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Weighted norm inequality for bilinear Calderón–Zygmund operators on Herz–Morrey spaces with variable exponents

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Abstract

In this paper, we obtain a weighted norm inequality of bilinear Calderón–Zygmund operators in Herz–Morrey spaces with variable exponents and weight in the variable Muckenhoupt class.

Introduction

We denote by \(\mathcal{S}(\mathbb{R}^{n})\) the space of all Schwartz functions on \(\mathbb{R}^{n}\) and by \(\mathcal{S}'(\mathbb{R}^{n})\) the space of all tempered distributions on \(\mathbb{R}^{n}\). Let T be a bilinear operator, which is originally defined on the 2-fold of Schwartz function space \(\mathcal{S}(\mathbb{R}^{n})\), and its value belongs to \(\mathcal{S}'(\mathbb{R}^{n})\):

$$ T:\mathcal{S}\bigl(\mathbb{R}^{n}\bigr) \times \mathcal{S}\bigl( \mathbb{R}^{n}\bigr) \rightarrow \mathcal{S}'\bigl( \mathbb{R}^{n}\bigr). $$

T is called bilinear Calderón–Zygmund operator, if it extends to a bounded bilinear operator from \(L^{p_{1}} \times L^{p_{2}}\) to \(L^{p}\) with \(1/p_{1}+1/p_{2}=1/p\), and for \(f_{1}\), \(f_{2}\in L^{ \infty }_{C}(\mathbb{R}^{n})\) (the space of compactly supported bounded functions), \(x \notin \mathrm{supp}(f_{1}) \cap \mathrm{supp}(f_{2})\)

$$ T(f_{1}, f_{2}) (x):= \int _{\mathbb{R}^{2n}}K(x,y_{1},y_{2}) f_{1}(y _{1}) f_{2}(y_{2}) \,dy_{1} \,dy_{2}, $$

where the kernel K is a function in \(\mathbb{R}^{3n}\) off from the diagonal \(x=y_{1}=y_{2}\) and there exist positive constants ε, A such that

$$ \bigl\vert K(x,y_{1},y_{2}) \bigr\vert \leq \frac{A}{ ( \vert x-y_{1} \vert + \vert x-y_{2} \vert + \vert y_{1}-y_{2} \vert )^{2n} } $$

and

$$ \bigl\vert K(x,y_{1},y_{2})-K\bigl(x',y_{1},y_{2} \bigr) \bigr\vert \leq \frac{A \vert x-x' \vert ^{\varepsilon }}{ ( \vert x-y_{1} \vert + \vert x-y_{2} \vert + \vert y_{1}-y_{2} \vert )^{2n+\varepsilon } } $$

whenever \(|x-x'| \leq \frac{1}{2}\max \{|x-y_{1}|,|x-y_{2}|\}\), and the two analogous difference estimates with respect to the variables \(y_{1}\) and \(y_{2}\) hold.

Recently, Cruz-Uribe and Guzman proved the boundedness of the bilinear Calderón–Zygmund operator on products of weighted variable Lebesgue spaces in [1]. As a generalization of variable Lebesgue spaces, variable and weighted variable Herz–Morrey (Herz) spaces have been introduced in the last decades; see [2,3,4,5,6,7,8,9,10,11]. Motivated by [1], in this paper, we will prove a weighted norm inequality on products of Herz–Morrey spaces with variable exponents and weight in the variable Muckenhoupt class. We only consider the bilinear Calderón–Zygmund operator for simplicity. The analogs of our result for m-linear Calderón–Zygmund operators also hold for \(m\geq 3\), because our argument and Lemma 8 in Sect. 3 also hold for m-linear Calderón–Zygmund operators with \(m\geq 3\), see Remark 2.7 for [1, Theorem 2.4] in [1]. We mention here that the theory of multilinear Calderón–Zygmund operators started in [12]. After that, the boundedness of multilinear Calderón–Zygmund operators on products of various spaces has been obtained; see [13,14,15,16,17,18,19].

The plan of the paper is as follows. In Sect. 2, we collect some notations and state main result. The proof of the main result will be given in Sect. 3.

Notations and main result

In this section, we firstly recall some definitions and notations, then we state our results. Let Ω be a positive measurable subset of \(\mathbb{R}^{n}\), given a measurable function \(p(\cdot ):\varOmega \rightarrow [1,\infty )\), the Lebesgue space with variable exponent \(L^{p(\cdot )}(\varOmega )\) is defined by

$$ L^{p(\cdot )}(\varOmega ): = \biggl\{ f \text{ is measurable:} \int _{{\mathbb{R}^{n}}} \biggl( \frac{ \vert f(x) \vert }{\lambda } \biggr)^{p(x)} \,dx < \infty \text{ for some }\lambda >0 \biggr\} . $$

The Lebesgue space \(L^{p(\cdot )}(\varOmega )\) becomes a Banach function space equipped with the norm

$$ { \Vert f \Vert _{{L^{p(\cdot )}}}}: = \inf \biggl\{ {\lambda > 0: \int _{\varOmega } {{{ \biggl( {\frac{{ \vert f(x) \vert }}{\lambda }} \biggr)}^{p(x)}}\,dx \leq 1} } \biggr\} . $$

The space \(L^{p(\cdot )} _{\mathrm{loc}}(\mathbb{R}^{n})\) is defined by \(L^{p(\cdot )} _{\mathrm{loc}}(\mathbb{R}^{n}):=\{f: f\chi _{K} \in L ^{p(\cdot )}(\mathbb{R}^{n})\text{ for all compact subsets} K \subset \mathbb{R}^{n}\}\), where and what follows, \(\chi _{S}\) denotes the characteristic function of a measurable set \(S\subset \mathbb{R}^{n}\). Let \(p(\cdot ):\mathbb{R}^{n}\rightarrow (0,\infty )\), we denote \({p_{-} }: = \mathrm{ess}\inf_{x \in {\mathbb{R}^{n}}} p(x)\), \({p_{+} }: = \mathrm{ess} \sup_{x \in {\mathbb{R}^{n}}} p(x)\). The set \(\mathcal{P}(\mathbb{R}^{n})\) consists of all \(p(\cdot )\) satisfying \(p_{-}>1\) and \(p_{+}<\infty \); \(\mathcal{P}_{0}(\mathbb{R}^{n})\) consists of all \(p(\cdot )\) satisfying \(p_{-}>0\) and \(p_{+}<\infty \). \(L^{p(\cdot )}\) can be similarly defined as above for \(p(\cdot ) \in \mathcal{P}_{0}(\mathbb{R}^{n})\). \(p'(\cdot )\) means that the conjugate exponent of \(p(\cdot ) \), that means \(1/p(\cdot )+1/p'( \cdot )=1\).

Let \(p(\cdot ) \in \mathcal{P}(\mathbb{R}^{n})\) and w be a weight which is a non-negative measurable function on \(\mathbb{R}^{n}\). Then the weighted variable exponent Lebesgue space \(L^{p(\cdot )}( w )\) is the set of all complex-valued measurable function f such that \(fw \in L^{p(\cdot )}\). The space \(L^{p(\cdot )}(w )\) is a Banach space equipped with the norm

$$ \Vert f \Vert _{L^{p(\cdot )}(w )}:= \Vert fw \Vert _{L^{p(\cdot )}}. $$

Let \(f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\). Then the standard Hardy–Littlewood maximal function of f is defined by

$$ Mf(x):= \mathop{\sup } _{B \ni x} \frac{1}{{ \vert B \vert }} \int _{B} { \bigl\vert {f(y)} \bigr\vert \,dy},\quad \forall x \in {\mathbb{R}^{n}}, $$

where the supremum is taken over all balls containing x in \(\mathbb{R}^{n}\). In general, the Hardy–Littlewood maximal operator is not bounded on weighted variable Lebesgue spaces. But if \(p(\cdot ) \in \mathcal{P}(\mathbb{R}^{n})\) and satisfies the following global log-Hölder continuous and \(w \in A_{p(\cdot )}\), then M is bounded on \(L^{p(\cdot )}(w)\).

Definition 1

Let \(\alpha (\cdot )\) be a real-valued measurable function on \(\mathbb{R}^{n}\).

  1. (i)

    The function \(\alpha (\cdot )\) is locally log-Hölder continuous if there exists a constant \(C_{1}\) such that

    $$ \bigl\vert \alpha (x) - \alpha (y) \bigr\vert \leq \frac{C_{1}}{ {\log ( {e + 1/ \vert {x - y} \vert } )}},\quad x,y \in {\mathbb{R}^{n}}, \vert x - y \vert < \frac{1}{2}. $$
  2. (ii)

    The function \(\alpha (\cdot )\) is log-Hölder continuous at the origin if there exists a constant \(C_{2}\) such that

    $$ \bigl\vert \alpha (x) - \alpha (0) \bigr\vert \leq \frac{C_{2}}{ {\log ( {e + 1/ \vert x \vert } )}}, \quad\forall x \in {\mathbb{R}^{n}}. $$

    Denote by \(\mathcal{P}^{\log }_{0}(\mathbb{R}^{n})\) the set of all log-Hölder continuous functions at the origin.

  3. (iii)

    The function \(\alpha (\cdot )\) is log-Hölder continuous at infinity if there exist \(\alpha _{\infty }\in \mathbb{R}\) and a constant \(C_{3}\) such that

    $$ \bigl\vert \alpha (x) - \alpha _{\infty } \bigr\vert \leq \frac{C _{3}}{{\log ( {e + \vert x \vert } )}},\quad \forall x \in {\mathbb{R}^{n}}. $$

    Denote by \(\mathcal{P}^{\log }_{\infty }(\mathbb{R}^{n})\) the set of all log-Hölder continuous functions at infinity.

  4. (iv)

    The function \(\alpha (\cdot )\) is global log-Hölder continuous if \(\alpha (\cdot )\) are both locally log-Hölder continuous and log-Hölder continuous at infinity. Denote by \(\mathcal{P}^{\log }(\mathbb{R}^{n})\) the set of all global log-Hölder continuous functions.

Definition 2

Let \(p(\cdot ) \in \mathcal{P}(\mathbb{R}^{n})\), a positive measurable function w is said to be in \(A_{p(\cdot )}\), if exists a positive constant C for all balls B in \(\mathbb{R}^{n}\) such that

$$ \sup_{B} \frac{1}{ \vert B \vert } \Vert w \chi _{B} \Vert _{L^{p(\cdot )}} \bigl\Vert w ^{-1}\chi _{B} \bigr\Vert _{L^{p'(\cdot )}} < \infty. $$

Remark 1

In [20], Cruz-Uribe, Fiorenza and Neugebauer found that if \(p(\cdot ) \in \mathcal{P}(\mathbb{R}^{n})\) and \(w \in A_{p(\cdot )}\), then \(w ^{-1} \in A_{p'(\cdot )}\).

The Muckenhoupt \(A_{p}\) class with constant exponent \(p \in (1,\infty )\) firstly proposed by Muckenhoupt in [21]. The variable Muckenhoupt \(A_{p(\cdot )}\) was considered in [20, 22,23,24,25].

Lemma 1

(see [20, Theorem 1.5])

If \(p(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n})\cap \mathcal{P}( \mathbb{R}^{n})\) and \(w \in A_{p(\cdot )}\), then there is a positive constant C such that, for each \(f\in L^{p(\cdot )}(w )\),

$$ \bigl\Vert (Mf)w \bigr\Vert _{L^{p(\cdot )}} \leq C \Vert fw \Vert _{L^{p(\cdot )}}. $$

To give the definitions of the Herz space and the Herz–Morrey space with variable exponents, we use the following notations. For each \(k \in \mathbb{Z}\) we define

$$\begin{aligned} &{B_{k}}: = \bigl\{ {x \in {\mathbb{R}^{n}}: \vert x \vert \leq {2^{k}}} \bigr\} , \qquad D_{k}: = B_{k} \backslash {B_{k - 1}},\\ & {\chi _{k}}: = {\chi _{{D_{k}}}}, \qquad \widetilde{\chi }_{m}=\chi _{m},\quad m \geq 1, \widetilde{\chi }_{0}= \chi _{B_{0}}. \end{aligned}$$

Definition 3

Let \(q \in (0, \infty ]\), \(p(\cdot )\in \mathcal{P}_{0} (\mathbb{R} ^{n})\), and \(\alpha (\cdot ):\mathbb{R}^{n}\rightarrow \mathbb{R}\) with \(\alpha \in L^{\infty }(\mathbb{R}^{n}) \).

