Open Access

Boundedness of rough singular integral operators and commutators on Morrey-Herz spaces with variable exponents

Journal of Inequalities and Applications20162016:222

https://doi.org/10.1186/s13660-016-1161-6

Received: 3 June 2016

Accepted: 2 September 2016

Published: 15 September 2016

Abstract

By decomposing functions, we establish some boundedness results for some rough singular integrals on the homogeneous Morrey-Herz spaces \(M\dot{K}_{q, p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\), where the two main indices are variable. The corresponding results as regards their commutators are also considered.

Keywords

variable exponent Morrey-Herz spaces commutators singular integrals

MSC

42B20 42B25

1 Introduction

In recent years, function spaces with variable exponents have been intensively studied; see [16] for example. The origin of such spaces is the study of PDE with non-standard growth conditions, fluid dynamics and image restoration; see [79]. By virtue of the fundamental work [10] by Kováčik and Rákosník appearing in 1991, the Lebesgue spaces and various other function spaces have been investigated in the variable exponent setting. Meanwhile, the boundedness of some classical operators, such as the Hardy-Littlewood maximal operator, singular integrals and commutators, has been proved on these spaces; see [1119] and the references therein.

Herz spaces have been playing a central role in harmonic analysis. After they were introduced in [20], the theory of these spaces had a remarkable development in part due to its useful applications. For instance, they are good substitutes of the ordinary Hardy spaces when considering the boundedness of non-translation invariant singular integral operators, they also appear in the characterization of multiplier on Hardy spaces and in the regularity theory for elliptic and parabolic equations in divergence form; see [2123] for example.

One of the important problems on Herz spaces is the boundedness of sublinear operators satisfying the size condition
$$ \bigl|Tf(x) \bigr|\lesssim \int_{{\Bbb {R}}^{n}}\frac{|f(y)|}{|x-y|^{n}}\,dy, \quad x\notin \operatorname{supp} {f}, $$
(1.1)
for integrable and compactly supported functions f. We mention that condition (1.1) was initially studied by Soria and Weiss [24] and it is satisfied by several classical operators, such as Calderón-Zymund operators, the Carleson maximal operator and the Hardy-Littlewood maximal operator. Hernández, Li, Lu, and Yang [2527] proved that if a sublinear operator T is bounded on \(L^{p}({\Bbb {R}}^{n})\) and satisfies the size condition (1.1), then T is bounded on the homogeneous Herz spaces \(\dot{K}_{p}^{\alpha,q}({\Bbb{ R}}^{n})\) and on the non-homogeneous Herz spaces \(K_{p}^{\alpha,q}({\Bbb{ R}}^{n})\). This result is extended to the generalized Herz spaces \(\dot{K}_{p(\cdot)}^{\alpha,q}({\Bbb{ R}}^{n})\) and \(K_{p(\cdot)}^{\alpha,q}({\Bbb{ R}}^{n})\) with variable exponent \(p(\cdot)\) by Izuki [28]. In 2012, Almeida and Drihem [11] made a further step and gave boundedness results for T on \(\dot{K}_{p(\cdot)}^{\alpha(\cdot),q}({\Bbb{ R}}^{n})\) and \(K_{p(\cdot)}^{\alpha(\cdot),q}({\Bbb{ R}}^{n})\), where the exponent α is variable as well.
Denote by \(S^{n-1}\) the unit sphere in \({\Bbb {R}}^{n}\) \((n \ge2)\) with normalized Lebesgue measure . Let \(\Omega\in L^{s}(S^{n-1})\) for some \(s\in(1,\infty]\) be homogeneous of degree zero. In this paper, we consider sublinear operators satisfying the size condition
$$ \bigl|T_{\Omega}f(x) \bigr|\lesssim \int_{{\Bbb {R}}^{n}}\frac{|\Omega (x-y)|}{|x-y|^{n}}\bigl|f(y)\bigr|\,dy, \quad x\notin \operatorname{supp} {f}, $$
(1.2)
for integrable and compactly supported functions f and commutators defined by
$$ [b,T_{\Omega}]f(x):=b(x)T_{\Omega}f(x)-T_{\Omega}(bf) (x), \quad b\in{BMO}\bigl({\Bbb {R}}^{n}\bigr). $$
(1.3)

