n})}\sum _{k=1}^{\infty} \Vert \chi _{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \bigl(2^{k-1}R\bigr)^{-2n+\alpha} \Vert f_{1}\chi_{2^{k+1}B} \Vert _{L^{1}(\mathbb {R}^{n})} \\ & \leq C \Vert f_{2}\chi_{2B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2B} \Vert _{L^{p'_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \\ &\quad \times\sum_{k=1}^{\infty}\bigl\vert 2^{k}B \bigr\vert ^{-2+\alpha/n} \Vert f_{1} \chi_{2^{k+1}B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2^{k+1}B} \Vert _{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb{R}^{n})} \vert 2B \vert ^{\lambda_{2}} \Vert \chi_{2B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{2B} \Vert _{L^{p'_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \vert B \vert ^{-2+\alpha/n} \\ & \quad\times\sum_{k=1}^{\infty}2^{kn(-2+\alpha/n)} \Vert f_{1} \Vert _{\dot{\mathcal {B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \bigl\vert 2^{k+1}B \bigr\vert ^{\lambda_{1}} \Vert \chi_{2^{k+1}B} \Vert _{L^{p_{1}(\cdot)}(\mathbb {R}^{n})} \Vert \chi_{2^{k+1}B} \Vert _{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert B \vert ^{\lambda_{2}+1-2+\alpha/n+\lambda_{1}+1} \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ &\quad \times\sum_{k=1}^{\infty}2^{-2kn+k\alpha+n\lambda _{2}+n+(k+1)n\lambda_{1}+(k+1)n} \\ & \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert B \vert ^{\lambda} \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\sum _{k=1}^{\infty}2^{kn(-1+\lambda_{1}+\alpha/n)} \\ & \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert B \vert ^{\lambda} \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\sum _{k=1}^{\infty}2^{kn(\lambda_{1}+\frac{\alpha}{2n})} \\ & \leq C \vert B \vert ^{\lambda} \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb{R}^{n})}. \end{aligned}$$
(3.5)

Similarly, we estimate \(I_{3}\). Noticing that \(|(x-y_{1}, x-y_{2})|^{2n-\alpha}\geq|x-y_{2}|^{2n-\alpha}\), by (3.2), \(\lambda _{2}<-\frac{\alpha}{2n}\), Lemma 2.3, the Minkowski inequality and the generalized Hölder inequality, we obtain

$$\begin{aligned} I_{3}& = \bigl\Vert T(f_{1} \chi_{2B},f_{2}\chi_{(2B)^{c}})\chi_{B} \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \\ & \leq\sum_{k=1}^{\infty}\bigl\Vert T(f_{1}\chi_{2B},f_{2}\chi_{2^{k+1}B\setminus 2^{k}B}) \chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ & \leq C\sum_{k=1}^{\infty}\biggl\Vert \int_{2^{k+1}B\setminus2^{k}B} \int _{2B}\frac{ \vert f_{1}(y_{1}) \vert \vert f_{2}(y_{2}) \vert }{ \vert (\cdot-y_{1}, \cdot-y_{2}) \vert ^{2n-\alpha }}\,dy_{1}\,dy_{2} \chi_{B}(\cdot) \biggr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \Vert f_{1}\chi_{2B} \Vert _{L^{1}(\mathbb{R}^{n})} \sum_{k=1}^{\infty}\biggl\Vert \int_{2^{k+1}B\setminus2^{k}B}\frac{ \vert f_{2}(y_{2}) \vert }{ \vert \cdot -y_{2} \vert ^{2n-\alpha}}\,dy_{2} \chi_{B}(\cdot) \biggr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})} \\ & \leq C \Vert f_{1}\chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2B} \Vert _{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}\sum _{k=1}^{\infty} \Vert \chi _{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \bigl(2^{k-1}R\bigr)^{-2n+\alpha} \Vert f_{2}\chi_{2^{k+1}B} \Vert _{L^{1}(\mathbb {R}^{n})} \\ & \leq C \Vert f_{1}\chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2B} \Vert _{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \\ &\quad \times\sum_{k=1}^{\infty}\bigl\vert 2^{k}B \bigr\vert ^{-2+\alpha/n} \Vert f_{2} \chi_{2^{k+1}B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2^{k+1}B} \Vert _{L^{p'_{2}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \vert 2B \vert ^{\lambda_{1}} \Vert \chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{2B} \Vert _{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \vert B \vert ^{-2+\alpha/n} \\ &\quad \times\sum_{k=1}^{\infty}2^{kn(-2+\alpha/n)} \Vert f_{2} \Vert _{\dot{\mathcal {B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb{R}^{n})} \bigl\vert 2^{k+1}B \bigr\vert ^{\lambda_{2}} \Vert \chi_{2^{k+1}B} \Vert _{L^{p_{2}(\cdot)}(\mathbb {R}^{n})} \Vert \chi_{2^{k+1}B} \Vert _{L^{p'_{2}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert B \vert ^{\lambda_{1}+1-2+\alpha/n+\lambda_{2}+1} \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ &\quad \times\sum_{k=1}^{\infty}2^{-2kn+k\alpha+n\lambda _{1}+n+(k+1)n\lambda_{2}+(k+1)n} \\ & \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert B \vert ^{\lambda} \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\sum _{k=1}^{\infty}2^{kn(-1+\lambda_{2}+\alpha/n)} \\ & \leq C \vert B \vert ^{\lambda} \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb{R}^{n})}. \end{aligned}$$
(3.6)

For the estimate of \(I_{4}\). Noting that \(|(x-y_{1}, x-y_{2})|^{2n-\alpha }\geq|x-y_{1}|^{n-\alpha/2}|x-y_{2}|^{n-\alpha/2}\), by (3.2), \(\lambda _{j}<-\frac{\alpha}{2n}\), \(j=1,2\), Lemma 2.3, the Minkowski inequality and the generalized Hölder inequality, we have

$$ \begin{aligned}[b] I_{4}& = \bigl\Vert T(f_{1} \chi_{(2B)^{c}},f_{2}\chi_{(2B)^{c}})\chi_{B} \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \\ & \leq\sum_{k_{1}=1}^{\infty}\sum _{k_{2}=1}^{\infty}\bigl\Vert T(f_{1}\chi _{2^{k_{1}+1}B\setminus2^{k_{1}}B},f_{2}\chi_{2^{k_{2}+1}B\setminus 2^{k_{2}}B})\chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ & \leq C\sum_{k=1}^{\infty}\sum _{k_{2}=1}^{\infty}\biggl\Vert \int _{2^{k_{2}+1}B\setminus2^{k_{2}}B} \int_{2^{k_{1}+1}B\setminus 2^{k_{1}}B}\frac{ \vert f_{1}(y_{1}) \vert \vert f_{2}(y_{2}) \vert }{ \vert (\cdot-y_{1}, \cdot -y_{2}) \vert ^{2n-\alpha}}\,dy_{1}\,dy_{2} \chi_{B}(\cdot) \biggr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \\ & \leq C\sum_{k=1}^{\infty}\sum _{k_{2}=1}^{\infty}\Biggl\Vert \prod _{j=1}^{2} \int_{2^{k_{j}+1}B\setminus2^{k_{j}}B}\frac{ \vert f_{j}(y_{j}) \vert }{ \vert \cdot -y_{j} \vert ^{n-\alpha/2}}\,dy_{j} \chi_{B}(\cdot) \Biggr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})} \\ & \leq C \prod_{j=1}^{2} \Biggl(\sum _{k_{j}=1}^{\infty}\bigl(2^{k_{j}-1}R \bigr)^{-n+\alpha/2} \int_{2^{k_{j}+1}B} \bigl\vert f_{j}(y_{j}) \bigr\vert \,dy_{j} \Biggr) \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\prod _{j=1}^{2} \Biggl(\sum_{k_{j}=1}^{\infty}\bigl\vert 2^{k_{j}}B \bigr\vert ^{-1+\frac{\alpha}{2n}} \Vert f_{j}\chi_{2^{k_{j}+1}B} \Vert _{L^{p_{j}(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2^{k_{j}+1}B} \Vert _{L^{p'_{j}(\cdot)}(\mathbb{R}^{n})} \Biggr) \\ & \leq C \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\prod _{j=1}^{2} \Vert f_{j} \Vert _{\dot{\mathcal{B}}^{p_{j}(\cdot),\lambda_{j}}(\mathbb{R}^{n})} \sum_{k_{j}=1}^{\infty}\bigl\vert 2^{k_{j}}B \bigr\vert ^{\lambda_{j}-1+\frac{\alpha}{2n}} \Vert \chi_{2^{k_{j}+1}B} \Vert _{L^{p_{j}(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2^{k_{j}+1}B} \Vert _{L^{p'_{j}(\cdot)}(\mathbb{R}^{n})}\hspace{-24pt} \\ & \leq C \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\prod _{j=1}^{2} \Vert f_{j} \Vert _{\dot{\mathcal{B}}^{p_{j}(\cdot),\lambda_{j}}(\mathbb{R}^{n})} \sum_{k_{j}=1}^{\infty}\bigl\vert 2^{k_{j}}B \bigr\vert ^{\lambda_{j}-1+\frac{\alpha }{2n}} \bigl\vert 2^{k_{j}}B \bigr\vert \\ & \leq C \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\prod _{j=1}^{2} \Vert f_{j} \Vert _{\dot{\mathcal{B}}^{p_{j}(\cdot),\lambda_{j}}(\mathbb {R}^{n})} \vert B \vert ^{\lambda} \sum _{k_{j}=1}^{\infty}2^{k_{j}n(\lambda_{j}+\frac{\alpha}{2n})} \\ & \leq C \vert B \vert ^{\lambda} \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb{R}^{n})}. \end{aligned} $$
(3.7)

Combining the estimates of (3.3)–(3.7), we have

$$\bigl\Vert T(f_{1},f_{2})\chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C \vert B \vert ^{\lambda} \Vert \chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb{R}^{n})} , $$

that is,

$$\bigl\Vert T(f_{1},f_{2}) \bigr\Vert _{\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb {R}^{n})} \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda _{1}}(\mathbb{R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda _{2}}(\mathbb{R}^{n})}. $$