  1. (1)

    The homogeneous weighted Herz space \(\dot{K}_{p( \cdot )}^{\alpha ( \cdot ),q}(w )\) is defined by

    $$ \dot{K}_{p( \cdot )}^{\alpha ( \cdot ),q}(w ):= \bigl\{ {f \in L_{\mathrm{loc}}^{p( \cdot )}\bigl({\mathbb{R}^{n}}\backslash \{ 0 \}, w \bigr):{{ \Vert f \Vert }_{\dot{K}_{p( \cdot )}^{\alpha ( \cdot ),q}(w )}} < \infty } \bigr\} , $$

    where

    $$ \Vert f \Vert _{\dot{K}_{p( \cdot )}^{\alpha ( \cdot ),q}(w )}:= { \Biggl\{ {\sum _{k = - \infty }^{\infty }{ \bigl\Vert {{2^{k \alpha ( \cdot )}}f{\chi _{k}}} \bigr\Vert _{L^{p( \cdot )}(w )} ^{q}} } \Biggr\} ^{1/q}}. $$
  2. (2)

    The inhomogeneous weighted Herz space \(K_{p( \cdot )}^{\alpha ( \cdot ),q}(w )\) is defined by

    $$ K_{p( \cdot )}^{\alpha ( \cdot ),q}(w ):= \bigl\{ {f \in L_{ \mathrm{loc}}^{p( \cdot )}( w ):{{ \Vert f \Vert }_{K_{p( \cdot )}^{\alpha ( \cdot ),q}(w )}} < \infty } \bigr\} , $$

    where

    $$ { \Vert f \Vert _{K_{p( \cdot )}^{\alpha ( \cdot ),q}(w )}} := { \Biggl\{ {\sum _{m = 0}^{\infty }{ \bigl\Vert {{2^{m\alpha ( \cdot )}}f{ \widetilde{\chi } _{m}}} \bigr\Vert _{L^{p( \cdot )}(w )}^{q}} } \Biggr\} ^{1/q}}. $$

Remark 2

If \(0 < q_{1} \leq q_{2} \leq \infty \) and \(w \equiv 1\), then \(\dot{K}^{\alpha (\cdot ),q_{1}}_{p(\cdot )}(\mathbb{R}^{n}) \subset \dot{K}^{\alpha (\cdot ),q_{2}}_{p(\cdot )}(\mathbb{R}^{n})\). If \(w \equiv 1\), \(\alpha (\cdot )\) and \(p(\cdot )\) are constants, then \(\dot{K}^{\alpha (\cdot ),q}_{p(\cdot )}(\mathbb{R}^{n})= \dot{K}^{ \alpha,q}_{p}(\mathbb{R}^{n})\) is the classical Herz spaces in [26, 27].

To generalize the above spaces to variable exponent \(q(\cdot )\), we need the notation of the variable mixed sequence space \(\ell ^{q(\cdot )}(L ^{p(\cdot )})\), which is firstly defined by Almeida and Hästö in [28]. Let w be a non-negative measurable function. Given a sequence of functions \(\{f_{j}\}_{j\in \mathbb{Z}}\), define the modular

$$ \rho _{\ell ^{q(\cdot )}(L^{p(\cdot )}(w ))}\bigl((f_{j})_{j}\bigr):= \sum _{j\in \mathbb{Z}}\inf \biggl\{ \lambda _{j}: \int _{\mathbb{R}^{n}} \biggl(\frac{ \vert f_{j}(x)w(x) \vert }{\lambda ^{\frac{1}{q(x)}}_{j}} \biggr)^{p(x)} \,dx \leq 1 \biggr\} , $$

where \(\lambda ^{1/\infty }=1\). If \(q^{+}<\infty \) or \(q(\cdot )\le p( \cdot )\), the above can be written as

$$ \rho _{\ell ^{q(\cdot )}(L^{p(\cdot )}(w ))}\bigl((f_{j})_{j}\bigr)= \sum _{j\in \mathbb{Z}}\bigl\| |f_{j}w|^{q( \cdot )} \bigr\| _{L^{\frac{p(\cdot )}{q(\cdot )}}} . $$

The norm is

$$ \bigl\Vert (f_{j})_{j} \bigr\Vert _{\ell ^{q(\cdot )}(L^{p(\cdot )}(w ))}:= \inf \bigl\{ \mu >0: \rho _{\ell ^{q(\cdot )}(L^{p(\cdot )}(w ))}\bigl((f_{j}/ \mu )_{j}\bigr)\le 1\bigr\} . $$

Now, spaces \(\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p(\cdot )}(w )\) and \(K^{\alpha (\cdot ),q(\cdot )}_{p(\cdot )}(w )\) are defined, respectively, by

$$ \dot{K}^{\alpha (\cdot ),q(\cdot )}_{p(\cdot )}(w ):= \bigl\{ f \in L^{p(\cdot )}_{\mathrm{loc}}\bigl(\mathbb{R}^{n}\backslash \{0\}, w \bigr): \Vert f \Vert _{\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p(\cdot )}( w )}< \infty \bigr\} , $$

where

$$ \Vert f \Vert _{\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p(\cdot )}(w )}:= \bigl\Vert \bigl(2^{j \alpha (\cdot )} f\chi _{j}\bigr)_{j} \bigr\Vert _{\ell ^{q(\cdot )}(L^{p(\cdot )}(w))} $$

and

$$ K^{\alpha (\cdot ),q(\cdot )}_{p(\cdot )}(w ):= \bigl\{ f\in L^{p( \cdot )}_{\mathrm{loc}}( w ): \Vert f \Vert _{K^{\alpha (\cdot ),q(\cdot )}_{p(\cdot )}(w )} = \bigl\Vert \bigl(2^{j\alpha (\cdot )} f\chi _{j}\bigr)_{j} \bigr\Vert _{\ell ^{q(\cdot )}(L^{p(\cdot )}(w))}< \infty \bigr\} . $$

For any quantities A and B, we shall write \(A \lesssim B\) to indicate that there exists a constant \(C>0\) such that \(A\leq CB\). If \(A\lesssim B\) and \(B \lesssim A\), we write \(A\approx B\).

The following lemma is a corollary of [29, Theorem 3].

Lemma 2

Let \(\alpha (\cdot )\in L^{\infty }(\mathbb{R}^{n})\), \(p(\cdot )\), \(q(\cdot )\in \mathcal{P}_{0}(\mathbb{R}^{n})\) and w be a weight. If \(\alpha (\cdot )\) and \(q(\cdot )\) are log-Hölder continuous at infinity, then

$$ K^{\alpha (\cdot ),q(\cdot )}_{p(\cdot )}(w )=K^{\alpha _{\infty },q_{\infty }}_{p(\cdot )}(w ). $$

Additionally, if \(\alpha (\cdot )\) and \(q(\cdot )\) are log-Hölder continuous at the origin, then

$$\begin{aligned} \Vert f \Vert _{\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p(\cdot )}(w )} \thickapprox{}& \biggl(\sum _{k\leqslant 0} \bigl\Vert 2^{k\alpha (0)}f\chi _{k} \bigr\Vert _{L^{p(\cdot )}(w )}^{q(0)} \biggr)^{1/q(0)} \\ &{} + \biggl(\sum_{k>0} \bigl\Vert 2^{k\alpha _{\infty }}f\chi _{k} \bigr\Vert _{L^{p(\cdot )}(w )}^{q_{\infty }} \biggr)^{1/q_{\infty }}. \end{aligned}$$

Definition 4

Let \(p(\cdot )\), \(q(\cdot )\in \mathcal{P}_{0}(\mathbb{R}^{n})\), \(\lambda \in [0, \infty )\). Let \(\alpha (\cdot )\) be a bounded real-valued measurable function on \(\mathbb{R}^{n}\). The homogeneous weighted Herz–Morrey space \(M\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p( \cdot ),\lambda }(w )\) and non-homogeneous weighted Herz–Morrey space \(MK^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ),\lambda }(w )\) are defined, respectively, by

$$ M\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ),\lambda }(w ):= \bigl\{ f\in L^{p(\cdot )}_{\mathrm{loc}}\bigl(\mathbb{R}^{n}\setminus \{0 \}, w \bigr): \Vert f \Vert _{M\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ), \lambda }(w )}< \infty \bigr\} $$

and

$$ MK^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ),\lambda }(w ):= \bigl\{ f\in L^{p(\cdot )}_{\mathrm{loc}} \bigl(\mathbb{R}^{n},w \bigr): \Vert f \Vert _{MK ^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ),\lambda }(w )}< \infty \bigr\} , $$

where

$$ \Vert f \Vert _{M\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ),\lambda }( w )}:= \sup_{L\in \mathbb{Z}}2^{-L\lambda } \bigl\Vert \bigl(2^{\alpha ( \cdot ) k}f\chi _{k}\bigr)_{k \leq L} \bigr\Vert _{ \ell ^{q(\cdot )}({L^{p(\cdot )}(w )})} $$

and

$$ \Vert f \Vert _{MK^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ),\lambda }(w )}:= \sup_{L\in \mathbb{N}_{0}}2^{-L\lambda } \bigl\Vert \bigl(2^{\alpha (\cdot ) k}f \tilde{\chi }_{k} \bigr)_{k=0}^{L} \bigr\Vert _{\ell ^{q(\cdot )}({L^{p(\cdot )}(w )})}. $$

Proposition 1

Let \(p(\cdot )\), \(q(\cdot )\in \mathcal{P}_{0}(\mathbb{R}^{n})\), w be a weight, \(\lambda \in [0,\infty )\), and \(\alpha (\cdot )\in L^{\infty }(\mathbb{R}^{n})\).

  1. (i)

    If \(\alpha (\cdot )\), \(q(\cdot )\in \mathcal{P}^{\log } _{0}(\mathbb{R}^{n})\cap \mathcal{P}^{\log }_{\infty }(\mathbb{R}^{n})\), then, for any \(f\in L^{p(\cdot )}_{\mathrm{loc}}(\mathbb{R}^{n}\backslash \{0\},w )\),

    $$\begin{aligned} & \Vert f \Vert _{M\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ),\lambda }( w )}\\ &\quad\approx\max \Bigl\{ \sup _{L\leqslant 0,L\in \mathbb{Z}} 2^{-L \lambda } \bigl\Vert \bigl(2^{k\alpha (0)}f \chi _{k}\bigr)_{k \leq L} \bigr\Vert _{\ell ^{q_{0}}({L^{p(\cdot )}( w )})}, \\ & \qquad\sup_{L>0,L\in \mathbb{Z}} \bigl[2^{-L\lambda } \bigl\Vert \bigl(2^{k\alpha (0)}f\chi _{k}\bigr)_{k< 0} \bigr\Vert _{\ell ^{q_{0}}({L^{p(\cdot )}(w )})}+2^{-L\lambda } \bigl\Vert \bigl(2^{k\alpha _{\infty }}f\chi _{k}\bigr)_{k= 0}^{L} \bigr\Vert _{\ell ^{q_{ \infty }}({L^{p(\cdot )}(w )})} \bigr] \Bigr\} , \end{aligned}$$

    where throughout \(q_{0}:=q(0)\).