Lu et al. first proved the boundedness for both \(T_{\Omega}\) and \([b,T_{\Omega}]\) on Herz spaces and Morrey-Herz spaces with constant exponent; see [29, 30]. Motivated by the work of Lu [29] and Almeida [11], we shall generalize these results to the case of variable exponent and also consider the boundedness on Morrey-Herz spaces with variable exponent. Our approach is mainly based on some properties of variable exponent and BMO norms obtained by the author [31].

For brevity, we denote \(B:=\{y\in{\Bbb {R}}^{n}:|x-y|< r\}\). \(f_{B}\) stands for the integral average of f on B, i.e. \(f_{B}=\frac{1}{|B|}\int_{B}f(x)\,dx\). \(p'(\cdot)\) means the conjugate exponent \(1/p(\cdot)+1/p'(\cdot)=1\). C denotes a positive constant, which may have different values even in the same line. \(f\lesssim g\) means that \(f\leq Cg\), and \(f\thickapprox g\) means \(f\lesssim g\lesssim f\).

2 Preliminaries and lemmas

Let \(E\subset{\Bbb{ R}}^{n}\) with the Lebesgue measure \(|E|>0\), \(p(\cdot): E\rightarrow[1,\infty)\) be a measurable function. The Lebesgue space with variable exponent \(L^{p(\cdot)}(E)\) is defined by
$$L^{p(\cdot)}(E)= \biggl\{ f \mbox{ is measurable}: \int_{E} \biggl(\frac{|f(x)|}{\lambda} \biggr)^{p(x)}\,dx< \infty \mbox{ for some constant }\lambda>0 \biggr\} . $$
This is a Banach space with the Luxemburg-Nakano norm
$$\|f\|_{L^{p(\cdot)}(E)}=\inf \biggl\{ \lambda>0: \int_{E} \biggl(\frac {|f(x)|}{\lambda} \biggr)^{p(x)}\,dx \leq1 \biggr\} . $$
The space with variable exponent \(L^{p(\cdot)}_{\mathrm{loc}}(E)\) is defined by
$$L^{p(\cdot)}_{\mathrm{loc}}(E)=\bigl\{ f: f\in L^{p(\cdot)}(K) \mbox{ for all compact subsets }K\subset E\bigr\} . $$
We denote
$$\begin{aligned}& p_{-}=\operatorname{ess\,inf}\bigl\{ p(x): x\in E\bigr\} ,\qquad p_{+}= \operatorname{ess\,sup}\bigl\{ p(x): x\in E\bigr\} , \\& \mathscr{P}(E)=\bigl\{ p(\cdot): p_{-}>1 \mbox{ and } p_{+}< \infty\bigr\} , \end{aligned}$$
and
$$\mathscr{B}(E)=\bigl\{ p(\cdot)\in\mathscr{P}(E): M \mbox{ is bounded on }L^{p(\cdot)}(E)\bigr\} , $$
where the Hardy-Littlewood maximal operator M is defined by
$$Mf(x)=\sup_{r>0}r^{-n} \int_{B(x,r)\cap E}\bigl\vert f(y)\bigr\vert \,dy. $$
A function \(\alpha(\cdot):{\Bbb{ R}}^{n} \rightarrow\Bbb {R}\) is called log-Hölder continuous at the origin (or has a log decay at the origin), if there exists a constant \(C_{\log}>0\) such that
$$\bigl\vert \alpha(x)-\alpha(0)\bigr\vert \leq\frac{C_{\log}}{\log(e+1/|x|)} $$
for all \(x\in{\Bbb{R}}^{n}\). If, for some \(\alpha_{\infty}\in{\Bbb{ R}}\) and \(C_{\log}>0\), we have
$$\bigl\vert \alpha(x)-\alpha_{\infty}\bigr\vert \leq\frac{C_{\log}}{\log(e+|x|)} $$
for all \(x\in{\Bbb{ R}}^{n}\), then \(\alpha(\cdot)\) is called log-Hölder continuous at infinity (or has a log decay at infinity).