This completes the proof of Theorem 3.1. □

Multilinear fractional integral commutators

Let \(m\in\mathbb{N}\), \(\vec{b}=(b_{1},b_{2},\ldots,b_{m})\) and \(b_{i}\in \mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})\), \(i=1,\ldots,m\). Then the multilinear commutators of fractional integral operator are defined by

$$[\vec{b},T_{\alpha,m}] (\vec{f}) (x)= \int_{(\mathbb {R}^{n})^{m}}\frac{\prod_{i=1}^{m}(b_{i}(x)-b_{i}(y_{i})f_{i}(y_{i})}{ \vert (x-y_{1},\ldots ,x-y_{m}) \vert ^{mn-\alpha}}\,dy_{1}\cdots dy_{m}. $$
(4.1)

Theorem 4.1

Let \(0< \alpha<mn\), \(0< v_{i}<1/n\), \(\lambda_{i}<-\frac{\alpha}{mn}\), \(\lambda =\sum_{i=1}^{m} v_{i}+\sum_{i=1}^{m}\lambda_{i}+\alpha/n\), \(v_{i}+\lambda _{i}<-\alpha/n\), \(p_{i}(\cdot)\) (\(i=1,\ldots,m\)), \(q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\)satisfy conditions (2.1) and (2.2) in Lemma 2.1 \(1/p(\cdot)=\sum_{i=1}^{m}1/p_{i}(\cdot)-\alpha/n>0\)and \(1/q(\cdot)=\sum_{i=1}^{m} 1/u_{i}(\cdot)+\sum_{i=1}^{m}1/p_{i}(\cdot )-\alpha/n\). Then \([\vec{b},T_{\alpha,m}]\)is also bounded from \(\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})\times \cdots\times\dot{\mathcal{B}}^{p_{m}(\cdot),\lambda_{m}}(\mathbb {R}^{n})\)into \(\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb{R}^{n})\)and the following inequality holds:

$$\bigl\Vert [\vec{b}, T_{\alpha,m}]\vec{f} \bigr\Vert _{\dot{\mathcal{B}}^{q(\cdot ),\lambda}(\mathbb{R}^{n})} \leq C\prod_{i=1}^{m}\bigl( \Vert b_{i} \Vert _{\mathrm {CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal {B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})}\bigr). $$

Proof

Without loss of generality, we still consider only the situation when \(m=2\). Let \(f_{1}\), \(f_{2}\) be functions in \(\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda _{1}}(\mathbb{R}^{n})\) and \(\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda _{2}}(\mathbb{R}^{n})\), respectively. For fixed \(R>0\), denote \(B(0,R)\) by B. We have the following decomposition:

$$ \begin{aligned}[b] [\vec{b},T_{\alpha,2}]\vec{f}(x)& = \bigl[b_{1}-\{b_{1}\}_{B}\bigr] \bigl[b_{2}-\{b_{2}\} _{B}\bigr]T_{\alpha,2}(f_{1},f_{2}) (x) \\ &\quad -\bigl[b_{1}-\{b_{1}\}_{B} \bigr]T_{\alpha,2}\bigl[f_{1},\bigl(b_{2}(\cdot)- \{b_{2}\}_{B}\bigr)f_{2}\bigr](x) \\ &\quad -\bigl[b_{2}-\{b_{2}\}_{B} \bigr]T_{\alpha,2}\bigl[\bigl(b_{1}(\cdot)-\{b_{1} \}_{B}\bigr)f_{1},f_{2}\bigr](x) \\ &\quad +T_{\alpha,2}\bigl[\bigl(b_{1}(\cdot)-\{b_{1} \}_{B}\bigr)f_{1},\bigl(b_{2}(\cdot)- \{b_{2}\} _{B}\bigr)f_{2}\bigr](x) \\ & =:J_{1}+J_{2}+J_{3}+J_{4}. \end{aligned} $$
(4.2)

Thus, by the Minkowski inequality we write

$$ \bigl\Vert [\vec{b}, T_{\alpha,2}]\vec{f}\chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\leq\sum_{i=1}^{4} \Vert J_{i}\chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}=:\sum _{i=1}^{4}L_{i}. $$
(4.3)

Because of the symmetry of \(f_{1}\) and \(f_{2}\), we can see that the estimate of \(L_{2}\) is analogous to that of \(L_{3}\). Then we will estimate \(L_{1}\), \(L_{2}\) and \(L_{4}\), respectively.

Next we will decompose \(f_{i}\) as \(f_{i}(y_{i})=f_{i}(y_{i})\chi _{2B}+f_{i}(y_{i})\chi_{(2B)^{c}}\), for \(i=1,2\).

(i) For \(L_{1}\), by using the Minkowski inequality we can write

$$ \begin{aligned}[b] L_{1}& \leq \bigl\Vert \bigl[b_{1}-\{b_{1}\}_{B}\bigr] \bigl[b_{2}-\{b_{2}\}_{B}\bigr]T_{\alpha,2}(f_{1} \chi _{2B},f_{2}\chi_{2B}) (\cdot) \chi_{B}(\cdot) \bigr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})} \\ &\quad + \bigl\Vert \bigl[b_{1}-\{b_{1}\}_{B}\bigr] \bigl[b_{2}-\{b_{2}\}_{B}\bigr]T_{\alpha,2}(f_{1} \chi_{2B},f_{2}\chi _{(2B)^{c}}) (\cdot) \chi_{B}(\cdot) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ &\quad + \bigl\Vert \bigl[b_{1}-\{b_{1}\}_{B}\bigr] \bigl[b_{2}-\{b_{2}\}_{B}\bigr]T_{\alpha,2}(f_{1} \chi _{(2B)^{c}},f_{2}\chi_{2B}) (\cdot) \chi_{B}(\cdot) \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \\ & \quad+ \bigl\Vert \bigl[b_{1}-\{b_{1}\}_{B}\bigr] \bigl[b_{2}-\{b_{2}\}_{B}\bigr]T_{\alpha,2}(f_{1} \chi _{(2B)^{c}},f_{2}\chi_{(2B)^{c}}) (\cdot) \chi_{B}(\cdot) \bigr\Vert _{L^{q(\cdot )}(\mathbb{R}^{n})} \\ & =:L_{11}+L_{12}+L_{13}+L_{14}. \end{aligned} $$
(4.4)

Firstly we estimate \(L_{11}\). Noticing that \(\frac{1}{p(\cdot)}=\sum_{i=1}^{2}\frac{1}{p_{i}(\cdot)}-\frac{\alpha}{n}\), then \(\frac {1}{q(\cdot)}=\sum_{i=1}^{2}\frac{1}{u_{i}(\cdot)}+\frac{1}{p(\cdot )}\). By Lemma 2.6 and using the boundedness of \(T_{\alpha,2}\) from \(L^{p_{1}(\cdot)}(\mathbb{R}^{n})\times L^{p_{2}(\cdot)}(\mathbb {R}^{n})\) into \(L^{q(\cdot)}(\mathbb{R}^{n})\) in Theorem B, we get

$$ \begin{aligned}[b] L_{11}& \leq C \bigl\Vert T_{\alpha,2}(f_{1}\chi_{2B},f_{2} \chi_{2B}) (\cdot)\chi _{B}(\cdot) \bigr\Vert _{L^{p(\cdot)}(\mathbb{R}^{n})}\prod_{i=1}^{2} \bigl\Vert \bigl[b_{i}-\{ b_{i}\}_{B}\bigr] \chi_{B} \bigr\Vert _{L^{u_{i}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \prod_{i=1}^{2} \Vert f_{i}\chi_{B} \Vert _{L^{p_{i}(\cdot)}(\mathbb {R}^{n})}\prod _{i=1}^{2} \bigl\Vert \bigl[b_{i}- \{b_{i}\}_{B}\bigr]\chi_{B} \bigr\Vert _{L^{u_{i}(\cdot )}(\mathbb{R}^{n})} \\ & \leq C \vert B \vert ^{\lambda_{1}+\lambda_{2}+v_{1}+v_{2}} \Vert \chi_{B} \Vert _{L^{p_{1}(\cdot )}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{u_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \\ & \quad\times\prod_{i=1}^{2} \bigl( \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot ),\lambda_{i}}(\mathbb{R}^{n})} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \bigr) \\ & \leq C \vert B \vert ^{\lambda_{1}+\lambda_{2}+v_{1}+v_{2}+\alpha/n+1/q(\cdot)}\prod_{i=1}^{2} \bigl( \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda _{i}}(\mathbb{R}^{n})} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \bigr) \\ & \leq C \vert B \vert ^{\lambda+1/q(\cdot)}\prod_{i=1}^{2} \bigl( \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \bigr), \end{aligned} $$
(4.5)

where

$$\begin{aligned} \Vert \chi_{B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{u_{1}(\cdot )}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})}& \approx \vert B \vert ^{1/p_{1}(\cdot)+1/p_{2}(\cdot)+1/u_{1}(\cdot)+1/u_{2}(\cdot)} \\ & = \vert B \vert ^{1/q(\cdot)+\alpha/n}. \end{aligned} $$

Now we estimate \(L_{12}\). Noticing that \(|(x-y_{1}, x-y_{2})|^{2n-\alpha }\geq|x-y_{2}|^{2n-\alpha}\). By \(\lambda_{2}<-\frac{\alpha}{2n}\), Lemma 2.3 and the generalized Hölder inequality, we obtain

$$\begin{aligned}& \bigl\vert T_{\alpha,2}(f_{1} \chi_{2B},f_{2}\chi_{(2B)^{c}}) (x) \bigr\vert \\& \quad = \biggl\vert \int_{(\mathbb{R}^{n})^{2}}\frac{[f_{1}(y_{1})\chi _{2B}(y_{1})][f_{2}(y_{2})\chi_{(2B)^{c}}(y_{2})]}{ \vert (x-y_{1},x-y_{2}) \vert ^{2n-\alpha }}\,dy_{1}\,dy_{2} \biggr\vert \\& \quad \leq C \int_{2B} \bigl\vert f_{1}(y_{1}) \bigr\vert \,dy_{1} \int_{(2B)^{c}}\frac { \vert f_{2}(y_{2}) \vert }{ \vert x-y_{2} \vert ^{2n-\alpha}}\,dy_{2} \\& \quad \leq C \Vert f_{1}\chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2B} \Vert _{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}\sum _{k=1}^{\infty}\int _{2^{k+1}B\setminus2^{k}B}\frac{ \vert f_{2}(y_{2}) \vert }{ \vert x-y_{2} \vert ^{2n-\alpha}}\,dy_{2} \\& \quad \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \vert 2B \vert ^{\lambda_{1}} \Vert \chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb {R}^{n})} \Vert \chi_{2B} \Vert _{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \\& \qquad{}\times\sum_{k=1}^{\infty}\bigl(2^{k-1}R\bigr)^{-2n+\alpha} \int _{2^{k+1}B} \bigl\vert f_{2}(y_{2}) \bigr\vert \,dy_{2} \\& \quad \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \vert 2B \vert ^{\lambda_{1}+1}\sum_{k=1}^{\infty}\bigl\vert 2^{k}B \bigr\vert ^{-2+\alpha/n} \Vert f_{2} \chi_{2^{k+1}B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2^{k+1}B} \Vert _{L^{p'_{2}(\cdot)}(\mathbb{R}^{n})} \\& \quad \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert B \vert ^{\lambda_{1}+1-2+\alpha/n+1+\lambda_{2}} \sum _{k=1}^{\infty}2^{kn(-2+\alpha/n+\lambda_{2}+1)} \\& \quad \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert B \vert ^{\lambda_{1}+\lambda_{2}+\alpha/n}\sum _{k=1}^{\infty}2^{kn(\frac{\alpha}{2n}+\lambda_{2})} \\& \quad \leq C \vert B \vert ^{\lambda_{1}+\lambda_{2}+\alpha/n}\prod_{i=1}^{2} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})}. \end{aligned}$$