  2. (ii)

    If \(\alpha (\cdot )\), \(q(\cdot )\in \mathcal{P}^{\log } _{\infty }(\mathbb{R}^{n})\), then

    $$ MK^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ),\lambda }(w )=MK^{ \alpha _{\infty },q_{\infty }}_{p(\cdot ),\lambda }( w ). $$

Proof

Obviously,

$$\begin{aligned} \Vert f \Vert _{M\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ),\lambda }( w )}={}&\max \Bigl\{ \sup_{L\leqslant 0,L\in \mathbb{Z}} 2^{-L\lambda } \bigl\Vert \bigl(2^{k\alpha (\cdot )}f\chi _{k} \bigr)_{k \leq L} \bigr\Vert _{\ell ^{q(\cdot )}({L^{p( \cdot )}(w )})}, \\ & \sup_{L>0,L\in \mathbb{Z}}2^{-L\lambda } \bigl\Vert \bigl(2^{k\alpha ( \cdot )}f\chi _{k}\bigr)_{k \leq L} \bigr\Vert _{ \ell ^{q(\cdot )}({L^{p(\cdot )(w )}})} \Bigr\} . \end{aligned}$$

When \(L \leq 0\), from Lemma 2 we know that

$$ \bigl\Vert \bigl(2^{k\alpha (\cdot )}f\chi _{k}\bigr)_{k \leq L} \bigr\Vert _{\ell ^{q(\cdot )}({L^{p(\cdot )}(w )})}\thickapprox \bigl\Vert \bigl(2^{k \alpha (0)}f\chi _{k}\bigr)_{k \leq L} \bigr\Vert _{\ell ^{q _{0}}({L^{p(\cdot )}(w )})}. $$

When \(L>0\), from Lemma 2 again we also obtain

$$\begin{aligned} \bigl\Vert \bigl(2^{k\alpha (\cdot )}f\chi _{k}\bigr)_{k< L} \bigr\Vert _{\ell ^{q(\cdot )}({L^{p( \cdot )}(w )})}\thickapprox{}& \bigl\Vert \bigl(2^{k\alpha (0)}f \chi _{k}\bigr)_{k< 0} \bigr\Vert _{\ell ^{q_{0}}({L^{p(\cdot )}(w )})} \\ &{} + \bigl\Vert \bigl(2^{k\alpha _{\infty }}f\chi _{k} \bigr)_{k= 0}^{L} \bigr\Vert _{ \ell ^{q_{\infty }}({L^{p(\cdot )}(w )})}. \end{aligned}$$

Thus we obtain (i). Similarly, we obtain (ii). □

Lemmas 3 and 4 below have been proved by Izuki and Noi in [30, 31].

Lemma 3

If \(p(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n}) \cap \mathcal{P}( \mathbb{R}^{n})\) and \(w \in A_{p(\cdot )}\), then there exists a constant \(C>0\) such that, for all balls B in \(\mathbb{R}^{n}\) and all measurable subsets \(S\subset B\),

$$\begin{aligned} \frac{{{{ \Vert {{\chi _{S}}} \Vert }_{L^{p( \cdot )}(w )}}}}{ {{{ \Vert {{\chi _{B}}} \Vert }_{L^{p( \cdot )}(w )}}}} \leq C {\frac{{ \vert S \vert }}{{ \vert B \vert }}}. \end{aligned}$$
(1)

Lemma 4

If \(p(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n})\cap \mathcal{P}( \mathbb{R}^{n})\) and \(w \in A_{p(\cdot )}\), then there exist constants \(\delta _{1}\), \(\delta _{2}\in (0,1)\) and \(C>0\) such that, for all balls B in \(\mathbb{R}^{n}\) and all measurable subsets \(S\subset B\),

$$\begin{aligned} & \frac{{{{ \Vert {{\chi _{S}}} \Vert }_{L^{p( \cdot )}(w )}}}}{ {{{ \Vert {{\chi _{B}}} \Vert }_{L^{p( \cdot )}(w )}}}} \leq C{ \biggl( {\frac{{ \vert S \vert }}{{ \vert B \vert }}} \biggr) ^{{\delta _{1}}}}, \end{aligned}$$
(2)
$$\begin{aligned} &\frac{{{{ \Vert {{\chi _{S}}} \Vert }_{L^{p'( \cdot )}(w ^{-1})}}}}{{{{ \Vert {{\chi _{B}}} \Vert }_{L^{p'( \cdot )}( w ^{-1})}}}} \leq C{ \biggl( {\frac{{ \vert S \vert }}{{ \vert B \vert }}} \biggr)^{{\delta _{2}}}}. \end{aligned}$$
(3)

Lemma 5

(see [30, Lemma 4])

If \(p(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n})\cap \mathcal{P}( \mathbb{R}^{n})\) and \(w \in A_{p(\cdot )}\), then there exists a positive constant C such that, for all balls B in \(\mathbb{R}^{n}\),

$$ {C^{ - 1}}\leq \frac{1}{{ \vert B \vert }}{ \Vert {{\chi _{B}}} \Vert _{L ^{p( \cdot )}(w )}} { \Vert {{\chi _{B}}} \Vert _{L^{p'( \cdot )}(w ^{-1})}} \leq C. $$

Our main result is as follows.

Theorem 1

Assume that T is a bilinear Calderón–Zygmund operator, \(p_{1}(\cdot )\), \(p_{2}(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n}) \cap \mathcal{P}(\mathbb{R}^{n})\) satisfying \(1/p(x)=1/{p_{1}(x)}+1/ {p_{2}(x)}\) for \(x \in \mathbb{R}^{n}\). Let \(w_{1}\), \(w_{2}\) be weights, \(w=w_{1} w_{2}\), \(w_{i} \in A_{p_{i}(\cdot )}\), \(i=1, 2\). Suppose that \(\alpha (\cdot ) \in L^{\infty }(\mathbb{R}^{n})\cap \mathcal{P}^{ \log }_{0}(\mathbb{R}^{n})\cap \mathcal{P}^{\log }_{\infty }( \mathbb{R}^{n})\), \(\alpha (0)=\alpha _{1}(0)+\alpha _{2}(0)\), \(\alpha _{\infty }=\alpha _{1\infty }+\alpha _{2\infty }\), \(q(\cdot ) \in \mathcal{P}^{\log }_{0}(\mathbb{R}^{n})\cap \mathcal{P}^{\log } _{\infty }(\mathbb{R}^{n})\), \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(1/q_{\infty }=1/{q_{1\infty }}+1/{q_{2\infty }}\), \(\lambda =\lambda _{1}+\lambda _{2}\), \(0 \leq \lambda _{i}< \infty \), \(\delta _{i1}\), \(\delta _{i2}\in (0,1)\) are the constants in Lemma 4 for exponents \(p_{i}(\cdot )\) and weights \(w_{i}\), \(i=1, 2\). If \(\lambda _{i}+n \delta _{i2}>\alpha _{i\infty }\geq \alpha _{i}(0)\) for \(i=1, 2\), then

$$ \bigl\Vert T(f_{1},f_{2}) \bigr\Vert _{M\dot{K}_{p(\cdot ),\lambda }^{\alpha (\cdot ),q( \cdot )}(w )} \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ), \lambda _{1}}^{\alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})} \Vert f _{2} \Vert _{M\dot{K}_{p_{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q _{2}( \cdot )}(w _{2})}. $$

From Theorem 1, we obtain the following corollary.

Corollary 1

Assume that T is a bilinear Calderón–Zygmund operator, \(p_{1}(\cdot )\), \(p_{2}(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n}) \cap \mathcal{P}(\mathbb{R}^{n})\) satisfying \(1/p(x)=1/{p_{1}(x)}+1/ {p_{2}(x)}\) for \(x \in \mathbb{R}^{n}\). Let \(w_{1}\), \(w_{2}\) be weights, \(w=w_{1} w_{2}\), \(w_{i} \in A_{p_{i}(\cdot )}\), \(i=1, 2\). Suppose that \(\alpha (\cdot ) \in L^{\infty }(\mathbb{R}^{n})\cap \mathcal{P}^{ \log }_{0}(\mathbb{R}^{n})\cap \mathcal{P}^{\log }_{\infty }( \mathbb{R}^{n})\), \(\alpha (0)=\alpha _{1}(0)+\alpha _{2}(0)\), \(\alpha _{\infty }=\alpha _{1\infty }+\alpha _{2\infty }\), \(q(\cdot ) \in \mathcal{P}^{\log }_{0}(\mathbb{R}^{n})\cap \mathcal{P}^{\log } _{\infty }(\mathbb{R}^{n})\), \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(1/q_{\infty }=1/{q_{1\infty }}+1/{q_{2\infty }}\), \(\delta _{i1}\), \(\delta _{i2}\in (0,1)\) are the constants in Lemma 4 for exponents \(p_{i}(\cdot )\) and weights \(w_{i}\), \(i=1, 2\). If \(\lambda _{i}+n \delta _{i2}>\alpha _{i\infty }\geq \alpha _{i}(0)\) for \(i=1, 2\), then

$$ \bigl\Vert T(f_{1},f_{2}) \bigr\Vert _{\dot{K}_{p(\cdot )}^{\alpha (\cdot ),q(\cdot )}( w )} \lesssim \Vert f_{1} \Vert _{\dot{K}_{p_{1}( \cdot )}^{\alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{\dot{K}_{p_{2}( \cdot )}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}(w _{2})}. $$

Proof of Theorem 1

To prove Theorem 1, we need a series of lemmas.

Lemma 6

(see [16, Theorem 2.3])

Let \(p(\cdot )\), \(p_{1}(\cdot )\), \(p_{2}(\cdot )\in \mathcal{P}_{0}( \mathbb{R}^{n})\) such that \(1/p(x)=1/p_{1}(x)+1/p_{2}(x)\) for \(x \in \mathbb{R}^{n}\). Then there exists a constant \(C_{p,p_{1}}\) independent of functions f and g such that

$$ { \Vert {fg} \Vert _{{L^{p( \cdot )}}}}\leq {C_{p,{p_{1}}}} { \Vert f \Vert _{{L^{{p_{1}}( \cdot )}}}} { \Vert g \Vert _{{L^{{p_{2}}( \cdot )}}}} $$

holds for every \(f\in L^{p_{1}(\cdot )}(\mathbb{R}^{n})\) and \(g\in L^{p_{2}(\cdot )}(\mathbb{R}^{n})\). If \(p \in \mathcal{P}( \mathbb{R}^{n})\), w be weight with \(w =w _{1} \times w _{2}\), then

$$ \Vert {fg} \Vert _{L^{p( \cdot )}(w)}\leq {C_{p, {p_{1}}}} { \Vert f \Vert _{{L^{{p_{1}}( \cdot )}(w _{1})}}} { \Vert g \Vert _{{L^{{p_{2}}( \cdot )}(w _{2})}}}. $$

Lemma 7

(see [32, Proposition 1.2])

Let \(0< p\leq \infty \), \(\delta >0\). Then there is a positive constant C such that

$$ \Biggl( {\sum_{j = -\infty }^{\infty }{{{ \Biggl( {\sum_{k = -\infty }^{\infty }{{2^{ - |k - j|\delta }} {a_{k}}} } \Biggr)} ^{p}}} } \Biggr)^{1/p} \leq C \Biggl( {\sum_{j = -\infty } ^{\infty }{a_{j}^{p}} } \Biggr)^{1/p} $$
(4)

for non-negative sequences \(\{a_{j}\}^{\infty }_{j= -\infty }\). Here, when \(p=\infty \), it is understood that (4) stands for

$$ \mathop{\sup } _{j \in \mathbb{Z}} \Biggl( {\sum_{k = -\infty }^{\infty }{{2^{ - |k - j|\delta }} {a_{k}}} } \Biggr) \leq C \mathop{\sup } _{j \in \mathbb{Z}} {a_{j}}. $$

The following lemma is a corollary of [1, Theorem 2.8].