By \(\mathscr{P}_{0}^{\log}({\Bbb{ R}}^{n})\) and \(\mathscr{P}_{\infty}^{\log}({\Bbb{ R}}^{n})\) we denote the class of all exponents \(p\in\mathscr{P}({\Bbb{ R}}^{n})\) which have a log decay at the origin and at infinity, respectively. It is worth noting that if \(p(\cdot)\in{\mathscr{P}_{0}^{\log}({\Bbb{ R}}^{n})\cap\mathscr{P}_{\infty}^{\log}({\Bbb{ R}}^{n})}\), then we have \(p(\cdot)\in\mathscr{B}({\Bbb{ R}}^{n})\); see [31] or [32].

Let \(B_{k} = \{x \in{\mathbb {R}} ^{n}: |x| \leq2^{k}\}\), \(R_{k} = B_{k} \backslash B_{k-1}\) and \(\chi_{k} = \chi_{R_{k}}\) be the characteristic function of the set \(R_{k}\) for \(k\in\mathbb{Z}\). Almeida and Direhem [11] first introduced the following Herz spaces with variable exponents.

Definition 2.1

Let \(0< q\leq\infty\), \(p(\cdot)\in \mathscr{P}({\Bbb{ R}}^{n})\) and \(\alpha(\cdot):{\Bbb{ R}}^{n} \rightarrow\Bbb {R}\) with \(\alpha\in L^{\infty}({\Bbb{ R}}^{n})\).
  1. (1)
    The homogeneous Herz space \(\dot{K}_{p(\cdot)}^{\alpha(\cdot),q}({\Bbb{ R}}^{n})\) is defined as the set of all \(f\in L_{\mathrm{loc}}^{p(\cdot)}({{\Bbb{ R}}^{n}}\backslash\{0\})\) such that
    $$\|f\|_{{\dot{K}_{p(\cdot)}^{\alpha(\cdot),q}}({\Bbb{ R}}^{n})}:= \biggl(\sum_{k\in\mathbb{Z}} \bigl\| 2^{k\alpha(\cdot)} f\chi_{k}\bigr\| ^{q}_{L^{p(\cdot)}({\Bbb{ R}}^{n})} \biggr)^{1/q}< \infty. $$
     
  2. (2)
    The non-homogeneous Herz space \(K_{p(\cdot)}^{\alpha(\cdot),q}({\Bbb{ R}}^{n})\) consists of all \(f\in L_{\mathrm{loc}}^{p(\cdot)}({{\Bbb{ R}}^{n}})\) such that
    $$\|f\|_{{K_{p(\cdot)}^{\alpha(\cdot),q}}({\Bbb{ R}}^{n})}:= \|f\chi_{B_{0}}\|_{L^{p(\cdot)}({\Bbb{ R}}^{n})}+ \biggl(\sum _{k\geq 1}\bigl\| 2^{k\alpha(\cdot)} f\chi_{k} \bigr\| ^{q}_{L^{p(\cdot)}({\Bbb{ R}}^{n})} \biggr)^{1/q}< \infty, $$
    with the usual modification when \(q=\infty\).
     