Let \(\frac{1}{u(\cdot)}=\frac{1}{u_{1}(\cdot)}+\frac{1}{u_{2}(\cdot )}\), by the fact \(1/p(\cdot)=\sum_{i=1}^{2}1/p_{i}(\cdot)-\alpha/n>0\) and \(1/q(\cdot)=\sum_{i=1}^{m} 1/u_{i}(\cdot)+\sum_{i=1}^{m}1/p_{i}(\cdot )-\alpha/n\), then \(u(\cdot)>q(\cdot)\). Thus by Lemma 2.5 and Lemma 2.6 we get

$$\begin{aligned} & \bigl\Vert \bigl[b_{1}- \{b_{1}\}_{B}\bigr] \bigl[b_{2}-\{b_{2} \}_{B}\bigr]\chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})} \\ &\quad \leq C \bigl\Vert \bigl[b_{1}-\{b_{1}\}_{B} \bigr]\chi_{B} \bigr\Vert _{L^{u_{1}(\cdot)}(\mathbb{R}^{n})} \bigl\Vert \bigl[b_{2}-\{b_{2}\}_{B}\bigr] \chi_{B} \bigr\Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})} \\ &\quad \leq C \Vert b_{1} \Vert _{\mathrm{CBMO}^{u_{1}(\cdot),v_{1}}(\mathbb {R}^{n})} \vert B \vert ^{v_{1}} \Vert \chi_{B} \Vert _{L^{u_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\qquad{}\times \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})} \vert B \vert ^{v_{2}} \Vert \chi_{B} \Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{B} \Vert _{L^{p(\cdot )}(\mathbb{R}^{n})} \\ &\quad \leq C \vert B \vert ^{v_{1}+v_{2}+\frac{1}{u_{1}(\cdot)}+\frac{1}{u_{2}(\cdot )}+\frac{1}{p(\cdot)}}\prod_{i=1}^{2} \Vert b_{i} \Vert _{\mathrm {CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \\ &\quad \leq C \vert B \vert ^{v_{1}+v_{2}+\frac{1}{q(\cdot)}}\prod_{i=1}^{2} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})}. \end{aligned} $$

This yields

$$ \begin{aligned}[b] L_{12} & \leq C \vert B \vert ^{\lambda_{1}+\lambda_{2}+\alpha/n+v_{1}+v_{2}+\frac{1}{q(\cdot )}}\prod_{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot ),v_{i}}(\mathbb{R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot ),\lambda_{i}}(\mathbb{R}^{n})} \bigr) \\ & \leq C \vert B \vert ^{\lambda+\frac{1}{q(\cdot)}}\prod_{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})} \bigr). \end{aligned} $$
(4.6)

Similarly, we have

$$ L_{13}\leq C \vert B \vert ^{\lambda+\frac{1}{q(\cdot)}}\prod _{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb {R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \bigr). $$
(4.7)

Now for the estimate of \(L_{14}\). Note that \(|(x-y_{1}, x-y_{2})|^{2n-\alpha}\geq|x-y_{1}|^{n-\alpha/2}|x-y_{2}|^{n-\alpha/2}\). Using \(\lambda_{j}<-\frac{\alpha}{2n}\), \(j=1,2\) and the generalized Hölder inequality, we have

$$\begin{aligned}& \bigl\vert T_{\alpha,2}(f_{1} \chi_{(2B)^{c}},f_{2}\chi_{(2B)^{c}}) (x) \bigr\vert \\& \quad \leq C \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}}\frac{ \vert f_{1}(y_{1})\chi _{(2B)^{c}}(y_{1}) \vert \vert f_{2}(y_{2})\chi _{(2B)^{c}}(y_{2}) \vert }{ \vert (x-y_{1},x-y_{2}) \vert ^{2n-\alpha}}\,dy_{1}\,dy_{2} \\& \quad \leq C\sum_{k_{1}=1}^{\infty}\sum _{k_{2}=1}^{\infty}\int _{2^{k_{2}+1}B\setminus2^{k_{2}}B} \int_{2^{k_{1}+1}B\setminus 2^{k_{1}}B}\frac{ \vert f_{1}(y_{1}) \vert \vert f_{2}(y_{2}) \vert }{ \vert (x-y_{1},x-y_{2}) \vert ^{2n-\alpha }}\,dy_{1}\,dy_{2} \\& \quad \leq C \sum_{k_{1}=1}^{\infty}\sum _{k_{2}=1}^{\infty}\prod_{i=1}^{2} \int _{2^{k_{i}+1}B\setminus2^{k_{i}}B}\frac{ \vert f_{i}(y_{i}) \vert }{ \vert (x-y_{i}) \vert ^{n-\alpha /2}}\,dy_{i} \\& \quad \leq C \prod_{i=1}^{2}\sum _{k_{i}=1}^{\infty}\bigl(2^{k_{i}-1}R \bigr)^{-n+\alpha /2} \int_{2^{k_{i}+1}B} \bigl\vert f_{i}(y_{i}) \bigr\vert \,dy_{i} \\& \quad \leq C \prod_{i=1}^{2}\sum _{k_{i}=1}^{\infty}\bigl\vert 2^{k_{j}}B \bigr\vert ^{-1+\frac {\alpha}{2n}} \Vert f_{j}\chi_{2^{k_{j}+1}B} \Vert _{L^{p_{j}(\cdot)}(\mathbb {R}^{n})} \Vert \chi_{2^{k_{j}+1}B} \Vert _{L^{p'_{j}(\cdot)}(\mathbb{R}^{n})} \\& \quad \leq C\prod_{i=1}^{2} \Vert f_{j} \Vert _{\dot{\mathcal{B}}^{p_{j}(\cdot),\lambda _{j}}(\mathbb{R}^{n})}\sum_{k_{i}=1}^{\infty}\bigl\vert 2^{k_{j}}B \bigr\vert ^{-1+\frac{\alpha }{2n}+\lambda_{j}+1} \\& \quad \leq C \vert B \vert ^{\lambda_{1}+\lambda_{2}+\alpha/n}\prod_{i=1}^{2} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})}. \end{aligned}$$

Similar to the estimates for \(L_{12}\), we have

$$ L_{14}\leq C \vert B \vert ^{\lambda+\frac{1}{q(\cdot)}}\prod _{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb {R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \bigr). $$
(4.8)

By the estimates of \(L_{1j}\), \(j=1,2,3,4\), we get

$$ L_{1}\leq C \vert B \vert ^{\lambda+\frac{1}{q(\cdot)}}\prod _{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb {R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \bigr). $$
(4.9)

(ii) For \(L_{2}\), we obtain

$$ \begin{aligned}[b] L_{2}& \leq \bigl\Vert \bigl[b_{1}-\{b_{1}\}_{B}\bigr]T_{\alpha,2} \bigl[f_{1}\chi_{2B}, \bigl(b_{2}(\cdot)-\{ b_{2}\}_{B}\bigr)f_{2}\chi_{2B}\bigr] \chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ &\quad + \bigl\Vert \bigl[b_{1}-\{b_{1}\}_{B} \bigr]T_{\alpha,2}\bigl[f_{1}\chi_{(2B)^{c}}, \bigl(b_{2}-\{b_{2}\} _{B}\bigr)f_{2} \chi_{2B}\bigr]\chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ &\quad + \bigl\Vert \bigl[b_{1}-\{b_{1}\}_{B} \bigr]T_{\alpha,2}\bigl[f_{1}\chi_{2B}, \bigl(b_{2}-\{b_{2}\}_{B}\bigr)f_{2}\chi _{(2B)^{c}}\bigr]\chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ &\quad + \bigl\Vert \bigl[b_{1}-\{b_{1}\}_{B} \bigr]T_{\alpha,2}\bigl[f_{1}\chi_{(2B)^{c}}, \bigl(b_{2}-\{b_{2}\} _{B}\bigr)f_{2} \chi_{(2B)^{c}}\bigr]\chi_{B}(\cdot) \bigr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})} \\ & =:L_{21}+L_{22}+L_{23}+L_{24}. \end{aligned} $$
(4.10)