Lemma 8

Let \(p_{1}(\cdot )\), \(p_{1}(\cdot ) \in \mathcal{P}(\mathbb{R}^{n})\), \(1<(p_{i})_{-}\leq (p_{i})_{+}<\infty \) and \(p_{i}(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n})\) satisfying \(1/p(x)=1/{p_{1}(x)}+1/{p_{2}(x)}\) for \(x \in \mathbb{R} ^{n}\), \(i=1, 2\). Let \(w_{1} \in A_{p_{1}(\cdot )}, w_{2} \in A_{p _{2}(\cdot )}\) and \(w=w_{1}w_{2}\). If T is a bilinear Calderón–Zygmund operator, then

$$ \bigl\Vert T(f_{1},f_{2}) \bigr\Vert _{L^{p(\cdot )}(w )} \lesssim \Vert f_{1} \Vert _{L^{p _{1}(\cdot )}(w _{1})} \Vert f_{2} \Vert _{L^{p_{2}(\cdot )}(w _{2})}. $$

Proof of Theorem 1

Let \(f_{1}\) and \(f_{2}\) be bounded functions with compact support and write

$$ f_{i}=\sum^{\infty }_{l=-\infty }f_{i} \chi _{l}=:\sum^{\infty }_{l=- \infty }f_{i l},\quad i=1,2. $$

By Proposition 1, we have

$$\begin{aligned} & \bigl\Vert T(f_{1}, f_{2}) \bigr\Vert _{M\dot{K}_{p(\cdot ),\lambda }^{\alpha ( \cdot ),q(\cdot )}(w )} \\ &\quad \approx \max \Bigl\{ \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L \lambda } \bigl\Vert \bigl(2^{k\alpha (0)} T(f_{1}, f_{2})\chi _{k}\bigr)_{k\leq L} \bigr\Vert _{l^{q_{0}}(L^{p(\cdot )}(w ))}, \\ & \qquad\sup_{L > 0,L \in \mathbb{Z}} \bigl[2^{ - L\lambda } \bigl\Vert \bigl(2^{k\alpha (0)} T(f_{1}, f_{2})\chi _{k} \bigr)_{k< 0} \bigr\Vert _{l^{q_{0}}(L^{p(\cdot )}(w ))} \\ &\qquad{} +2^{ - L\lambda } \bigl\Vert \bigl(2^{k\alpha _{\infty }} T(f_{1}, f_{2})\chi _{k}\bigr)_{k=0} ^{L} \bigr\Vert _{l^{q_{\infty }}(L^{p(\cdot )}(w ))} \bigr] \Bigr\} \\ &\quad:=\max \{E,F+G\}, \end{aligned}$$

where

$$\begin{aligned} &E: = \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \bigl\Vert \bigl(2^{k\alpha (0)} T(f_{1},f_{2})\chi _{k} \bigr)_{k\leq L} \bigr\Vert _{l^{q_{0}}(L^{p(\cdot )}(w ))}, \\ &F:= \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \bigl\Vert \bigl(2^{k\alpha (0)} T(f_{1},f_{2})\chi _{k} \bigr)_{k< 0} \bigr\Vert _{l^{q_{0}}(L^{p(\cdot )}(w ))}, \\ &G:= \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \bigl\Vert \bigl(2^{k\alpha _{ \infty }} T(f_{1},f_{2})\chi _{k} \bigr)_{k=0}^{L} \bigr\Vert _{l^{q_{\infty }}(L^{p( \cdot )}(w ))}. \end{aligned}$$

Since to estimate F is essentially similar to estimate E, so it suffices for us to show that E and G are bounded in weighted Herz–Morrey space with variable exponents. It is easy to see that

$$ E \leq C \sum^{9}_{i=i} E_{i},\qquad G \leq C \sum^{9}_{i=i}G_{i}, $$

where

$$\begin{aligned} &E_{1}:= \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L}_{k=-\infty } 2^{k\alpha (0) q(0)} \Biggl\Vert \sum^{k-2}_{l=-\infty } \sum ^{k-2}_{j=-\infty } T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q(0)} _{L^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q(0)}}, \\ &E_{2}:= \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L}_{k=-\infty } 2^{k\alpha (0) q(0)} \Biggl\Vert \sum^{k-2}_{l=-\infty } \sum ^{k+1}_{j=k-1} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q(0)}_{L^{p( \cdot )}(w)} \Biggr)^{\frac{1}{q(0)}}, \\ &E_{3}:= \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L}_{k=-\infty } 2^{k\alpha (0) q(0)} \Biggl\Vert \sum^{k-2}_{l=-\infty } \sum ^{\infty }_{j=k+2} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q(0)} _{L^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q(0)}}, \\ &E_{4}:= \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L}_{k=-\infty } 2^{k\alpha (0) q(0)} \Biggl\Vert \sum^{k+1}_{l=k-1} \sum ^{k-2}_{j=-\infty } T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q(0)}_{L ^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q(0)}}, \\ &E_{5}:= \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L}_{k=-\infty } 2^{k\alpha (0) q(0)} \Biggl\Vert \sum^{k+1}_{l=k-1} \sum ^{k+1}_{j=k-1} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q(0)}_{L^{p( \cdot )}(w)} \Biggr)^{\frac{1}{q(0)}}, \\ &E_{6}:= \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L}_{k=-\infty } 2^{k\alpha (0) q(0)} \Biggl\Vert \sum^{k+1}_{l=k-1} \sum ^{\infty }_{j=k+2} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q(0)}_{L ^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q(0)}}, \\ &E_{7}:= \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L}_{k=-\infty } 2^{k\alpha (0) q(0)} \Biggl\Vert \sum^{\infty }_{l=k+2} \sum ^{k-2}_{j=-\infty } T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q(0)}_{L ^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q(0)}}, \\ &E_{8}:= \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L}_{k=-\infty } 2^{k\alpha (0) q(0)} \Biggl\Vert \sum^{\infty }_{l=k+2} \sum ^{k+1}_{j=k-1} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q(0)}_{L^{p( \cdot )}(w)} \Biggr)^{\frac{1}{q(0)}}, \\ &E_{9}:= \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L}_{k=-\infty } 2^{k\alpha (0) q(0)} \Biggl\Vert \sum^{\infty }_{l=k+2} \sum ^{\infty }_{j=k+2} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q(0)}_{L ^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q(0)}}, \\ &G_{1}:= \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L} _{k=0} 2^{k\alpha _{\infty }q_{\infty}} \Biggl\Vert \sum^{k-2}_{l=-\infty } \sum ^{k-2}_{j=-\infty } T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q_{\infty }}_{L^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q_{\infty }}}, \\ &G_{2}:= \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L} _{k=0} 2^{k\alpha _{\infty }q_{\infty}} \Biggl\Vert \sum^{k-2}_{l=-\infty } \sum ^{k+1}_{j=k-1} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q_{\infty }} _{L^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q_{\infty }}}, \\ &G_{3}:= \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L} _{k=0} 2^{k\alpha _{\infty }q_{\infty}} \Biggl\Vert \sum^{k-2}_{l=-\infty } \sum ^{\infty }_{j=k+2} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q_{\infty }}_{L^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q_{\infty }}}, \\ &G_{4}:= \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L} _{k=0} 2^{k\alpha _{\infty }q_{\infty}} \Biggl\Vert \sum^{k+1}_{l=k-1} \sum ^{k-2} _{j=-\infty } T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q_{\infty }}_{L^{p( \cdot )}(w)} \Biggr)^{\frac{1}{q_{\infty }}}, \\ &G_{5}:= \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L} _{k=0} 2^{k\alpha _{\infty }q_{\infty}} \Biggl\Vert \sum^{k+1}_{l=k-1} \sum ^{k+1} _{j=k-1} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q_{\infty }}_{L^{p( \cdot )}(w)} \Biggr)^{\frac{1}{q_{\infty }}}, \\ &G_{6}:= \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L} _{k=0} 2^{k\alpha _{\infty }q_{\infty}} \Biggl\Vert \sum^{k+1}_{l=k-1} \sum ^{ \infty }_{j=k+2} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q_{\infty }}_{L ^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q_{\infty }}}, \\ &G_{7}:= \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L} _{k=0} 2^{k\alpha _{\infty }q_{\infty}} \Biggl\Vert \sum^{\infty }_{l=k+2} \sum ^{k-2}_{j=-\infty } T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q_{\infty }} _{L^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q_{\infty }}}, \\ &G_{8}:= \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L} _{k=0} 2^{k\alpha _{\infty }q_{\infty}} \Biggl\Vert \sum^{\infty }_{l=k+2} \sum ^{k+1}_{j=k-1} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q_{\infty }}_{L ^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q_{\infty }}}, \\ &G_{9}:= \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L} _{k=0} 2^{k\alpha _{\infty }q_{\infty}} \Biggl\Vert \sum^{\infty }_{l=k+2} \sum ^{\infty }_{j=k+2} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q_{\infty }}_{L^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q_{\infty }}}. \end{aligned}$$

We shall use the following estimates. If \(l\leq k-1\), then, by Hölder’s inequality and Lemmas 4 and 5, we have

$$\begin{aligned} & \biggl\Vert 2^{-kn} \int _{\mathbb{R}^{n}} f_{il}\,dy_{i} \chi _{k} \biggr\Vert _{L^{p_{i}(\cdot )}(w_{i})} \\ &\quad \leq C2^{-kn} \Vert \chi _{B_{k}} \Vert _{L^{p_{i}(\cdot )}(w_{i})} \Vert f_{il} w _{i} \chi _{l} \Vert _{L^{p_{i}(\cdot )}} \bigl\Vert \chi _{l}w_{i}^{-1} \bigr\Vert _{L^{p'_{i}( \cdot )}} \\ &\quad \leq C2^{-kn} \vert B_{k} \vert \Vert \chi _{B_{k}} \Vert ^{-1}_{L^{p'_{i}(\cdot )}(w^{-1} _{i})} \Vert \chi _{B_{l}} \Vert _{L^{p'_{i}(\cdot )}(w^{-1}_{i})} \Vert f_{il} \chi _{l} \Vert _{L^{p_{i}(\cdot )}(w_{i})} \\ & \quad\leq C2^{(l-k)n\delta _{2i}} \Vert f_{il} \chi _{l} \Vert _{L^{p_{i}(\cdot )}(w _{i})}. \end{aligned}$$
(5)

If \(l=k\), then

$$\begin{aligned} & \biggl\Vert 2^{-kn} \int _{\mathbb{R}^{n}} f_{il}\,dy_{i} \chi _{k} \biggr\Vert _{L^{p_{i}(\cdot )}(w_{i})} \\ &\quad \leq C2^{-kn} \Vert \chi _{B_{k}} \Vert _{L^{p_{i}(\cdot )}(w_{i})} \Vert f_{il} w _{i} \chi _{l} \Vert _{L^{p_{i}(\cdot )}} \bigl\Vert \chi _{l}w_{i}^{-1} \bigr\Vert _{L^{p'_{i}( \cdot )}} \\ & \quad\leq C2^{-kn} \Vert \chi _{B_{k}} \Vert _{L^{p_{i}(\cdot )}(w_{i})} \Vert \chi _{B _{l}} \Vert _{L^{p'_{i}(\cdot )}(w^{-1}_{i})} \Vert f_{il} \chi _{l} \Vert _{L^{p _{i}(\cdot )}(w_{i})} \\ & \quad\leq \Vert f_{il} \chi _{l} \Vert _{L^{p_{i}(\cdot )}(w_{i})}. \end{aligned}$$
(6)

If \(l\geq k+1\), then

$$\begin{aligned} & \biggl\Vert 2^{-kn} \int _{\mathbb{R}^{n}} f_{il}\,dy_{i} \chi _{k} \biggr\Vert _{L^{p_{i}(\cdot )}(w_{i})} \\ &\quad \leq C2^{-kn} \Vert \chi _{B_{k}} \Vert _{L^{p_{i}(\cdot )}(w_{i})} \Vert f_{il} w _{i} \chi _{l} \Vert _{L^{p_{i}(\cdot )}} \bigl\Vert \chi _{l}w_{i}^{-1} \bigr\Vert _{L^{p'_{i}( \cdot )}} \\ & \quad\leq C2^{-kn} \Vert \chi _{B_{k}} \Vert _{L^{p_{i}(\cdot )}(w_{i})} \Vert \chi _{B _{l}} \Vert _{L^{p_{i}(\cdot )}(w_{i})} \Vert \chi _{B_{l}} \Vert ^{-1}_{L^{p_{i}( \cdot )}(w_{i})} \\ &\qquad{} \times \Vert \chi _{B_{l}} \Vert _{L^{p'_{i}(\cdot )}(w^{-1}_{i})} \Vert f_{il} \chi _{l} \Vert _{L^{p_{i}(\cdot )}(w_{i})} \\ &\quad \leq C2^{(l-k)n(1-\delta _{1i})} \Vert f_{il} \chi _{l} \Vert _{L^{p_{i}(\cdot )}(w _{i})}. \end{aligned}$$
(7)

By the symmetry of \(f_{1}\) and \(f_{2}\), it is only necessary to estimate \(E_{1}\), \(E_{2}\), \(E_{3}\), \(E_{5}\), \(E_{6}\), and \(E_{9}\).