Definition 2.2

Let \(0< q\leq\infty\), \(p(\cdot)\in \mathscr{P}({\Bbb{ R}}^{n})\), \(0\leq\lambda<\infty\) and \(\alpha(\cdot):{\Bbb{ R}}^{n} \rightarrow\Bbb {R}\) with \(\alpha\in L^{\infty}({\Bbb{ R}}^{n})\). The homogeneous Morrey-Herz space \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\) is defined as the set of all \(f\in L_{\mathrm{loc}}^{p(\cdot)}({{\Bbb{ R}}^{n}}\backslash\{0\})\) such that
$$\|f\|_{{M\dot{K}_{q, p(\cdot)}^{\alpha(\cdot),\lambda}}({\Bbb{ R}}^{n})}:= \sup_{L\in\mathbb{Z}}2^{-L\lambda} \Biggl( \sum_{k=-\infty}^{L}\bigl\| 2^{k\alpha(\cdot)} f \chi_{k}\bigr\| ^{q}_{L^{p(\cdot)}({\Bbb{ R}}^{n})} \Biggr)^{1/q}< \infty, $$
with the usual modification when \(q=\infty\).

Remark 2.3

It is easy to see that \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),0}({\Bbb{ R}}^{n})=\dot{K}_{p(\cdot)}^{\alpha(\cdot),q}({\Bbb{ R}}^{n})\). If \(\alpha(\cdot)\) is constant, then \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})= M\dot{K}_{q,p(\cdot)}^{\alpha,\lambda}({\Bbb{ R}}^{n})\), which is first defined by Izuki [33]. If both \(\alpha(\cdot)\) and \(p(\cdot)\) are constant, then \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),0}({\Bbb{ R}}^{n})=\dot{K}_{q}^{\alpha,p}({\Bbb{ R}}^{n})\) are classical Herz spaces in [22], and \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})=M\dot{K}_{p,q}^{\alpha,\lambda}({\Bbb{ R}}^{n})\) are classical Morrey-Herz spaces in [30].

In [32, 34], Lu and Zhu obtained the following result:

Proposition 2.4

Let \(0< q\leq\infty\), \(p(\cdot)\in \mathscr{P}({\Bbb{ R}}^{n})\), \(0\leq\lambda<\infty\) and \(\alpha\in L^{\infty}({\Bbb{ R}}^{n})\). If α is log-Hölder continuous both at origin and at infinity, then
$$\begin{aligned}& \|f\|_{{M\dot{K}_{q, p(\cdot)}^{\alpha(\cdot),\lambda}}({\Bbb{ R}}^{n})}\approx\max \Biggl\{ \sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda} \Biggl(\sum_{k=-\infty}^{L}2^{k\alpha(0)q}\| f \chi_{k}\|^{q}_{L^{p(\cdot)}({\Bbb{ R}}^{n})} \Biggr)^{1/q}, \\& \sup_{L\geq 0,L\in\mathbb{Z}} \Biggl[2^{-L\lambda} \Biggl(\sum _{k=-\infty }^{-1}2^{k\alpha(0)q}\| f\chi_{k} \|^{q}_{L^{p(\cdot)}({\Bbb{ R}}^{n})} \Biggr)^{1/q}+2^{-L\lambda} \Biggl( \sum_{k=0}^{L}2^{k\alpha _{\infty} q}\| f \chi_{k}\|^{q}_{L^{p(\cdot)}({\Bbb{ R}}^{n})} \Biggr)^{1/q} \Biggr] \Biggr\} . \end{aligned}$$

Before stating our main results, we introduce some key lemmas which will be used later.

Lemma 2.5

([10]) (Generalized Hölder inequality)

Let \(p(\cdot)\in\mathscr{P}({\Bbb{ R}}^{n})\), if \(f\in L^{p(\cdot)}{({\Bbb {R}}^{n})}\) and \(g\in L^{p'(\cdot)}{({\Bbb {R}}^{n})}\), then
$$\int_{{\Bbb {R}}^{n}}\bigl|f(x)g(x)\bigr|\,dx\leq r_{p}\|f\|_{L^{p(\cdot)}({\Bbb {R}}^{n})} \| g\|_{L^{p'(\cdot)}({\Bbb {R}}^{n})}, $$
where \(r_{p}=1+1/p_{-}-1/p_{+}\).