Let \(\frac{1}{q(\cdot)}=\frac{1}{u_{1}(\cdot)}+\frac{1}{q_{1}(\cdot )}\), \(\frac{1}{q_{1}(\cdot)}=\frac{1}{p_{1}(\cdot)}+\frac{1}{g(\cdot )}-\frac{\alpha}{n}\), \(\frac{1}{g(\cdot)}=\frac{1}{p_{2}(\cdot )}+\frac{1}{u_{2}(\cdot)}\). By \(\lambda_{j}<-\frac{\alpha}{2n}\), \(j=1,2\) and boundedness of \(T_{\alpha,2}\) from \(L^{p_{1}(\cdot)}(\mathbb {R}^{n})\times L^{g(\cdot)}(\mathbb{R}^{n})\) into \(L^{q_{1}(\cdot )}(\mathbb{R}^{n})\), we get

$$\begin{aligned} L_{21}& \leq C \bigl\Vert \bigl[b_{1}-\{b_{1}\}_{B}\bigr] \chi_{B} \bigr\Vert _{L^{u_{1}(\cdot)}(\mathbb {R}^{n})} \bigl\Vert T_{\alpha,2} \bigl[f_{1}\chi_{2B},\bigl(b_{2}(\cdot)- \{b_{2}\}_{B}\bigr)f_{2}\chi _{2B} \bigr]\chi_{B} \bigr\Vert _{L^{q_{1}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \bigl\Vert \bigl[b_{1}-\{b_{1}\}_{B} \bigr]\chi_{B} \bigr\Vert _{L^{u_{1}(\cdot)}(\mathbb {R}^{n})} \Vert f_{1} \chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \bigl\Vert \bigl[b_{2}- \{ b_{2}\}_{B}\bigr]f_{2}\chi_{2B} \bigr\Vert _{L^{g(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \bigl\Vert \bigl[b_{1}-\{b_{1}\}_{B} \bigr]\chi_{B} \bigr\Vert _{L^{u_{1}(\cdot)}(\mathbb {R}^{n})} \Vert f_{1} \chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert f_{2}\chi _{2B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \\ &\quad \times \bigl\Vert \bigl[b_{2}-\{b_{2}\}_{2B}+ \{b_{2}\}_{2B}-\{b_{2}\}_{B}\bigr] \chi_{2B} \bigr\Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \Vert b_{1} \Vert _{\mathrm{CBMO}^{u_{1}(\cdot),v_{1}}(\mathbb {R}^{n})} \vert B \vert ^{v_{1}} \Vert \chi_{B} \Vert _{L^{u_{1}(\cdot)}(\mathbb{R}^{n})} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \vert 2B \vert ^{\lambda_{1}} \Vert \chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \\ & \quad\times \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert 2B \vert ^{\lambda_{2}} \Vert \chi_{2B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})} \vert 2B \vert ^{v_{2}} \Vert \chi_{2B} \Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \vert B \vert ^{v_{1}+v_{2}+\lambda_{1}+\lambda_{2}+\frac{1}{u_{1}(\cdot)}+\frac {1}{u_{2}(\cdot)}+\frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}} \prod_{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb {R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \bigr) \\ & \leq C \vert B \vert ^{\lambda+1/q(\cdot)}\prod_{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm {CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal {B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})} \bigr), \end{aligned}$$
(4.11)

where

$$\begin{aligned} \bigl\vert \{b_{2}\}_{2B}- \{b_{2}\}_{B} \bigr\vert & \leq\frac{1}{ \vert B \vert } \bigl\Vert \bigl(b_{2}-\{b_{2}\}_{2B}\bigr)\chi _{2B} \bigr\Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{2B} \Vert _{L^{u'_{2}(\cdot )}(\mathbb{R}^{n})} \\ & \leq C \bigl\Vert \bigl(b_{2}-\{b_{2}\}_{2B} \bigr)\chi_{2B} \bigr\Vert _{L^{u_{2}(\cdot)}(\mathbb {R}^{n})}\frac{1}{ \Vert \chi_{2B} \Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})}}. \end{aligned} $$

For the estimate of \(L_{22}\), we have \(|(x-y_{1}, x-y_{2})|^{2n-\alpha }\approx|x-y_{1}|^{2n-\alpha}\). Using \(\frac{1}{p'_{2}(\cdot)}=\frac {1}{g'(\cdot)}+\frac{1}{u_{2}(\cdot)}\), \(\lambda_{1}<-\frac{\alpha }{2n}\), Lemmas 2.3, 2.6 and the generalized Hölder inequality, we have

$$\begin{gathered} \bigl\vert T_{\alpha,2}\bigl[f_{1} \chi_{(2B)^{c}},\bigl(b_{2}(\cdot)-\{b_{2} \}_{B}\bigr)f_{2}\chi _{2B}\bigr](x) \bigr\vert \\ \quad \leq C \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}}\frac{ \vert f_{1}(y_{1})\chi _{(2B)^{c}}(y_{1}) \vert \vert f_{2}(y_{2})\chi_{2B}(y_{2}) \vert \vert b_{2}(y_{2})-\{b_{2}\} _{B} \vert }{ \vert (x-y_{1},x-y_{2}) \vert ^{2n-\alpha}}\,dy_{1}\,dy_{2} \\ \quad \leq C \int_{(2B)^{c}}\frac{ \vert f_{1}(y_{1}) \vert }{ \vert x-y_{1} \vert ^{2n-\alpha}}\,dy_{1} \int _{2B} \bigl\vert f_{2}(y_{2}) \bigr\vert \bigl\vert b_{2}(y_{2})-\{b_{2} \}_{B} \bigr\vert \,dy_{2} \\ \quad \leq C \Vert f_{2}\chi_{2B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \bigl\Vert \bigl[b_{2}-\{ b_{2}\}_{B}\bigr] \chi_{2B} \bigr\Vert _{L^{p'_{2}(\cdot)}(\mathbb{R}^{n})}\sum _{k=1}^{\infty}\int_{2^{k+1}B\setminus2^{k}B}\frac { \vert f_{1}(y_{1}) \vert }{ \vert x-y_{1} \vert ^{2n-\alpha}}\,dy_{1} \\ \quad \leq C \Vert f_{2}\chi_{2B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \bigl\Vert \bigl[b_{2}-\{ b_{2}\}_{B}\bigr] \chi_{2B} \bigr\Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{2B} \Vert _{L^{g'(\cdot)}(\mathbb{R}^{n})} \\ \qquad{}\times\sum_{k=1}^{\infty}\int_{2^{k+1}B\setminus2^{k}B}\frac { \vert f_{1}(y_{1}) \vert }{ \vert x-y_{1} \vert ^{2n-\alpha}}\,dy_{1} \\ \quad \leq C \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert 2B \vert ^{\lambda_{2}} \Vert \chi_{2B} \Vert _{L^{p_{2}(\cdot)}(\mathbb {R}^{n})} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb {R}^{n})} \vert 2B \vert ^{v_{2}} \Vert \chi_{2B} \Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \\ \quad\quad{} \times \Vert \chi_{2B} \Vert _{L^{g'(\cdot)}(\mathbb{R}^{n})} \sum _{k=1}^{\infty}\bigl(2^{k-1}R \bigr)^{-2n+\alpha} \int _{2^{k+1}B} \bigl\vert f_{1}(y_{1}) \bigr\vert \,dy_{1} \\ \quad \leq C \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert 2B \vert ^{\lambda_{2}+v_{2}+1} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot ),v_{2}}(\mathbb{R}^{n})} \\ \quad\quad{} \times\sum_{k=1}^{\infty}\bigl\vert 2^{k}B \bigr\vert ^{-2+\alpha/n} \Vert f_{1} \chi_{2^{k+1}B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{2^{k+1}B} \Vert _{L^{p'_{1}(\cdot )}(\mathbb{R}^{n})} \\ \quad \leq C \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \vert B \vert ^{\lambda_{2}+v_{2}+1-2+\alpha/n+1+\lambda_{1}} \\ \quad\quad{} \times \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})} \sum _{k=1}^{\infty}2^{kn(-2+\alpha/n+\lambda_{1}+1)} \\ \quad \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert B \vert ^{\lambda_{1}+\lambda_{2}+v_{2}+\alpha/n} \Vert b_{2} \Vert _{\mathrm {CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})}\sum_{k=1}^{\infty}2^{kn(\frac{\alpha}{2n}+\lambda_{1})} \\ \quad \leq C \vert B \vert ^{\lambda_{1}+\lambda_{2}+v_{2}+\alpha/n} \Vert b_{2} \Vert _{\mathrm {CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})}\prod_{i=1}^{2} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})}. \end{gathered} $$

Thus, from \(\frac{1}{q(\cdot)}=\frac{1}{u_{1}(\cdot)}+\frac {1}{q_{1}(\cdot)}\) and Lemmas 2.5, 2.6, we get

$$ \begin{aligned}[b] L_{22}& \leq C \vert B \vert ^{\lambda_{1}+\lambda_{2}+v_{2}+\alpha/n} \Vert b_{2} \Vert _{\mathrm {CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})}\prod _{i=1}^{2} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})} \bigl\Vert \bigl[b_{1}-\{b_{1} \}_{B}\bigr]\chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\hspace{-12pt} \\ & \leq C \vert B \vert ^{\lambda_{1}+\lambda_{2}+v_{2}+\alpha/n} \Vert b_{2} \Vert _{\mathrm {CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})} \bigl\Vert \bigl[b_{1}-\{b_{1} \}_{B}\bigr]\chi_{B} \bigr\Vert _{L^{u_{1}(\cdot)}(\mathbb{R}^{n})} \\ &\quad \times \Vert \chi_{B} \Vert _{L^{q_{1}(\cdot)}(\mathbb{R}^{n})}\prod _{i=1}^{2} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \\ & \leq C \vert B \vert ^{\lambda_{1}+\lambda_{2}+v_{2}+\alpha/n} \Vert b_{2} \Vert _{\mathrm {CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})} \Vert b_{1} \Vert _{\mathrm {CBMO}^{u_{1}(\cdot),v_{1}}(\mathbb{R}^{n})} \vert B \vert ^{v_{1}} \Vert \chi_{B} \Vert _{L^{u_{1}(\cdot)}(\mathbb{R}^{n})} \\ &\quad \times \Vert \chi_{B} \Vert _{L^{q_{1}(\cdot)}(\mathbb{R}^{n})}\prod _{i=1}^{2} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \\ & \leq C \vert B \vert ^{\lambda+1/q(\cdot)}\prod_{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm {CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal {B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})} \bigr). \end{aligned} $$
(4.12)