To estimate \(E_{1}\), since l, \(j \leq k-2\), we deduce that, for \(i=1,2\),

$$ \vert x-y_{i} \vert \geq \vert x \vert - \vert y_{i} \vert >2^{k-1}-2^{ \min \{l,j\} } \geq 2^{k-2},\quad x\in D_{k}, y_{1} \in D_{l}, y_{2} \in D_{j}. $$

Therefore, for \(x\in D_{k}\), we have

$$ \bigl\vert K(x,y_{1},y_{2}) \bigr\vert \leq C\bigl( \vert x-y_{1} \vert + \vert x-y_{2} \vert \bigr)^{-2n} \leq C2^{-2kn}. $$

Thus, \(\forall x \in D_{k}\) and l, \(j \leq k-2\), we have

$$\begin{aligned} \bigl\vert T(f_{1l},f_{2j}) (x) \bigr\vert &\lesssim \int _{\mathbb{R}^{n}} \int _{\mathbb{R}^{n}} \frac{ \vert f_{1l}(y_{1}) \vert \vert f_{2j}(y_{2}) \vert }{( \vert x-y_{1} \vert + \vert x-y _{2} \vert )^{2n}} \,dy_{1} \,dy_{2} \\ & \lesssim 2^{-2kn} \int _{\mathbb{R}^{n}} \int _{\mathbb{R}^{n}} \bigl\vert f _{1l}(y_{1}) \bigr\vert \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy_{1}\,dy_{2}. \end{aligned}$$

Therefore, by Hölder’s inequality, we obtain

$$\begin{aligned} & \Biggl\Vert \sum ^{k-2}_{l=-\infty } \sum^{k-2}_{j=-\infty } T(f_{1l},f _{2j})\chi _{k} \Biggr\Vert _{L^{p(\cdot )}(w)} \\ &\quad \lesssim 2^{-2kn} \Biggl\Vert \sum^{k-2}_{l=-\infty } \sum^{k-2} _{j=-\infty } \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p( \cdot )}(w)} \\ & \quad\lesssim \Biggl\Vert \sum^{k-2}_{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p _{1}(\cdot )}(w_{1})} \\ &\qquad{} \times \Biggl\Vert \sum^{k-2}_{j=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}. \end{aligned}$$
(8)

Since \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(\lambda =\lambda _{1}+ \lambda _{2}\), by Hölder’s inequality, we have

$$\begin{aligned} E_{1} \lesssim{}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl( \sum^{L}_{k=-\infty } 2^{k\alpha (0)q(0)} \Biggl\Vert \sum^{k-2} _{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q(0)} \\ & {}\times \Biggl\Vert \sum^{k-2}_{j=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q(0)} \Biggr)^{\frac{1}{q(0)}} \\ \lesssim {}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{1} } \\ &{} \times \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha _{1}(0)q_{1}(0)} \Biggl\Vert \sum^{k-2}_{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy _{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{1}(0)} \Biggr)^{\frac{1}{q _{1}(0)}} \\ &{} \times \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \\ &{} \times \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha _{2}(0)q_{2}(0)} \Biggl\Vert \sum^{k-2}_{j=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy _{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2}(0)} \Biggr)^{\frac{1}{q _{2}(0)}} \\ :={}&E_{1,1} \times E_{1,2}, \end{aligned}$$

where

$$\begin{aligned} E_{1,i}:={}&\sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{i} } \\ &{} \times \Biggl\{ \sum^{L}_{k=-\infty } 2^{k\alpha _{i}(0)q_{i}(0)} \Biggl\Vert \sum^{k-2}_{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{il}(y _{i}) \bigr\vert \,dy_{i} \chi _{k} \Biggr\Vert _{L^{p_{i}(\cdot )}(w_{i})}^{q_{i}(0)} \Biggr\} ^{\frac{1}{q_{i}(0)}}. \end{aligned}$$

Since \(n\delta _{2i}-\alpha _{i}(0)>0\), by (5) and Lemma 7 we obtain

$$\begin{aligned} E_{1,i}\lesssim{}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L{\lambda _{i}}} \Biggl\{ \sum^{L}_{k=-\infty } 2^{k\alpha _{i}(0)q_{i}(0)} \Biggl(\sum^{k-2}_{l=-\infty } 2^{(l-k)n\delta _{2i}} \Vert f_{il} \Vert _{L ^{p_{i}(\cdot )}(w_{i})} \Biggr)^{q_{i}(0)} \Biggr\} ^{ \frac{1}{q_{i}(0)}} \\ ={}&\sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L{\lambda _{i}}} \\ &{} \times \Biggl\{ \sum^{L}_{k=-\infty } \Biggl( \sum^{k-2}_{l=- \infty } 2^{l \alpha _{i}(0)} \Vert f_{il} \Vert _{L^{p_{i}(\cdot )}(w _{i})} 2^{(l-k)(n\delta _{2i}-\alpha _{i}(0))} \Biggr)^{q_{i}(0)} \Biggr\} ^{\frac{1}{q_{i}(0)}} \\ \lesssim{}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L{\lambda _{i}}} \Biggl(\sum ^{L-2}_{l=-\infty } 2^{l \alpha _{i}(0)q_{i}(0)} \Vert f _{il} \Vert _{L^{p_{i}(\cdot )}(w_{i})}^{q_{i}(0)} \Biggr)^{\frac{1}{q _{i}(0)}} \\ \lesssim {}&\Vert f_{i} \Vert _{M\dot{K}_{p_{i}( \cdot ),\lambda _{i}}^{\alpha _{i} ( \cdot ),q_{i}( \cdot )}(w _{i})}, \end{aligned}$$

where we wrote \(2^{-|k-l|(n\delta _{2i}-\alpha _{i}(0))}\lesssim 2^{-|k-l| \varepsilon _{i}}\) for some \(\varepsilon _{i} \in (0,n\delta _{2i}-\alpha _{i}(0))\). Thus, we obtain

$$ E_{1} \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ),\lambda _{1}}^{ \alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{M \dot{K}_{p_{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}(w _{2})}. $$

To estimate \(E_{2}\), since \(l \leq k-2\), \(k-1 \leq j \leq k+1\) for \(i=1, 2\), we have

$$ \vert x-y_{1} \vert \geq \vert x \vert - \vert y_{1} \vert \geq 2^{k-2},\quad x\in D_{k}, y_{1} \in D _{l}. $$

Therefore, by Hölder’s inequality, we obtain

$$\begin{aligned} & \Biggl\Vert \sum ^{k-2}_{l=-\infty } \sum^{k+1}_{j=k-1} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert _{L^{p(\cdot )}(w)} \\ & \quad\lesssim 2^{-2kn} \Biggl\Vert \sum^{k-2}_{l=-\infty } \sum^{k+1} _{j=k-1} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \int _{\mathbb{R} ^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p(\cdot )}(w)} \\ & \quad\lesssim \Biggl\Vert \sum^{k-2}_{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p _{1}(\cdot )}(w_{1})} \\ &\qquad{} \times \Biggl\Vert \sum^{k+1}_{j=k-1} 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}. \end{aligned}$$
(9)

Since \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(\lambda =\lambda _{1}+ \lambda _{2}\), by Hölder’s inequality, we have

$$\begin{aligned} E_{2} \lesssim {}&\sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl( \sum^{L}_{k=-\infty } 2^{k\alpha (0)q(0)} \Biggl\Vert \sum^{k-2} _{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q(0)} \\ & {}\times \Biggl\Vert \sum^{k+1}_{j=k-1} 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q(0)} \Biggr)^{\frac{1}{q(0)}} \\ \lesssim{}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{1} } \\ &{} \times \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha _{1}(0)q_{1}(0)} \Biggl\Vert \sum^{k-2}_{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy _{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{1}(0)} \Biggr)^{\frac{1}{q _{1}(0)}} \\ &{} \times \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \\ & {}\times \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha _{2}(0)q_{2}(0)} \Biggl\Vert \sum^{k+1}_{j=k-1} 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy _{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2}(0)} \Biggr)^{\frac{1}{q _{2}(0)}} \\ :={}&E_{2,1} \times E_{2,2}. \end{aligned}$$

It is obvious that

$$ E_{2,1} = E_{1,1} \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ), \lambda _{1}}^{\alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})}. $$

Now we estimate \(E_{2,2}\). Taking (5), (6) and (7) together, we have

$$\begin{aligned} E_{2,2} &\lesssim \sup_{L \le 0,L \in \mathbb{Z}} 2^{-L\lambda _{2}} \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha _{2}(0)q_{2}(0)} \Biggl\Vert \sum^{k+1}_{j=k-1} 2^{(j-k)n} f_{2j} \Biggr\Vert _{L^{p_{2}(\cdot )}(w _{2})}^{q_{2}(0)} \Biggr)^{\frac{1}{q_{2}(0)}} \\ & \lesssim \sup_{L \le 0,L \in \mathbb{Z}} 2^{-L\lambda _{2}} \Biggl( \sum ^{L}_{k=-\infty } 2^{k\alpha _{2}(0)q_{2}(0)} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2}(0)} \Biggr)^{ \frac{1}{q_{2}(0)}} \\ & \lesssim \Vert f_{2} \Vert _{M\dot{K}_{p_{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}(w _{2})}, \end{aligned}$$

where we used \(2^{-n\delta _{22}} <1\) and \(2^{(j-k)n(1-\delta _{12})} < 2^{(j-k)n} \), \(j\in \{ k-1, k, k+1\}\) for (5) and (7), respectively. Thus, we obtain

$$ E_{2} \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ),\lambda _{1}}^{ \alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{M \dot{K}_{p_{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}(w _{2})}. $$

To estimate \(E_{3}\), since \(l \leq k-2\), \(j \geq k+2\), then we have

$$ \vert x-y_{1} \vert \geq \vert x \vert - \vert y_{1} \vert \geq 2^{k-2},\qquad \vert x-y_{2} \vert \geq \vert y_{2} \vert - \vert x \vert > 2^{j-2},\quad x\in D_{k}, y_{1} \in D_{l}, y_{2} \in D_{j}. $$

Therefore, \(\forall x \in D_{k}\), \(l \leq k-2\), \(j \geq k+2\), we get

$$\begin{aligned} \bigl\vert T(f_{1l},f_{2j}) (x) \bigr\vert &\lesssim \int _{\mathbb{R}^{n}} \int _{\mathbb{R}^{n}} \frac{ \vert f_{1l}(y_{1}) \vert \vert f_{2j}(y_{2}) \vert }{( \vert x-y_{1} \vert + \vert x-y _{2} \vert )^{2n}} \,dy_{1} \,dy_{2} \\ & \lesssim 2^{-kn} 2^{-jn} \int _{\mathbb{R}^{n}} \int _{\mathbb{R} ^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy_{1}\,dy_{2}. \end{aligned}$$

Thus, by Hölder’s inequality, we have

$$\begin{aligned} & \Biggl\Vert \sum ^{k-2}_{l=-\infty } \sum^{\infty }_{j=k+2} T(f_{1l},f _{2j})\chi _{k} \Biggr\Vert _{L^{p(\cdot )}(w)} \\ & \quad\lesssim 2^{-kn} 2^{-jn} \Biggl\Vert \sum ^{k-2}_{l=-\infty } \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \sum^{\infty }_{j=k+2} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p( \cdot )}(w)} \\ & \quad\lesssim \Biggl\Vert \sum^{k-2}_{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p _{1}(\cdot )}(w_{1})} \\ & \qquad{}\times \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}. \end{aligned}$$
(10)

Since \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(\lambda =\lambda _{1}+ \lambda _{2}\), by Hölder’s inequality, we have

$$\begin{aligned} E_{3} \lesssim{} & \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl( \sum^{L}_{k=-\infty } 2^{k\alpha (0)q(0)} \Biggl\Vert \sum^{k-2} _{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q(0)} \\ & {}\times \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q(0)} \Biggr)^{\frac{1}{q(0)}} \\ \lesssim{}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{1} } \\ &{} \times \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha _{1}(0)q_{1}(0)} \Biggl\Vert \sum^{k-2}_{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy _{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{1}(0)} \Biggr)^{\frac{1}{q _{1}(0)}} \\ & {}\times \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \\ &{} \times \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha _{2}(0)q_{2}(0)} \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy _{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2}(0)} \Biggr)^{\frac{1}{q _{2}(0)}} \\ :={}&E_{3,1} \times E_{3,2}. \end{aligned}$$

It is obvious that

$$ E_{3,1} = E_{1,1} \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ), \lambda _{1}}^{\alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})}. $$

Since \(n\delta _{12}+\alpha _{2}(0)>0\), by (7) and Lemma 7 we obtain

$$\begin{aligned} E_{3,2}\lesssim{}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha _{2}(0)q_{2}(0)} \\ &{} \times \Biggl(\sum^{\infty }_{j=k+2} 2^{(k-j)n\delta _{12}} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w_{2})} \Biggr)^{q_{2}(0)} \Biggr)^{\frac{1}{q_{2}(0)}} \\ \lesssim{}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \\ &{} \times \Biggl(\sum^{L}_{k=-\infty } \Biggl( \sum^{\infty }_{j=k+2} 2^{j\alpha _{2}(0)} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w_{2})} 2^{(k-j)(n \delta _{12}+\alpha _{2}(0))} \Biggr)^{q_{2}(0)} \Biggr)^{ \frac{1}{q_{2}(0)}} \\ \lesssim{}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L{\lambda _{2}}} \Biggl(\sum ^{L+2}_{j=-\infty } 2^{j\alpha _{2}(0)q_{2}(0)} \Vert f _{2j} \Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2}(0)} \Biggr)^{\frac{1}{q _{2}(0)}} \\ \lesssim {}& \Vert f_{2} \Vert _{M\dot{K}_{p_{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}(w _{2})}, \end{aligned}$$

where we wrote \(2^{-|k-j|(n\delta _{12}+\alpha _{2}(0))}\lesssim 2^{-|k-j| \eta _{2}}\) for some \(\eta _{2} \in (0,n\delta _{12}+\alpha _{2}(0))\). Thus, we have

$$ E_{3} \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ),\lambda _{1}}^{ \alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{M \dot{K}_{p_{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}(w _{2})}. $$

To estimate \(E_{5}\), using Hölder’s inequality and Lemma 8, we have

$$\begin{aligned} E_{5}\lesssim{}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha (0) q(0)} \Biggl\Vert \sum^{k+1} _{l=k-1} \sum ^{k+1}_{j=k-1} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q(0)} _{L^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q(0)}} \\ \lesssim{}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl( \sum ^{L}_{k=-\infty } 2^{k\alpha (0) q(0)} \bigl( \Vert f_{1} \Vert _{L^{p_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{L^{p_{2}(\cdot )}(w _{2})} \bigr)^{q(0)} \Biggr)^{\frac{1}{q(0)}} \\ \lesssim{} &\sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{1} } \Biggl(\sum ^{L}_{k=-\infty } 2^{k\alpha _{1}(0)q_{1}(0)} \Vert f_{1} \Vert _{L ^{p_{1}(\cdot )}(w _{1})}^{q_{1}(0)} \Biggr)^{\frac{1}{q_{1}(0)}} \\ &{} \times 2^{ - L\lambda _{2} } \Biggl(\sum^{L}_{k=-\infty } 2^{k \alpha _{2}(0)q_{2}(0)} \Vert f_{2} \Vert _{L^{p_{2}(\cdot )}(w _{2})}^{q _{2}(0)} \Biggr)^{\frac{1}{q_{2}(0)}} \\ \lesssim{}& \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ),\lambda _{1}}^{\alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{M\dot{K}_{p _{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}( w _{2})}. \end{aligned}$$

To estimate \(E_{6}\), since \(k-1 \leq l \leq k+1\) and \(j \geq k+2\), we obtain

$$ \vert x-y_{1} \vert >2^{k-2}, \qquad \vert x-y_{2} \vert > 2^{j-2},\quad x\in D_{k}, y_{1} \in D _{l}, y_{2} \in D_{j}. $$

Thus, \(\forall x \in D_{k}\), \(k-1 \leq l \leq k+1\) and \(j \geq k+2\), we obtain

$$\begin{aligned} \bigl\vert T(f_{1l},f_{2j}) (x) \bigr\vert &\lesssim \int _{\mathbb{R}^{n}} \int _{\mathbb{R}^{n}} \frac{ \vert f_{1l}(y_{1}) \vert \vert f_{2j}(y_{2}) \vert }{( \vert x-y_{1} \vert + \vert x-y _{2} \vert )^{2n}} \,dy_{1} \,dy_{2} \\ & \lesssim 2^{-kn} 2^{-jn} \int _{\mathbb{R}^{n}} \int _{\mathbb{R} ^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy_{1}\,dy_{2}. \end{aligned}$$

Therefore, by Hölder’s inequality, we obtain

$$\begin{aligned} & \Biggl\Vert \sum ^{k+1}_{l=k-1} \sum^{\infty }_{j=k+2} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert _{L^{p(\cdot )}(w)} \\ & \quad\lesssim 2^{-kn} 2^{-jn} \Biggl\Vert \sum ^{k+1}_{l=k-1} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \sum^{k-2}_{j=-\infty } \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p( \cdot )}(w)} \\ & \quad\lesssim \Biggl\Vert \sum^{k+1}_{l=k-1} 2^{-kn} \int _{\mathbb{R} ^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w _{1})} \\ &\qquad{} \times \Biggl\Vert \sum^{k-2}_{j=-\infty } 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}. \end{aligned}$$
(11)

Since \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(\lambda =\lambda _{1}+ \lambda _{2}\), by Hölder’s inequality, we have

$$\begin{aligned} E_{6} \lesssim {}&\sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl( \sum^{L}_{k=-\infty } 2^{k\alpha (0)q(0)} \Biggl\Vert \sum^{k+1} _{l=k-1} 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q(0)} \\ &{} \times \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q(0)} \Biggr)^{\frac{1}{q(0)}} \\ \lesssim{}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{1} } \\ & {}\times \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha _{1}(0)q_{1}(0)} \Biggl\Vert \sum^{k+1}_{l=k-1} 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy _{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{1}(0)} \Biggr)^{\frac{1}{q _{1}(0)}} \\ & {}\times \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \\ &{} \times \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha _{2}(0)q_{2}(0)} \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy _{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2}(0)} \Biggr)^{\frac{1}{q _{2}(0)}} \\ :={}&E_{6,1} \times E_{6,2}. \end{aligned}$$

By the symmetry of \(f_{1}\) and \(f_{2}\), we can know that the estimate \(E_{6,1}\) is similar to the estimated \(E_{2,2}\) and \(E_{6,2} =E_{3,2}\).

To estimate \(E_{9}\), since l, \(j \geq k+2\), for \(i=1,2\), we get

$$ \vert x-y_{i} \vert > 2^{k-2},\quad x\in D_{k}, y_{1} \in D_{l}, y_{2} \in D _{j}. $$

Therefore, \(\forall x \in D_{k}\), l, \(j \geq k+2\), we have

$$\begin{aligned} \bigl\vert T(f_{1l},f_{2j}) (x) \bigr\vert &\lesssim \int _{\mathbb{R}^{n}} \int _{\mathbb{R}^{n}} \frac{ \vert f_{1l}(y_{1}) \vert \vert f_{2j}(y_{2}) \vert }{( \vert x-y_{1} \vert + \vert x-y _{2} \vert )^{2n}} \,dy_{1} \,dy_{2} \\ & \lesssim 2^{-ln } 2^{-jn} \int _{\mathbb{R}^{n}} \int _{\mathbb{R} ^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy_{1}\,dy_{2}. \end{aligned}$$

Thus, by Hölder’s inequality, we have

$$\begin{aligned} & \Biggl\Vert \sum ^{\infty }_{l=k+2} \sum^{\infty }_{j=k+2} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert _{L^{p(\cdot )}(w)} \\ & \quad\lesssim 2^{-ln } 2^{-jn} \Biggl\Vert \sum ^{\infty }_{l=k+2} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \sum^{\infty }_{j=k+2} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p( \cdot )}(w)} \\ & \quad\lesssim \Biggl\Vert \sum^{\infty }_{l=k+2} 2^{-ln } \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p _{1}(\cdot )}(w_{1})} \\ &\qquad{} \times \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}. \end{aligned}$$
(12)

Since \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(\lambda =\lambda _{1}+ \lambda _{2}\), by Hölder’s inequality, we have

$$\begin{aligned} E_{9} \lesssim{}& \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl( \sum^{L}_{k=-\infty } 2^{k\alpha (0)q(0)} \Biggl\Vert \sum^{\infty }_{l=k+2} 2^{-ln } \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q(0)} \\ & {}\times \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q(0)} \Biggr)^{\frac{1}{q(0)}} \\ \lesssim {}&\sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{1} } \\ &{} \times \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha _{1}(0)q_{1}(0)} \Biggl\Vert \sum^{\infty }_{l=k+2} 2^{-ln} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy _{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{1}(0)} \Biggr)^{\frac{1}{q _{1}(0)}} \\ &{} \times \sup_{L \le 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \\ & {}\times \Biggl(\sum^{L}_{k=-\infty } 2^{k\alpha _{2}(0)q_{2}(0)} \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy _{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2}(0)} \Biggr)^{\frac{1}{q _{2}(0)}} \\ :={}&E_{9,1} \times E_{9,2}. \end{aligned}$$

Obviously, the estimate \(E_{9,i}\) is similar to the estimated \(E_{3,2}\) for \(i=1,2\).

Taking all estimates for \(E_{i}\) together, \(i=1,2,\ldots,9\), we obtain

$$ E \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ),\lambda _{1}}^{\alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{M\dot{K}_{p _{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}( w _{2})}. $$

To go on, we need some further preparation.

If \(l<0\), by Proposition 1, we have

$$\begin{aligned} \Vert f_{il} \chi _{l} \Vert _{L^{p_{i}(\cdot )}(w_{i})} &=2^{-l\alpha _{i}(0)} \bigl(2^{l \alpha _{i}(0)q_{i}(0)} \Vert f_{il}\chi _{l} \Vert _{L^{p_{i}(\cdot )}(w_{i})}^{q_{i}(0)} \bigr)^{\frac{1}{q_{i}(0)}} \\ &\lesssim 2^{-l\alpha _{i}(0)} \Biggl(\sum_{t=-\infty }^{l} 2^{t\alpha _{i}(0)q_{i}(0)} \Vert f_{it}\chi _{t} \Vert _{L^{p_{i}(\cdot )}(w_{i})}^{q_{i}(0)} \Biggr)^{\frac{1}{q_{i}(0)}} \\ &\lesssim 2^{l(\lambda _{i}-\alpha _{i}(0))} \Biggl( 2^{-l\lambda _{i}} \Biggl(\sum _{t=-\infty }^{l} \bigl\Vert 2^{t\alpha _{i}(0)}f_{it} \chi _{t} \bigr\Vert _{L ^{p_{i}(\cdot )}(w_{i})}^{q_{i}(0)} \Biggr) \Biggr)^{\frac{1}{q_{i}(0)}} \\ & \lesssim 2^{l(\lambda _{i}-\alpha _{i}(0))} \Vert f_{i} \Vert _{M\dot{K}_{p _{i}( \cdot ),\lambda _{i}}^{\alpha _{i} ( \cdot ),q_{i}( \cdot )}(w_{i})}. \end{aligned}$$
(13)

If \(l\geq 0\), we have

$$\begin{aligned} \Vert f_{il}\chi _{l} \Vert _{L^{p_{i}(\cdot )}(w_{i})} &=2^{-l\alpha _{i\infty }} \bigl(2^{l\alpha _{i\infty }q_{i\infty }} \Vert f_{i}\chi _{il} \Vert _{L^{p_{i}( \cdot )}(w_{i})}^{q_{i\infty }} \bigr)^{1/q_{i\infty }} \\ &\lesssim 2^{-l\alpha _{i\infty }} \Biggl(\sum_{t=-\infty }^{l} 2^{t \alpha _{i\infty }q_{i\infty }} \Vert f_{it}\chi _{t} \Vert _{L^{p_{i}(\cdot )}(w_{i})} ^{q_{i\infty }} \Biggr)^{1/q_{i\infty }} \\ &\lesssim 2^{l(\lambda _{i}-\alpha _{i\infty })} \Biggl( 2^{-l\lambda _{i}} \Biggl(\sum _{t=-\infty }^{l} \bigl\Vert 2^{t\alpha _{i\infty }}f_{it} \chi _{t} \bigr\Vert _{L^{p_{i}(\cdot )}(w_{i})}^{q_{i\infty }} \Biggr) \Biggr)^{1/q_{i\infty }} \\ & \lesssim 2^{l(\lambda _{i}-\alpha _{i\infty })} \Vert f_{i} \Vert _{M\dot{K} _{p_{i}( \cdot ),\lambda _{i}}^{\alpha _{i} ( \cdot ),q_{i}(\cdot )}(w_{i})}. \end{aligned}$$
(14)

Finally, we estimate G. By the symmetry of \(f_{1}\) and \(f_{2}\), it is only necessary to estimate \(G_{1}\), \(G_{2}\), \(G_{3}\), \(G_{5}\), \(G_{6}\), and \(G_{9}\).