For \(L_{23}\), noticing that \(|(x-y_{1}, x-y_{2})|^{2n}\geq|x-y_{2}|^{2n}\) and \(\frac{1}{p'_{2}(\cdot)}=\frac{1}{g'(\cdot)}+\frac{1}{u_{2}(\cdot )}\). By Lemmas 2.3, 2.6, \(v_{2}+\lambda_{2}+\alpha/n<0\), the generalized Hölder inequality and the Minkowski inequality, we get

$$\begin{aligned}& \bigl\vert T_{\alpha,2}\bigl[f_{1} \chi_{2B},\bigl(b_{2}(\cdot)-\{b_{2} \}_{B}\bigr)f_{2}\chi _{(2B)^{c}}\bigr](x) \bigr\vert \\& \quad \leq C \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}}\frac{ \vert f_{1}(y_{1})\chi _{2B}(y_{1}) \vert \vert f_{2}(y_{2})\chi_{(2B)^{c}}(y_{2}) \vert \vert b_{2}(y_{2})-\{b_{2}\}_{B} \vert }{ \vert (x-y_{1},x-y_{2}) \vert ^{2n-\alpha}}\,dy_{1}\,dy_{2} \\& \quad \leq C \int_{2B} \bigl\vert f_{1}(y_{1}) \bigr\vert \,dy_{1} \int_{(2B)^{c}}\frac { \vert f_{2}(y_{2}) \vert \vert b_{2}(y_{2})-\{b_{2}\}_{B} \vert }{ \vert x-y_{2} \vert ^{2n-\alpha}}\,dy_{2} \\& \quad \leq C \Vert f_{1}\chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2B} \Vert _{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \int_{(2B)^{c}}\frac { \vert f_{2}(y_{2}) \vert \vert b_{2}(y_{2})-\{b_{2}\}_{B} \vert }{ \vert x-y_{2} \vert ^{2n-\alpha}}\,dy_{2} \\& \quad \leq C \vert B \vert ^{\lambda_{1}+1} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot ),\lambda_{1}}(\mathbb{R}^{n})} \Biggl(\sum_{k=1}^{\infty}\bigl\vert 2^{k}B \bigr\vert ^{-2+\alpha/n} \\& \quad\quad{} \times \int_{2^{k+1}B} \bigl\vert f_{2}(y_{2}) \bigr\vert \bigl\vert b_{2}(y_{2})-\{b_{2} \}_{2^{k+1}B}+\{b_{2}\} _{2^{k+1}B}-\{b_{2} \}_{B} \bigr\vert \,dy_{2} \Biggr) \\& \quad \leq C \vert B \vert ^{\lambda_{1}+1} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot ),\lambda_{1}}(\mathbb{R}^{n})} \Biggl[\sum_{k=1}^{\infty}\bigl\vert 2^{k}B \bigr\vert ^{-2+\alpha/n} \Vert f_{2} \chi_{2^{k+1}B} \Vert _{L^{p_{2}(\cdot)}(\mathbb {R}^{n})} \\& \quad\quad{} \times \bigl( \bigl\Vert \bigl(b_{2}-\{b_{2} \}_{2^{k+1}B}\bigr)\chi_{2^{k+1}B} \bigr\Vert _{L^{p'_{2}(\cdot)}(\mathbb{R}^{n})}+ \bigl\vert \{b_{2}\}_{2^{k+1}B}-\{b_{2}\}_{B} \bigr\vert \Vert \chi_{2^{k+1}B} \Vert _{L^{p'_{2}(\cdot)}(\mathbb{R}^{n})} \bigr) \Biggr] \\& \quad \leq C \vert B \vert ^{\lambda_{1}+1} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot ),\lambda_{1}}(\mathbb{R}^{n})} \Biggl(\sum_{k=1}^{\infty}k \bigl\vert 2^{k}B \bigr\vert ^{-2+\alpha/n+\lambda_{2}+v_{2}} \Vert f_{2} \Vert _{\dot{\mathcal {B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb{R}^{n})} \\& \quad\quad{} \times \Vert \chi_{2^{k+1}B} \Vert _{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})} \Vert \chi_{2^{k+1}B} \Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{2^{k+1}B} \Vert _{L^{g'(\cdot )}(\mathbb{R}^{n})} \Biggr) \\& \quad \leq C \vert B \vert ^{\lambda_{1}+1-2+\alpha/n+\lambda_{2}+v_{2}+1} \Vert f_{1} \Vert _{\dot {\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \Vert f_{2} \Vert _{\dot {\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb{R}^{n})} \\& \quad\quad{} \times \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})} \sum _{k=1}^{\infty}k2^{kn(-2+\alpha/n+\lambda_{2}+v_{2}+1)} \\& \quad \leq C \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb {R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \vert B \vert ^{\lambda_{1}+\lambda_{2}+v_{2}+\alpha/n} \Vert b_{2} \Vert _{\mathrm {CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})}\sum_{k=1}^{\infty}k2^{-kn} \\& \quad \leq C \vert B \vert ^{\lambda_{1}+\lambda_{2}+v_{2}+\alpha/n} \Vert b_{2} \Vert _{\mathrm {CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})}\prod_{i=1}^{2} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})}, \end{aligned}$$

where

$$\begin{aligned} \bigl\vert \{b_{2}\}_{2^{k+1}B}- \{b_{2}\}_{B} \bigr\vert & \leq\sum _{j=0}^{k} \bigl\vert \{b_{2} \}_{2^{j+1}B}-\{ b_{2}\}_{2^{j}B} \bigr\vert \\ & \leq\sum_{j=0}^{k}\frac{1}{ \vert 2^{j}B \vert } \int_{2^{j}B} \bigl\vert b_{2}(y)-\{b_{2}\} _{2^{j+1}B} \bigr\vert \,dy \\ & \leq C\sum_{j=0}^{k}\frac{1}{ \vert 2^{j}B \vert } \bigl\Vert \bigl(b_{2}(\cdot)-\{b_{2}\} _{2^{j+1}B} \bigr)\chi_{2^{j+1}B} \bigr\Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2^{j+1}B} \Vert _{L^{u'_{2}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C\sum_{j=0}^{k}\frac{1}{ \vert 2^{j}B \vert } \bigl\Vert \bigl(b_{2}(\cdot)-\{b_{2}\} _{2^{j+1}B} \bigr)\chi_{2^{j+1}B} \bigr\Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})}\frac { \vert 2^{j+1}B \vert }{ \Vert \chi_{2^{j+1}B} \Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})}} \\ & \leq C \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})}\sum _{j=0}^{k} \bigl\vert 2^{j+1}B \bigr\vert ^{v_{2}} \Vert \chi_{2^{j+1}B} \Vert _{L^{u_{2}(\cdot)}(\mathbb {R}^{n})} \frac{1}{ \Vert \chi_{2^{j+1}B} \Vert _{L^{u_{2}(\cdot)}(\mathbb {R}^{n})}} \\ & \leq C \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb {R}^{n})}(k+1) \bigl\vert 2^{k+1}B \bigr\vert ^{v_{2}}, \end{aligned} $$

for \(v_{2}>0\).

Hence,

$$ L_{23}\leq C \vert B \vert ^{\lambda+1/q(\cdot)}\prod _{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb {R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \bigr). $$
(4.13)

For \(L_{24}\), using Lemmas 2.3, 2.6, \(v_{2}+\lambda_{2}+\alpha/n<0\) and the generalized Hölder inequality, we obtain

$$\begin{aligned}& \bigl\vert T_{\alpha,2}\bigl[f_{1} \chi_{(2B)^{c}},\bigl(b_{2}(\cdot)-\{b_{2} \}_{B}\bigr)f_{2}\chi _{(2B)^{c}}\bigr](x) \bigr\vert \\& \quad \leq C \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}}\frac{ \vert f_{1}(y_{1})\chi _{(2B)^{c}}(y_{1}) \vert \vert f_{2}(y_{2})\chi_{(2B)^{c}}(y_{2}) \vert \vert b_{2}(y_{2})-\{b_{2}\}_{B} \vert }{ \vert (x-y_{1},x-y_{2}) \vert ^{2n-\alpha}}\,dy_{1}\,dy_{2} \\& \quad \leq C \int_{(2B)^{c}}\frac{ \vert f_{1}(y_{1}) \vert }{ \vert x-y_{1} \vert ^{n-\alpha/2}}\,dy_{1} \int _{(2B)^{c}}\frac{ \vert f_{2}(y_{2}) \vert \vert b_{2}(y_{2})-\{b_{2}\}_{B} \vert }{ \vert x-y_{2} \vert ^{n-\alpha /2}}\,dy_{2} \\& \quad \leq C\sum_{k_{1}=1}^{\infty}\bigl\vert 2^{k_{1}}B \bigr\vert ^{-1+\frac{\alpha}{2n}} \int _{2^{k_{1}+1}B} \bigl\vert f_{1}(y_{1}) \bigr\vert \,dy_{1} \\& \quad\quad{} \times\sum_{k_{2}=1}^{\infty}\bigl\vert 2^{k_{2}}B \bigr\vert ^{-1+\frac{\alpha}{2n}} \int _{2^{k_{2}+1}B} \bigl\vert f_{2}(y_{2}) \bigr\vert \bigl\vert b_{2}(y_{2})-\{b_{2} \}_{2^{k_{2}+1}B}+\{b_{2}\} _{2^{k_{2}+1}B}-\{b_{2} \}_{B} \bigr\vert \,dy_{2} \\& \quad \leq C\sum_{k_{1}=1}^{\infty}\bigl\vert 2^{k_{1}}B \bigr\vert ^{-1+\frac{\alpha }{2n}} \bigl\vert 2^{k_{1}}B \bigr\vert ^{\lambda_{1}+1} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot ),\lambda_{1}}(\mathbb{R}^{n})} \\& \quad\quad{} \times\sum_{k_{2}=1}^{\infty}k_{2} \bigl\vert 2^{k_{2}}B \bigr\vert ^{v_{2}+\lambda_{2}+\frac {\alpha}{2n}} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb {R}^{n})} \Vert f_{2} \Vert _{\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb {R}^{n})} \\& \quad \leq C \vert B \vert ^{\lambda_{1}+\lambda_{2}+v_{2}+\alpha/n} \Vert b_{2} \Vert _{\mathrm {CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})}\prod_{i=1}^{2} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})}. \end{aligned}$$

Thus

$$ L_{24}\leq C \vert B \vert ^{\lambda+1/q(\cdot)}\prod _{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb {R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \bigr). $$
(4.14)

Combining the estimates of \(L_{2j}\), \(j=1,2,3,4\), we can deduce that

$$ L_{2}\leq C \vert B \vert ^{\lambda+1/q(\cdot)}\prod _{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb {R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \bigr). $$
(4.15)

(iii) For \(L_{4}\), we have

$$ \begin{aligned}[b] L_{4}& \leq \bigl\Vert T_{\alpha,2} \bigl[\bigl(b_{1}-\{b_{1}\}_{B} \bigr)f_{1}\chi_{2B}, \bigl(b_{2}(\cdot)-\{ b_{2}\}_{B}\bigr)f_{2}\chi_{2B}\bigr] \chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ &\quad + \bigl\Vert T_{\alpha,2}\bigl[\bigl(b_{1}-\{b_{1} \}_{B}\bigr)f_{1}\chi_{2B}, \bigl(b_{2}- \{b_{2}\}_{B}\bigr)f_{2}\chi _{(2B)^{c}} \bigr]\chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ &\quad + \bigl\Vert T_{\alpha,2}\bigl[b_{1}-\{b_{1} \}_{B}\bigr] \bigl[f_{1}\chi_{(2B)^{c}}, \bigl(b_{2}-\{b_{2}\} _{B}\bigr)f_{2} \chi_{2B}\bigr]\chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ &\quad + \bigl\Vert T_{\alpha,2}\bigl[\bigl(b_{1}-\{b_{1} \}_{B}\bigr)f_{1}\chi_{(2B)^{c}}, \bigl(b_{2}- \{b_{2}\} _{B}\bigr)f_{2}\chi_{(2B)^{c}} \bigr]\chi_{B}(\cdot) \bigr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})} \\ & =:L_{41}+L_{42}+L_{43}+L_{44}. \end{aligned} $$
(4.16)

For \(L_{41}\), let \(\frac{1}{h_{i}(\cdot)}=\frac{1}{p_{i}(\cdot)}+\frac {1}{u_{i}(\cdot)}\), \(i=1,2\), then \(1/q(\cdot)=\sum_{i=1}^{2} 1/h_{i}(\cdot )-\alpha/n\). Using Lemmas 2.3, 2.5, 2.6 and the boundedness of \(T_{\alpha,2}\) from \(L^{h_{1}(\cdot)}(\mathbb{R}^{n})\times L^{h_{2}(\cdot)}(\mathbb{R}^{n})\) into \(L^{q(\cdot)}(\mathbb {R}^{n})\), we have

$$ \begin{aligned}[b] L_{41}& \leq C\prod _{i=1}^{2} \bigl\Vert \bigl(b_{i}( \cdot)-\{b_{i}\}_{B}\bigr)f_{i}(\cdot)\chi _{2B}(\cdot) \bigr\Vert _{L^{h_{i}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C\prod_{i=1}^{2} \bigl\Vert f_{i}(\cdot)\chi_{2B}(\cdot) \bigr\Vert _{L^{p_{i}(\cdot )}(\mathbb{R}^{n})} \bigl\Vert \bigl[b_{i}-\{b_{i}\}_{2B}+ \{b_{i}\}_{2B}-\{b_{i}\}_{B}\bigr] \chi _{2B} \bigr\Vert _{L^{u_{i}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \vert B \vert ^{v_{1}+v_{2}+\lambda_{1}+\lambda_{2}+\frac{1}{u_{1}(\cdot)}+\frac {1}{u_{2}(\cdot)}+\frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}} \prod_{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb {R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \bigr) \\ & \leq C \vert B \vert ^{\lambda+1/q(\cdot)}\prod_{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm {CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal {B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})} \bigr). \end{aligned} $$
(4.17)

For \(L_{42}\), using Lemmas 2.3, 2.6, \(v_{2}+\lambda_{2}+\alpha/n<0\) and the generalized Hölder inequality, we get

$$\begin{gathered} \bigl\vert T_{\alpha,2}\bigl[ \bigl(b_{1}(\cdot)-\{b_{1}\}_{B} \bigr)f_{1}\chi_{2B},\bigl(b_{2}(\cdot)-\{ b_{2}\}_{B}\bigr)f_{2}\chi_{(2B)^{c}} \bigr](x) \bigr\vert \\ \quad \leq C \int_{(2B)^{c}} \int_{2B}\frac{ \vert b_{1}(y_{1})-\{b_{1}\} _{B} \vert \vert f_{1}(y_{1}) \vert \vert b_{2}(y_{2})-\{b_{2}\}_{B} \vert \vert f_{2}(y_{2}) \vert }{ \vert (x-y_{1},x-y_{2}) \vert ^{2n-\alpha}}\,dy_{1}\,dy_{2} \\ \quad \leq C \biggl( \int_{2B} \bigl\vert b_{1}(y_{1})- \{b_{1}\}_{B} \bigr\vert \bigl\vert f_{1}(y_{1}) \bigr\vert \,dy_{1} \biggr) \biggl( \int_{(2B)^{c}}\frac{ \vert b_{2}(y_{2})-\{b_{2}\} _{B} \vert \vert f_{2}(y_{2}) \vert }{ \vert x-y_{2} \vert ^{2n-\alpha}}\,dy_{2} \biggr) \\ \quad \leq C \Vert f_{1}\chi_{2B} \Vert _{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \bigl\Vert \bigl(b_{1}-\{ b_{1}\}_{B}\bigr) \chi_{2B} \bigr\Vert _{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \int _{(2B)^{c}}\frac{ \vert b_{2}(y_{2})-\{b_{2}\}_{B} \vert \vert f_{2}(y_{2}) \vert }{ \vert x-y_{2} \vert ^{2n-\alpha }}\,dy_{2} \\ \quad \leq C \vert B \vert ^{\lambda_{1}+v_{1}+1} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot ),\lambda_{1}}(\mathbb{R}^{n})} \Vert b_{1} \Vert _{\mathrm{CBMO}^{u_{1}(\cdot ),v_{1}}(\mathbb{R}^{n})} \Biggl(\sum _{k=1}^{\infty}\bigl\vert 2^{k}B \bigr\vert ^{-2+\alpha /n} \\ \quad\quad{} \times \int_{2^{k+1}B} \bigl\vert b_{2}(y_{2})- \{b_{2}\}_{B} \bigr\vert \bigl\vert f_{2}(y_{2}) \bigr\vert \,dy_{2} \Biggr) \\ \quad \leq C \vert B \vert ^{\lambda_{1}+v_{1}+1-2+\alpha/n+\lambda_{2}+v_{2}+1} \Vert f_{1} \Vert _{\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \Vert b_{1} \Vert _{\mathrm{CBMO}^{u_{1}(\cdot),v_{1}}(\mathbb{R}^{n})} \Vert f_{2} \Vert _{\dot {\mathcal{B}}^{p_{2}(\cdot),\lambda_{2}}(\mathbb{R}^{n})} \\ \quad\quad{} \times \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})} \sum _{k=1}^{\infty}k2^{kn(-2+\alpha/n+\lambda_{2}+v_{2}+1)} \\ \quad \leq C \vert B \vert ^{\lambda}\prod_{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm {CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal {B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})} \bigr). \end{gathered} $$

This implies that

$$ L_{42}\leq C \vert B \vert ^{\lambda+1/q(\cdot)}\prod _{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb {R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \bigr). $$
(4.18)

Similarly,

$$ L_{43}\leq C \vert B \vert ^{\lambda+1/q(\cdot)}\prod _{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb {R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \bigr). $$
(4.19)

For \(L_{44}\), using Lemmas 2.3, 2.6, \(v_{i}+\lambda_{i}+\alpha/n<0\), \(i=1,2\) and the generalized Hölder inequality, we get

$$\begin{aligned}& \bigl\vert T_{\alpha,2}\bigl[ \bigl(b_{1}(\cdot)-\{b_{1}\}_{B} \bigr)f_{1}\chi_{(2B)^{c}},\bigl(b_{2}(\cdot )- \{b_{2}\}_{B}\bigr)f_{2}\chi_{(2B)^{c}} \bigr](x) \bigr\vert \\& \quad \leq C \int_{(2B)^{c}} \int_{(2B)^{c}}\frac{ \vert b_{1}(y_{1})-\{b_{1}\} _{B} \vert \vert f_{1}(y_{1}) \vert \vert b_{2}(y_{2})-\{b_{2}\}_{B} \vert \vert f_{2}(y_{2}) \vert }{ \vert (x-y_{1},x-y_{2}) \vert ^{2n-\alpha }}\,dy_{1}\,dy_{2} \\& \quad \leq C\prod_{i=1}^{2} \biggl( \int_{(2B)^{c}}\frac{ \vert b_{i}(y_{i})-\{b_{i}\} _{B} \vert \vert f_{i}(y_{i}) \vert }{ \vert x-y_{i} \vert ^{n-\alpha/2}}\,dy_{i} \biggr) \\& \quad \leq C\prod_{i=1}^{2} \Biggl(\sum _{k_{i}=1}^{\infty}\bigl\vert 2^{k_{i}}B \bigr\vert ^{-1+\frac {\alpha}{2n}} \int_{2^{k_{i}+1}B} \bigl\vert b_{i}(y_{i})- \{b_{i}\} _{B} \bigr\vert \bigl\vert f_{i}(y_{i}) \bigr\vert \,dy_{i} \Biggr) \\& \quad \leq C \vert B \vert ^{\lambda}\prod_{i=1}^{2} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot ),v_{i}}(\mathbb{R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot ),\lambda_{i}}(\mathbb{R}^{n})} \Biggl(\sum_{k_{i}=1}^{\infty}k_{i}2^{k_{i}n(v_{i}+\lambda_{i}+\frac{\alpha}{2n})} \Biggr) \\& \quad \leq C \vert B \vert ^{\lambda}\prod_{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm {CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal {B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})} \bigr). \end{aligned}$$

So

$$ L_{44}\leq C \vert B \vert ^{\lambda+1/q(\cdot)}\prod _{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb {R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \bigr). $$
(4.20)

From the estimates of \(L_{4j}\), \(j=1,2,3,4\), we have

$$ L_{4}\leq C \vert B \vert ^{\lambda+1/q(\cdot)}\prod _{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb {R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb {R}^{n})} \bigr). $$
(4.21)

Furthermore, we obtain

$$\bigl\Vert [\vec{b}, T_{\alpha,2}]\vec{f}\chi_{B} \bigr\Vert _{L^{q(\cdot)}(\mathbb {R}^{n})}\leq C \vert B \vert ^{\lambda+1/q(\cdot)}\prod _{i=1}^{2} \bigl( \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal{B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})} \bigr). $$

Thus, we have

$$\bigl\Vert [\vec{b}, T_{\alpha,2}]\vec{f} \bigr\Vert _{\dot{\mathcal{B}}^{q(\cdot ),\lambda}(\mathbb{R}^{n})} \leq C\prod_{i=1}^{2}\bigl( \Vert b_{i} \Vert _{\mathrm {CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})} \Vert f_{i} \Vert _{\dot{\mathcal {B}}^{p_{i}(\cdot),\lambda_{i}}(\mathbb{R}^{n})}\bigr). $$

This completes the proof of Theorem 4.1. □

Multilinear fractional integral commutators of the second kind

There is another kind of multilinear commutators \([\vec{b}, T_{\alpha }]\), which was introduced by Pérez and Trujillo-González [25] in 2002, with the vector symbol \(\vec{b}=(b_{1},b_{2},\ldots,b_{m})\) defined by

$$[\vec{b},T_{\alpha}] f(x)= \int_{\mathbb{R}^{n}}\frac {\prod_{i=1}^{m}(b_{i}(x)-b_{i}(y))f(y)}{ \vert (x-y) \vert ^{n-\alpha}}\,dy, $$
(5.1)

where \(b_{i}\in\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})\), \(i=1,\ldots,m\). We have the following result.