To estimate \(G_{1}\), since l, \(j \leq k-2\), \(1/q_{\infty }=1/{q_{1 \infty }}+1/{q_{2\infty }}\), \(\lambda =\lambda _{1}+\lambda _{2}\), by (8) and Hölder’s inequality, we have

$$\begin{aligned} G_{1} \lesssim{}& \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl( \sum^{L}_{k=0} 2^{k\alpha _{\infty }q_{\infty }} \Biggl\Vert \sum^{k-2} _{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{\infty }} \\ &{} \times \Biggl\Vert \sum^{k-2}_{j=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q _{\infty }} \Biggr)^{\frac{1}{q_{\infty }}} \\ \lesssim{}& \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{1} } \\ &{} \times \Biggl(\sum^{L}_{k=0} 2^{k\alpha _{1\infty }q_{1\infty }} \Biggl\Vert \sum^{k-2}_{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy _{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{1\infty }} \Biggr)^{\frac{1}{q_{1\infty }}} \\ &{} \times \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \\ &{} \times \Biggl(\sum^{L}_{k=0} 2^{k\alpha _{2\infty }q_{2\infty }} \Biggl\Vert \sum^{k-2}_{j=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy _{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2\infty }} \Biggr)^{\frac{1}{q_{2\infty }}} \\ :={}&G_{1,1} \times G_{1,2}, \end{aligned}$$

where

$$\begin{aligned} G_{1,i}:=\sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{i} } \Biggl( \sum ^{L}_{k=0} 2^{k\alpha _{i\infty }q_{i\infty }} \Biggl\Vert \sum^{k-2} _{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{il}(y_{i}) \bigr\vert \,dy_{i} \chi _{k} \Biggr\Vert _{L^{p_{i}(\cdot )}(w_{i})}^{q_{i\infty }} \Biggr)^{\frac{1}{q _{i\infty }}} . \end{aligned}$$

Since \(\lambda _{i}+n\delta _{2i}>\alpha _{i\infty } \geq \alpha _{i}(0)\), by (5), (13), (14) and Lemma 7 we obtain

$$\begin{aligned} G_{1,i}\lesssim{}& \sup_{L > 0,L \in \mathbb{Z}} \Vert f_{i} \Vert _{M\dot{K} _{p_{i}( \cdot ),\lambda _{i}}^{\alpha _{i} ( \cdot ),q_{i}( \cdot )}(w_{i})} 2^{ - L \lambda _{i}} \Biggl\{ \sum _{k = 0 }^{L} 2^{k\alpha _{i \infty }q _{i \infty }} \\ & {}\times \Biggl( \sum_{l = - \infty }^{-1 } 2^{(l - k)n\delta _{2i}} 2^{l(\lambda _{i}-\alpha _{i}(0))} +\sum_{l = 0 }^{k } 2^{(l - k)n \delta _{12}} 2^{i(\lambda _{1}-\alpha _{1\infty })} \Biggr)^{q_{i \infty }} \Biggr\} ^{\frac{1}{q_{i \infty }}} \\ \lesssim {}&\sup_{L > 0,L \in \mathbb{Z}} \Vert f_{i} \Vert _{M\dot{K}_{p_{i}( \cdot ),\lambda _{i}}^{\alpha _{i} ( \cdot ),q_{i}( \cdot )}(w_{i})} 2^{ - L \lambda _{i}} \Biggl\{ \sum_{k = 0 }^{L} 2^{k\alpha _{i \infty } q_{i \infty }} \\ & {}\times \Biggl( \sum_{l = - \infty }^{-1 } 2^{(l - k)(n\delta _{2i}+\lambda _{i}-\alpha _{i\infty }) } 2^{l(\alpha _{i\infty }-\alpha _{i}(0))} 2^{k(\lambda _{i}-\alpha _{i\infty })} \\ &{} +\sum_{l = 0 }^{k } 2^{(l - k)(n\delta _{2i}+\lambda _{i}-\alpha _{i \infty })} 2^{k(\lambda _{i}-\alpha _{i\infty })} \Biggr)^{q_{i\infty }} \Biggr\} ^{\frac{1}{q_{i \infty }}} \\ \lesssim{}& \sup_{L > 0,L \in \mathbb{Z}} \Vert f_{i} \Vert _{M\dot{K}_{p_{i}( \cdot ),\lambda _{i}}^{\alpha _{i} ( \cdot ),q_{i}( \cdot )}(w_{i})} \\ & {}\times 2^{ - L{\lambda _{i}}} \Biggl\{ \sum_{k = 0 }^{L} 2^{k \lambda _{i} q_{i\infty }} \Biggl( \sum_{l = - \infty }^{k} 2^{(l - k)(n \delta _{2i} - \alpha _{i\infty } +\lambda _{i} )} \Biggr)^{q_{i\infty }} \Biggr\} ^{\frac{1}{q_{i \infty }}} \\ \lesssim {}&\sup_{L > 0,L \in \mathbb{Z}} \Vert f_{i} \Vert _{M\dot{K}_{p_{i}( \cdot ),\lambda _{i}}^{\alpha _{i} ( \cdot ),q_{i}( \cdot )}(w_{i})} 2^{ - L {\lambda _{i}}} \Biggl( \sum_{k = 0 }^{L} 2^{k\lambda _{i} q_{i\infty }} \Biggr)^{\frac{1}{q_{i \infty }}} \\ \lesssim{}& \Vert f_{i} \Vert _{M\dot{K}_{p_{i}( \cdot ),\lambda _{i}}^{\alpha _{i} ( \cdot ),q_{i}( \cdot )}(w_{i})}. \end{aligned}$$

Thus, we get

$$ G_{1} \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ),\lambda _{1}}^{ \alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{M \dot{K}_{p_{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}(w _{2})}. $$

To estimate \(G_{2}\), since \(l \leq k-2\), \(k-1 \leq j \leq k+1\), \(1/q_{\infty }=1/{q_{1\infty }}+1/{q_{2\infty }}\), \(\lambda =\lambda _{1}+\lambda _{2}\), by (9) and Hölder’s inequality, we have

$$\begin{aligned} G_{2} \lesssim{}& \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl( \sum^{L}_{k=0} 2^{k\alpha _{\infty }q_{\infty }} \Biggl\Vert \sum^{k-2} _{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{\infty }} \\ & {}\times \Biggl\Vert \sum^{k+1}_{j=k-1} 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q _{\infty }} \Biggr)^{\frac{1}{q_{\infty }}} \\ \lesssim{}& \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{1} } \\ &{} \times \Biggl(\sum^{L}_{k=0} 2^{k\alpha _{1\infty }q_{1\infty }} \Biggl\Vert \sum^{k-2}_{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy _{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{1\infty }} \Biggr)^{\frac{1}{q_{1\infty }}} \\ &{} \times \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \\ & {}\times \Biggl(\sum^{L}_{k=0} 2^{k\alpha_{2\infty}q_{2\infty}} \Biggl\Vert \sum ^{k+1}_{j=k-1} 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy _{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2\infty }} \Biggr)^{\frac{1}{q_{2\infty }}} \\ :={}&G_{2,1} \times G_{2,2}. \end{aligned}$$

It is obvious that

$$ G_{2,1} = G_{1,1} \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ), \lambda _{1}}^{\alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})}. $$

Now we estimate \(G_{2,2}\). Combining (5), (6) and (7), we have

$$\begin{aligned} G_{2,2} &\lesssim \sup_{L > 0,L \in \mathbb{Z}} 2^{-L\lambda _{2}} \Biggl(\sum^{L}_{k=0} 2^{k\alpha _{2\infty }q_{2\infty }} \Biggl\Vert \sum^{k+1}_{j=k-1} 2^{(j-k)n} \vert f_{2j} \vert \Biggr\Vert _{L^{p_{2}(\cdot )}(w _{2})}^{q_{2\infty }} \Biggr)^{\frac{1}{q_{2\infty }}} \\ & \lesssim \sup_{L > 0,L \in \mathbb{Z}} 2^{-L\lambda _{2}} \Biggl(\sum ^{L}_{k=0} 2^{k\alpha _{2\infty }q_{2\infty }} \bigl\| |f_{2j}| \bigr\| _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2\infty }} \Biggr)^{\frac{1}{q _{2\infty }}} \\ & \lesssim \Vert f_{2} \Vert _{M\dot{K}_{p_{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}(w _{2})}, \end{aligned}$$

where we used \(2^{-n\delta _{22}} <1\) and \(2^{(j-k)n(1-\delta _{12})} < 2^{(j-k)n}\) for (5) and (7), respectively. Thus, we obtain

$$ G_{2} \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ),\lambda _{1}}^{ \alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{M \dot{K}_{p_{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}(w _{2})}. $$

To estimate \(G_{3}\), since \(l \leq k-2\), \(j \geq k+2\), \(1/q_{\infty }=1/ {q_{1\infty }}+1/{q_{2\infty }}\), \(\lambda =\lambda _{1}+\lambda _{2}\), by (10) and Hölder’s inequality, we have

$$\begin{aligned} G_{3} \lesssim{}& \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl( \sum^{L}_{k=0} 2^{k\alpha _{\infty }q_{\infty }} \Biggl\Vert \sum^{k-2} _{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{\infty }} \\ &{} \times \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q _{\infty }} \Biggr)^{\frac{1}{q_{\infty }}} \\ \lesssim{}& \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{1} } \\ & {}\times \Biggl(\sum^{L}_{k=0} 2^{k\alpha _{1\infty }q_{1\infty }} \Biggl\Vert \sum^{k-2}_{l=-\infty } 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy _{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{1\infty }} \Biggr)^{\frac{1}{q_{1\infty }}} \\ &{} \times \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \\ & {}\times \Biggl(\sum^{L}_{k=0} 2^{k\alpha _{2\infty }q_{2\infty }} \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy _{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2\infty }} \Biggr)^{\frac{1}{q_{2\infty }}} \\ :={}&G_{3,1} \times G_{3,2}. \end{aligned}$$

It is obvious that

$$ G_{3,1} = G_{1,1} \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ), \lambda _{1}}^{\alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})}. $$

Since \(n\delta _{12}+\alpha _{2\infty }>0\), by (7) and Lemma 7 we obtain

$$\begin{aligned} G_{3,2} &\lesssim \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \Biggl(\sum^{L}_{k=0} 2^{k\alpha _{2\infty }q_{2\infty }} \Biggl( \sum^{\infty }_{j=k+2} 2^{(k-j)n\delta _{12}} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w_{2})} \Biggr)^{q_{2\infty }} \Biggr)^{\frac{1}{q _{2\infty }}} \\ &\lesssim \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \Biggl( \sum ^{L}_{k=0} \Biggl(\sum^{\infty }_{j=k+2} 2^{j\alpha _{2\infty }} \Vert f_{2j} \Vert _{L^{p_{2}(\cdot )}(w_{2})} 2^{(k-j)(n\delta _{12}+\alpha _{2\infty })} \Biggr)^{q_{2\infty }} \Biggr)^{\frac{1}{q _{2\infty }}} \\ & \lesssim \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L{\lambda _{2}}} \Biggl(\sum ^{L+2}_{j=2} 2^{j\alpha _{2\infty }q_{2\infty }} \Vert f _{2j} \Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2\infty }} \Biggr)^{\frac{1}{q _{2\infty }}} \\ & \lesssim \Vert f_{2} \Vert _{M\dot{K}_{p_{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}(w _{2})}, \end{aligned}$$

where we wrote \(2^{-|k-j|(n\delta _{12}+\alpha _{2\infty })}\lesssim 2^{-|k-j| \vartheta _{2}}\) for some \(\vartheta _{2} \in (0,n\delta _{12}+ \alpha _{2\infty })\). Thus, we get

$$ G_{3} \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ),\lambda _{1}}^{ \alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{M \dot{K}_{p_{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}(w _{2})}. $$