Theorem 5.1

Let \(0< \alpha<n\), \(0< v_{i}<1/n\), \(\lambda=\sum_{i=1}^{m} v_{i}+\mu+\alpha /n<0\), \(p(\cdot), q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\)satisfy conditions (2.1) and (2.2) in Lemma 2.1 \(1/p(\cdot)>\alpha/n\)and \(1/q(\cdot)=\sum_{i=1}^{m} 1/u_{i}(\cdot)+1/p(\cdot)-\alpha/n\). Then \([\vec{b},T_{\alpha}]\)is bounded from \(\dot{\mathcal{B}}^{p(\cdot),\mu}(\mathbb{R}^{n})\)into \(\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb{R}^{n})\)and the following inequality holds:

$$\bigl\Vert [\vec{b}, T_{\alpha}]f \bigr\Vert _{\dot{\mathcal{B}}^{q(\cdot),\lambda }(\mathbb{R}^{n})}\leq C \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu }(\mathbb{R}^{n})}\prod_{i=1}^{m} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot ),v_{i}}(\mathbb{R}^{n})}. $$

Proof

Without loss of generality, we can assume that \(m=2\). For any fixed \(R>0\), denote \(B(0,R)\) by B and \(B(0,kR)\) by kB for \(k\in\mathbb{N}\). Let \(\{b\}_{E}\) denote the integral average of the function b over the set E. For \(f\in\dot{\mathcal{B}}^{p(\cdot ),\mu}(\mathbb{R}^{n})\) and any \(x\in\mathbb{R}^{n}\), we write

$$f(x)=f(x)\chi_{2B}+f(x)\chi_{(2B)^{c}}=: f_{1}(x)+f_{2}(x), $$

and then we may decompose \([\vec{b}, T_{\alpha}]\vec{f}(x)\) into four parts as follows:

$$ \begin{aligned}[b] [\vec{b},T_{\alpha}]f(x)& = \bigl[b_{1}-\{b_{1}\}_{B}\bigr] \bigl[b_{2}-\{b_{2}\}_{B}\bigr]T_{\alpha }(f) (x) \\ & \quad-\bigl[b_{1}-\{b_{1}\}_{B} \bigr]T_{\alpha}\bigl[\bigl(b_{2}(\cdot)-\{b_{2} \}_{B}\bigr)f\bigr](x) \\ & \quad-\bigl[b_{2}-\{b_{2}\}_{B} \bigr]T_{\alpha}\bigl[\bigl(b_{1}(\cdot)-\{b_{1} \}_{B}\bigr)f\bigr](x) \\ &\quad +T_{\alpha}\bigl[\bigl(b_{1}(\cdot)-\{b_{1} \}_{B}\bigr) \bigl(b_{2}(\cdot)-\{b_{2} \}_{B}\bigr)f\bigr](x) \\ & =:M_{1}(x)+M_{2}(x)+M_{3}(x)+M_{4}(x). \end{aligned} $$
(5.2)

Now we will give the estimates of four functions above, respectively.

(i) For \(M_{1}(x)\), using the Minkowski inequality we write

$$ \begin{aligned}[b] \Vert M_{1}\chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}& \leq \bigl\Vert \bigl[b_{1}(\cdot)-\{ b_{1}\}_{B}\bigr] \bigl[b_{2}(\cdot)- \{b_{2}\}_{B}\bigr]T_{\alpha}f_{1}(\cdot) \chi_{B}(\cdot) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ &\quad + \bigl\Vert \bigl[b_{1}(\cdot)-\{b_{1} \}_{B}\bigr] \bigl[b_{2}(\cdot)-\{b_{2} \}_{B}\bigr]T_{\alpha}f_{2}(\cdot ) \chi_{B}(\cdot) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ & =:M_{11}+M_{12}. \end{aligned} $$
(5.3)

Firstly we estimate \(M_{11}\). Let \(\frac{1}{r(\cdot)}=\frac {1}{p(\cdot)}-\frac{\alpha}{n}\), then \(\frac{1}{q(\cdot)}=\sum_{i=1}^{2}\frac{1}{u_{i}(\cdot)}+\frac{1}{r(\cdot)}\). By Lemmas 2.3, 2.6 and the boundedness of \(T_{\alpha}\) from \(L^{p(\cdot)}(\mathbb {R}^{n})\) into \(L^{r(\cdot)}(\mathbb{R}^{n})\) in Theorem 1.8 of [2], we have

$$ \begin{aligned}[b] M_{11}& \leq C \bigl\Vert T_{\alpha}f_{1}(\cdot)\chi_{B}(\cdot) \bigr\Vert _{L^{r(\cdot )}(\mathbb{R}^{n})}\prod_{i=1}^{2} \bigl\Vert \bigl[b_{i}-\{b_{i}\}_{B}\bigr] \chi_{B} \bigr\Vert _{L^{u_{i}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \vert B \vert ^{\lambda+1/q(\cdot)} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot ),\mu}(\mathbb{R}^{n})} \prod_{i=1}^{2} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})}. \end{aligned} $$
(5.4)

Since

$$\begin{aligned} \bigl\vert T_{\alpha}f_{2}(x) \bigr\vert & = \biggl\vert \int_{\mathbb{R}^{n}}\frac{f_{2}(y)}{ \vert x-y \vert ^{n-\alpha }}\,dy \biggr\vert \\ & \leq\sum_{k=1}^{\infty}\int_{2^{k+1}B\setminus2^{k}B} \bigl\vert f(y) \bigr\vert \,dy \\ & \leq C\sum_{k=1}^{\infty}\bigl\vert 2^{k}B \bigr\vert ^{-1+\alpha/n} \Vert f\chi_{2^{k+1}B} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{2^{k+1}B} \Vert _{L^{p'(\cdot )}(\mathbb{R}^{n})} \\ & \leq C \vert B \vert ^{\mu+\alpha/n} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu }(\mathbb{R}^{n})}, \end{aligned} $$

using Lemma 2.6 and the generalized Hölder inequality, we obtain

$$ M_{12}\leq C \vert B \vert ^{\lambda+1/q(\cdot)} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu}(\mathbb{R}^{n})}\prod_{i=1}^{2} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})}. $$
(5.5)

(ii) For \(M_{2}(x)\), let \(\frac{1}{q_{1}(\cdot)}=\frac{1}{u_{2}(\cdot )}+\frac{1}{p(\cdot)}-\frac{\alpha}{n}\) and then \(\frac{1}{q(\cdot )}=\frac{1}{u_{1}(\cdot)}+\frac{1}{q_{1}(\cdot)}\). Using Lemma 2.5 and Lemma 2.6 we get

$$ \begin{aligned}[b] \Vert M_{2}\chi_{B} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}& \leq C \vert B \vert ^{v_{1}+\frac {1}{u_{1}(\cdot)}} \Vert b_{1} \Vert _{\mathrm{CBMO}^{u_{1}(\cdot),v_{1}}(\mathbb {R}^{n})} \bigl\Vert T_{\alpha} \bigl[b_{2}-\{b_{2}\}_{B}\bigr]f(\cdot) \chi_{B}(\cdot) \bigr\Vert _{L^{q_{1}(\cdot)}(\mathbb{R}^{n})}\hspace{-12pt} \\ & \leq C \vert B \vert ^{v_{1}+\frac{1}{u_{1}(\cdot)}} \Vert b_{1} \Vert _{\mathrm {CBMO}^{u_{1}(\cdot),v_{1}}(\mathbb{R}^{n})}(M_{21}+M_{22}), \end{aligned} $$
(5.6)

where

$$M_{21}=: \bigl\Vert T_{\alpha}\bigl[b_{2}- \{b_{2}\}_{B}\bigr]f_{1}(\cdot) \chi_{B}(\cdot) \bigr\Vert _{L^{q_{1}(\cdot)}(\mathbb{R}^{n})} $$

and

$$M_{22}=: \bigl\Vert T_{\alpha}\bigl[b_{2}- \{b_{2}\}_{B}\bigr]f_{2}(\cdot) \chi_{B}(\cdot) \bigr\Vert _{L^{q_{1}(\cdot)}(\mathbb{R}^{n})}. $$

Let \(\frac{1}{l(\cdot)}=\frac{1}{u_{2}(\cdot)}+\frac{1}{p(\cdot)}\), then \(\frac{1}{q_{1}(\cdot)}=\frac{1}{l(\cdot)}-\frac{\alpha}{n}\). Using the \((L^{l(\cdot)},L^{q_{1}(\cdot)})\)-boundedness of \(T_{\alpha}\), Lemma 2.5 and Lemma 2.6 we have

$$ \begin{aligned}[b] M_{21}& \leq C \bigl\Vert \bigl(b_{2}(\cdot)-\{b_{2}\}_{B} \bigr)f_{1} \bigr\Vert _{L^{l(\cdot)}(\mathbb {R}^{n})} \\ & \leq C \bigl\Vert \bigl(b_{2}(\cdot)-\{b_{2} \}_{B}\bigr)\chi_{2B} \bigr\Vert _{L^{u_{2}(\cdot)}(\mathbb {R}^{n})} \Vert f\chi_{2B} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \vert B \vert ^{\mu} \Vert \chi_{2B} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu}(\mathbb{R}^{n})} \bigl\Vert \bigl[b_{2}-\{b_{2}\} _{2B}+\{b_{2} \}_{2B}-\{b_{2}\}_{B}\bigr]\chi_{2B} \bigr\Vert _{L^{u_{2}(\cdot)}(\mathbb {R}^{n})}\hspace{-12pt} \\ & \leq C \vert B \vert ^{\mu} \Vert \chi_{2B} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu}(\mathbb{R}^{n})} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})} \vert 2B \vert ^{v_{2}} \Vert \chi_{2B} \Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \vert B \vert ^{\mu+v_{2}} \Vert \chi_{2B} \Vert _{L^{l(\cdot)}(\mathbb{R}^{n})} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu}(\mathbb{R}^{n})} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})}. \end{aligned} $$
(5.7)