To estimate \(G_{5}\), using Hölder’s inequality and Lemma 8

$$\begin{aligned} G_{5}\lesssim{}& \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl( \sum^{L}_{k=0} 2^{k\alpha _{\infty }q_{\infty }} \Biggl\Vert \sum^{k+1} _{l=k-1} \sum ^{k+1}_{j=k-1} T(f_{1l},f_{2j}) \chi _{k} \Biggr\Vert ^{q_{ \infty }}_{L^{p(\cdot )}(w)} \Biggr)^{\frac{1}{q_{\infty }}} \\ \lesssim {}&\sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl(\sum ^{L}_{k=0} 2^{k\alpha _{\infty }q_{\infty }} \bigl( \Vert f_{1} \Vert _{L^{p_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{L^{p_{2}(\cdot )}(w _{2})} \bigr)^{q_{\infty }} \Biggr)^{\frac{1}{q_{\infty }}} \\ \lesssim {}&\sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{1} } \Biggl( \sum ^{L}_{k=0} 2^{k\alpha _{1\infty }q_{1\infty }} \Vert f_{1} \Vert _{L^{p_{1}( \cdot )}(w _{1})}^{q_{1\infty }} \Biggr)^{\frac{1}{q_{1\infty }}} \\ &{} \times 2^{ - L\lambda _{2} } \Biggl(\sum^{L}_{k=0} 2^{k \alpha _{2\infty }q_{2\infty }} \Vert f_{2} \Vert _{L^{p_{2}(\cdot )}(w _{2})}^{q_{2\infty }} \Biggr)^{\frac{1}{q_{2\infty }}} \\ \lesssim{}& \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ),\lambda _{1}}^{\alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{M\dot{K}_{p _{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}( w _{2})}. \end{aligned}$$

To estimate \(G_{6}\), since \(k-1 \leq l \leq k+1\) and \(j \geq k+2\), \(1/q_{\infty }=1/{q_{1\infty }}+1/{q_{2\infty }}\), \(\lambda =\lambda _{1}+\lambda _{2}\), by (11) and Hölder’s inequality, we have

$$\begin{aligned} G_{6} \lesssim{}& \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl( \sum^{L}_{k=0} 2^{k\alpha _{\infty }q_{\infty }} \Biggl\Vert \sum^{k+1} _{l=k-1} 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{\infty }} \\ &{} \times \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q _{\infty }} \Biggr)^{\frac{1}{q_{\infty }}} \\ \lesssim{}& \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{1} } \\ & {}\times \Biggl(\sum^{L}_{k=0} 2^{k\alpha _{1\infty }q_{1\infty }} \Biggl\Vert \sum^{k+1}_{l=k-1} 2^{-kn} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy _{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{1\infty }} \Biggr)^{\frac{1}{q_{1\infty }}} \\ &{} \times \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \\ &{} \times \Biggl(\sum^{L}_{k=0} 2^{k\alpha _{2\infty }q_{2\infty }} \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy _{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2\infty }} \Biggr)^{\frac{1}{q_{2\infty }}} \\ :={}&G_{6,1} \times G_{6,2}. \end{aligned}$$

By the symmetry of \(f_{1}\) and \(f_{2}\), we can know that the estimate \(G_{6,1}\) is similar to the estimated \(G_{2,2}\) and \(G_{6,2} =G_{3,2}\).

To estimate \(G_{9}\), since l, \(j \geq k+2\), \(1/q_{\infty }=1/{q_{1 \infty }}+1/{q_{2\infty }}\), \(\lambda =\lambda _{1}+\lambda _{2}\), by (12) and Hölder’s inequality, we have

$$\begin{aligned} G_{9} \lesssim{}& \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda } \Biggl( \sum^{L}_{k=0} 2^{k\alpha _{\infty }q_{\infty }} \Biggl\Vert \sum^{\infty }_{l=k+2} 2^{-ln } \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy_{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{\infty }} \\ &{} \times \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f _{2j}(y_{2}) \bigr\vert \,dy_{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q _{\infty }} \Biggr)^{\frac{1}{q_{\infty }}} \\ \lesssim {}&\sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{1} } \\ &{} \times \Biggl(\sum^{L}_{k=0} 2^{k\alpha _{1\infty }q_{1\infty }} \Biggl\Vert \sum^{\infty }_{l=k+2} 2^{-ln} \int _{\mathbb{R}^{n}} \bigl\vert f_{1l}(y_{1}) \bigr\vert \,dy _{1} \chi _{k} \Biggr\Vert _{L^{p_{1}(\cdot )}(w_{1})}^{q_{1\infty }} \Biggr)^{\frac{1}{q_{1\infty }}} \\ &{} \times \sup_{L > 0,L \in \mathbb{Z}} 2^{ - L\lambda _{2} } \\ & {}\times \Biggl(\sum^{L}_{k=0} 2^{k\alpha _{2\infty }q_{2\infty }} \Biggl\Vert \sum^{\infty }_{j=k+2} 2^{-jn} \int _{\mathbb{R}^{n}} \bigl\vert f_{2j}(y_{2}) \bigr\vert \,dy _{2} \chi _{k} \Biggr\Vert _{L^{p_{2}(\cdot )}(w_{2})}^{q_{2\infty }} \Biggr)^{\frac{1}{q_{2\infty }}} \\ :={}&G_{9,1} \times G_{9,2}. \end{aligned}$$

Obviously, the estimate \(G_{9,i}\) is similar to the estimated \(G_{3,2}\) for \(i=1,2\).

Taking all estimates for \(G_{i}\) together, \(i=1,2,\ldots,9\), we obtain

$$ G \lesssim \Vert f_{1} \Vert _{M\dot{K}_{p_{1}( \cdot ),\lambda _{1}}^{\alpha _{1} ( \cdot ),q_{1}( \cdot )}(w _{1})} \Vert f_{2} \Vert _{M\dot{K}_{p _{2}( \cdot ),\lambda _{2}}^{\alpha _{2} ( \cdot ),q_{2}( \cdot )}( w _{2})}. $$

This completes the proof. □

References

  1. 1.

    Cruz-Uribe, D., Guzman, O.M.: Weighted norm inequalities for the bilinear maximal operator on variable Lebesgue spaces (2018) https://arxiv.org/abs/1811.00618

  2. 2.

    Izuki, M.: Herz and amalgam spaces with variable exponent, the Haar wavelets and greediness of the wavelet system. East J. Approx. 15(1), 87–109 (2009)

  3. 3.

    Izuki, M.: Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Anal. Math. 36(1), 33–50 (2010)

  4. 4.

    Izuki, M.: Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent. Rend. Circ. Mat. Palermo 59(3), 461–472 (2010)

  5. 5.

    Izuki, M.: Fractional integrals on Herz–Morrey spaces with variable exponent. Hiroshima Math. J. 40(3), 343–355 (2010)

  6. 6.

    Izuki, M., Noi, T.: Hardy spaces associated to critical Herz spaces with variable exponent. Mediterr. J. Math. 13(5), 2981–3013 (2016)

  7. 7.

    Izuki, M., Noi, T.: Boundedness of fractional integrals on weighted Herz spaces with variable exponent. J. Inequal. Appl. 2016(199), 15 (2016)

  8. 8.

    Izuki, M.: Boundedness of commutators on Herz spaces with variable exponent. Rend. Circ. Mat. Palermo 59(2), 199–213 (2010)

  9. 9.

    Izuki, M., Noi, T.: An intrinsic square function on weighted Herz spaces with variable exponent. J. Math. Inequal. 11(3), 799–816 (2017)

  10. 10.

    Wang, L.W., Shu, L.S.: Boundedness of some sublinear operators on weighted variable Herz–Morrey spaces. J. Math. Inequal. 12(1), 31–42 (2018)

  11. 11.

    Wang, P., Liu, Z.: Weighted norm inequalities for multilinear Calderón–Zygmund operators in generalized Morrey spaces. J. Inequal. Appl. 2017, Article ID 48 (2017)

  12. 12.

    Coifman, R.R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975)

  13. 13.

    Bui, T.A., Duong, X.T.: Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers. Bull. Sci. Math. 137(1), 63–75 (2013)

  14. 14.

    Grafakos, L., Kalton, N.: Multilinear Calderón–Zygmund operators on Hardy spaces. Collect. Math. 52, 169–179 (2001)

  15. 15.

    Grafakos, L., Torres, R.H.: Multilinear Calderón–Zygmund theory. Adv. Math. 165(1), 124–164 (2002)

  16. 16.

    Huang, A.W., Xu, J.S.: Multilinear singular integral and commutators in variable exponent Lebesgue space. Appl. Math. J. Chin. Univ. Ser. B 25(1), 69–77 (2010)

  17. 17.

    Pérez, C., Torres, R.H.: Sharp maximal function estimates for multilinear singular integrals. Contemp. Math. 320, 323–331 (2003)

  18. 18.

    Xu, J.S.: Boundedness of multilinear singular integrals for non-doubling measures. J. Math. Anal. Appl. 327(1), 471–480 (2007)

  19. 19.

    Lu, Y., Zhu, Y.P.: Boundedness of multilinear Calderón–Zygmund singular operators on Morrey–Herz spaces with variable exponents. Acta Math. Sin. 30(7), 1180–1194 (2014)

  20. 20.

    Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.J.: Weighted norm inequalities for the maximal operator on variable Lebesgue spaces. J. Math. Anal. Appl. 394(2), 744–760 (2012)

  21. 21.

    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)

  22. 22.

    Cruz-Uribe, D., Diening, L., Hästö, P.: The maximal operator on weighted variable Lebesgue spaces. Fract. Calc. Appl. Anal. 14(3), 361–374 (2011)

  23. 23.

    Cruz-Uribe, D., Wang, L.A.: Extrapolation and weighted norm inequalities in the variable Lebesgue spaces. Trans. Am. Math. Soc. 369(2), 1205–1235 (2017)

  24. 24.

    Izuki, M.: Remarks on Muckenhoupt weights with variable exponent. J. Anal. Appl. 11(1), 27–41 (2013)

  25. 25.

    Izuki, M., Nakai, E., Sawano, Y.: Wavelet characterization and modular inequalities for weighted Lebesgue spaces with variable exponent. Ann. Acad. Sci. Fenn., Math. 40(2), 551–571 (2015)

  26. 26.

    Li, X.W., Yang, D.C.: Boundedness of some sublinear operators on Herz spaces. Ill. J. Math. 40(3), 484–501 (1996)

  27. 27.

    Lu, S.Z., Yang, D.C., Hu, G.E.: Herz Type Spaces and Their Applications. Science Press, Beijing (2008)

  28. 28.

    Almeida, A., Hästö, P.: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258(5), 1628–1655 (2010)

  29. 29.

    Izuki, M., Noi, T.: Two weighted Herz spaces with variable exponents. Bull. Malays. Math. Sci. Soc. (2018). https://doi.org/10.1007/s40840-018-0671-4

  30. 30.

    Izuki, M., Noi, T.: Boundedness of fractional integrals on weighted Herz spaces with variable exponent. J. Inequal. Appl. 2016, Article ID 199 (2016)

  31. 31.

    Izuki, M., Noi, T.: An intrinsic square function on weighted Herz spaces with variable exponent. J. Math. Inequal. 11(3), 799–816 (2016)

  32. 32.

    Sawano, Y.: Theory of Besov Spaces. Springer, Singapore (2018)

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Acknowledgements

The authors would like to thank the referees for their careful reading and suggestions.

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Funding

The work is partially supported by Hainan Province Natural Science Foundation of China (2018CXTD338) and the National Natural Science Foundation of China (Grant No. 11761026 and 11761027).

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All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Correspondence to Jingshi Xu.

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Keywords

  • Bilinear Calderón–Zygmund operator
  • Muckenhoupt weight
  • Variable exponent
  • Herz–Morrey space