For \(M_{22}\), by using \(\frac{1}{u'_{2}(\cdot)}=\frac{1}{l'(\cdot )}+\frac{1}{p(\cdot)}\), \(\mu+v_{2}+\frac{\alpha}{n}<0\), Lemmas 2.3, 2.5, 2.6 and the generalized Hölder inequality, we get

$$\begin{gathered} \bigl\vert T_{\alpha}\bigl(b_{2}( \cdot)-\{b_{2}\}_{B}\bigr)f_{2}(x) \bigr\vert \\ \quad \leq C \int_{(2B)^{c}}\frac{ \vert b_{2}(y)-\{b_{2}\}_{B} \vert \vert f(y) \vert }{ \vert x-y \vert ^{n-\alpha }}\,dy \\ \quad \leq C\sum_{k=1}^{\infty}\int_{2^{k+1}B\setminus2^{k}B}\frac { \vert b_{2}(y)-\{b_{2}\}_{B} \vert \vert f(y) \vert }{ \vert x-y \vert ^{n-\alpha}}\,dy \\ \quad \leq C\sum_{k=1}^{\infty}\bigl\vert 2^{k}B \bigr\vert ^{-1+\alpha/n} \bigl\Vert \bigl(b_{2}( \cdot)-\{b_{2}\} _{B}\bigr)\chi_{2^{k+1}B} \bigr\Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \Vert f\chi _{2^{k+1}B} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})} \Vert \chi_{2^{k+1}B} \Vert _{L^{l'(\cdot)}(\mathbb{R}^{n})} \\ \quad \leq C \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu}(\mathbb{R}^{n})} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})}\sum_{k=1}^{\infty}k \bigl\vert 2^{k}B \bigr\vert ^{\alpha/n+\mu+v_{2}} \\ \quad \leq C \vert B \vert ^{\alpha/n+\mu+v_{2}} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot ),\mu}(\mathbb{R}^{n})} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot ),v_{2}}(\mathbb{R}^{n})}\sum_{k=1}^{\infty}k2^{kn(\alpha/n+\mu+v_{2})} \\ \quad \leq C \vert B \vert ^{\alpha/n+\mu+v_{2}} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot ),\mu}(\mathbb{R}^{n})} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot ),v_{2}}(\mathbb{R}^{n})}. \end{gathered} $$

Thus

$$ \begin{aligned}[b] M_{22} & \leq C \vert B \vert ^{\alpha/n+\mu+v_{2}+\frac{1}{q_{1}(\cdot)}} \Vert f \Vert _{\dot {\mathcal{B}}^{p(\cdot),\mu}(\mathbb{R}^{n})} \Vert b_{2} \Vert _{\mathrm {CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})} \\ & \leq C \vert B \vert ^{\mu+v_{2}+\frac{1}{u_{2}(\cdot)}+\frac{1}{p(\cdot)}} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu}(\mathbb{R}^{n})} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})}. \end{aligned} $$
(5.8)

Combining the estimates for (5.6)–(5.8), we obtain

$$ \Vert M_{2} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C \vert B \vert ^{\lambda+1/q(\cdot)} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu }(\mathbb{R}^{n})}\prod _{i=1}^{2} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot ),v_{i}}(\mathbb{R}^{n})}. $$
(5.9)

(iii) Observing that \(M_{3}\) is symmetric to \(M_{2}\), we have

$$ \Vert M_{3} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C \vert B \vert ^{\lambda+1/q(\cdot)} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu }(\mathbb{R}^{n})}\prod _{i=1}^{2} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot ),v_{i}}(\mathbb{R}^{n})}. $$
(5.10)

(iv) Finally, we split \(M_{4}\) as follows:

$$ \begin{aligned}[b] \Vert M_{4} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} & \leq \bigl\Vert T_{\alpha}\bigl[b_{1}(\cdot)-\{b_{1} \}_{B}\bigr] \bigl[b_{2}(\cdot)-\{b_{2}\} _{B}\bigr]f_{1}(\cdot)\chi_{B}(\cdot) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ &\quad + \bigl\Vert T_{\alpha}\bigl[b_{1}(\cdot)-\{b_{1} \}_{B}\bigr] \bigl[b_{2}(\cdot)-\{b_{2} \}_{B}\bigr]f_{2}(\cdot )\chi_{B}(\cdot) \bigr\Vert _{L^{q(\cdot)}(\mathbb{R}^{n})} \\ & =:M_{41}+M_{42}. \end{aligned} $$
(5.11)

Let \(\frac{1}{t(\cdot)}=\frac{1}{u_{1}(\cdot)}+\frac{1}{u_{2}(\cdot )}+\frac{1}{p(\cdot)}\), then by the \((L^{t(\cdot)}(\mathbb{R}^{n}), L^{q(\cdot)}(\mathbb{R}^{n}))\)-boundedness of \(T_{\alpha}\) we have

$$ \begin{aligned}[b] M_{41}& \leq C \bigl\Vert \bigl(b_{1}(\cdot)-\{b_{1}\}_{B}\bigr) \bigl(b_{2}(\cdot)-\{b_{2}\}_{B} \bigr)f_{1} \bigr\Vert _{L^{t(\cdot)}(\mathbb{R}^{n})} \\ & \leq C \bigl\Vert \bigl(b_{1}(\cdot)-\{b_{1} \}_{B}\bigr)\chi_{2B} \bigr\Vert _{L^{u_{1}(\cdot)}(\mathbb {R}^{n})} \bigl\Vert \bigl(b_{2}(\cdot)-\{b_{2}\}_{B}\bigr) \chi_{2B} \bigr\Vert _{L^{u_{2}(\cdot)}(\mathbb {R}^{n})} \Vert f\chi_{2B} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})}\hspace{-12pt} \\ & \leq C \vert B \vert ^{\mu+v_{2}} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu }(\mathbb{R}^{n})} \prod_{i=1}^{2} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})}. \end{aligned} $$
(5.12)

For \(M_{42}\), using \(\mu+v_{1}+v_{2}+\alpha/n<0\), Lemmas 2.3, 2.5, 2.6 and the generalized Hölder inequality, we get

$$\begin{aligned}& \bigl\vert T_{\alpha}\bigl(b_{1}( \cdot)-\{b_{1}\}_{B}\bigr) \bigl(b_{2}(\cdot)- \{b_{2}\}_{B}\bigr)f_{2}(x) \bigr\vert \\& \quad \leq C \int_{(2B)^{c}}\frac{ \vert b_{1}(y)-\{b_{1}\}_{B} \vert \vert b_{2}(y)-\{b_{2}\} _{B} \vert \vert f(y) \vert }{ \vert x-y \vert ^{n-\alpha}}\,dy \\& \quad \leq C\sum_{k=1}^{\infty}\int_{2^{k+1}B\setminus2^{k}B}\frac { \vert b_{1}(y)-\{b_{1}\}_{B} \vert \vert b_{2}(y)-\{b_{2}\}_{B} \vert \vert f(y) \vert }{ \vert x-y \vert ^{n-\alpha}}\,dy \\& \quad \leq C\sum_{k=1}^{\infty}\bigl\vert 2^{k}B \bigr\vert ^{-1+\alpha/n} \bigl\Vert \bigl(b_{1}( \cdot)-\{b_{1}\} _{B}\bigr)\chi_{2^{k+1}B} \bigr\Vert _{L^{u_{1}(\cdot)}(\mathbb{R}^{n})} \bigl\Vert \bigl(b_{2}(\cdot )-\{b_{2} \}_{B}\bigr)\chi_{2^{k+1}B} \bigr\Vert _{L^{u_{2}(\cdot)}(\mathbb{R}^{n})} \\& \quad\quad{} \times \Vert f\chi_{2^{k+1}B} \Vert _{L^{p(\cdot)}(\mathbb{R}^{n})} \Vert \chi _{2^{k+1}B} \Vert _{L^{t'(\cdot)}(\mathbb{R}^{n})} \\& \quad \leq C \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu}(\mathbb{R}^{n})} \Vert b_{1} \Vert _{\mathrm{CBMO}^{u_{1}(\cdot),v_{1}}(\mathbb{R}^{n})} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot),v_{2}}(\mathbb{R}^{n})}\sum _{k=1}^{\infty}k^{2} \bigl\vert 2^{k}B \bigr\vert ^{\alpha/n+\mu+v_{1}+v_{2}} \\& \quad \leq C \vert B \vert ^{\alpha/n+\mu+v_{1}+v_{2}} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot ),\mu}(\mathbb{R}^{n})} \Vert b_{1} \Vert _{\mathrm{CBMO}^{u_{1}(\cdot ),v_{1}}(\mathbb{R}^{n})} \Vert b_{2} \Vert _{\mathrm{CBMO}^{u_{2}(\cdot ),v_{2}}(\mathbb{R}^{n})}\\& \qquad{}\times\sum_{k=1}^{\infty}k^{2}2^{kn(\alpha/n+\mu +v_{1}+v_{2})} \\& \quad \leq C \vert B \vert ^{\lambda} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu }(\mathbb{R}^{n})} \prod_{i=1}^{2} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot ),v_{i}}(\mathbb{R}^{n})}. \end{aligned}$$

Thus

$$ M_{42} \leq C \vert B \vert ^{\lambda+\frac{1}{q(\cdot)}} \Vert f \Vert _{\dot{\mathcal {B}}^{p(\cdot),\mu}(\mathbb{R}^{n})}\prod_{i=1}^{2} \Vert b_{i} \Vert _{\mathrm {CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})}. $$
(5.13)

In combination with the estimates of \(M_{41}\) and \(M_{42}\), we have

$$ \Vert M_{4} \Vert _{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C \vert B \vert ^{\lambda+1/q(\cdot)} \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu }(\mathbb{R}^{n})}\prod _{i=1}^{2} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot ),v_{i}}(\mathbb{R}^{n})}. $$
(5.14)

To sum up, combining the estimates of (i)–(iv),

$$\bigl\Vert [\vec{b}, T_{\alpha}]f \bigr\Vert _{\dot{\mathcal{B}}^{q(\cdot),\lambda }(\mathbb{R}^{n})}\leq C \Vert f \Vert _{\dot{\mathcal{B}}^{p(\cdot),\mu }(\mathbb{R}^{n})}\prod_{i=1}^{2} \Vert b_{i} \Vert _{\mathrm{CBMO}^{u_{i}(\cdot ),v_{i}}(\mathbb{R}^{n})}. $$

 □

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Acknowledgements

The authors are very grateful to the referees for their valuable comments.

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Funding

This work was supported by Shandong Provincial Natural Science Foundation (Grant No. ZR2017MA041), Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J18KA225) and National Natural Science Foundation of China (Grant No. 11761026).

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HW put forward the ideas of the paper, and the authors completed the paper together. All authors read and approved the final manuscript.

Correspondence to Hongbin Wang.

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Wang, H., Xu, J. Multilinear fractional integral operators on central Morrey spaces with variable exponent. J Inequal Appl 2019, 311 (2019). https://doi.org/10.1186/s13660-019-2264-7

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MSC

  • 42B35
  • 42B20
  • 46E30

Keywords

  • Multilinear fractional integral operator
  • Commutator
  • Variable exponent
  • λ-central BMO spaces
  • Central Morrey space