Lemma 2.6

([35])

Let \(\Omega\in L^{s}(S^{n-1})\), \(s\in [1,\infty]\). If \(a>0\), \(d\in(0, s]\) and \(-n+\frac{(n-1)d}{s}<\nu<\infty\), then
$$\biggl( \int_{|y|\leq a|x|}|y|^{\nu}\bigl|\Omega(x-y)\bigr|^{d}\,dy \biggr)^{1/d} \lesssim\|\Omega\|_{L^{s}(S^{n-1})}|x|^{(\nu+n)/d}. $$

Lemma 2.7

([36])

Let \(p(\cdot)\in\mathscr{P}({\Bbb{ R}}^{n})\). If \(q\in(p_{+}, \infty)\) and \(\frac{1}{p(x)}=\frac{1}{\widetilde{q}(x)}+\frac{1}{q} \) (\(x\in {\Bbb{ R}}^{n}\)), then we have
$$\|fg\|_{L^{p(\cdot)}({\Bbb {R}}^{n})} \lesssim\|f\|_{L^{\widetilde{q}(\cdot)}({\Bbb {R}}^{n})}\|g\|_{L^{q}({\Bbb {R}}^{n})} $$
for all measurable functions f and g.

We remark that Lemmas 2.8-2.10 were shown in Izuki [14, 31], and Lemma 2.11 was considered by Almeida and Drihem in [11].

Lemma 2.8

Let \(p(\cdot)\in\mathscr{B} ({\Bbb{ R}}^{n})\), then we see that for all balls B in \({\Bbb{ R}}^{n}\),
$$\frac{1}{|B|}\|\chi_{B}\|_{L^{p(\cdot)}({\Bbb {R}}^{n})}\|\chi_{B} \| _{L^{p'(\cdot)}({\Bbb {R}}^{n})}\lesssim1. $$

Lemma 2.9

Let \(p(\cdot)\in\mathscr{B} ({\Bbb{ R}}^{n})\), then we see that for all balls B in \({\Bbb{ R}}^{n}\) and all measurable subsets \(S\subset B\),
$$\frac{\|\chi_{S}\|_{L^{p(\cdot)}({\Bbb {R}}^{n})}}{\|\chi_{B}\|_{L^{p(\cdot )}({\Bbb {R}}^{n})}} \lesssim \biggl(\frac{|S|}{|B|} \biggr)^{\delta_{1}}, \qquad \frac{\|\chi_{S}\|_{L^{p'(\cdot)}({\Bbb {R}}^{n})}}{\|\chi_{B}\|_{L^{p'(\cdot)}({\Bbb {R}}^{n})}}\lesssim \biggl(\frac{|S|}{|B|} \biggr)^{\delta_{2}}, $$
where \(\delta_{1},\delta_{2}\) are constants with \(0<\delta_{1}, \delta_{2}<1\).

Lemma 2.10

Let \(b\in BMO({\Bbb{ R}}^{n})\), \(k>j\) (\(k,j\in \mathbb{N}\)), then we have
$$\sup_{B\subset{\Bbb{R}}^{n}} \frac{1}{\|\chi_{B}\|_{L^{p(\cdot)}({\Bbb {R}}^{n})}} \bigl\| (b-b_{B}) \chi_{B}\bigr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}\thickapprox \|b\|_{BMO} $$
and
$$\bigl\| (b-b_{B_{j}})\chi_{B_{k}}\bigr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}\lesssim (k-j)\|b\|_{BMO} \|\chi_{B_{k}}\|_{L^{p(\cdot)}({\Bbb {R}}^{n})}. $$

Lemma 2.11

Let \(\alpha\in L^{\infty}({\Bbb{ R}}^{n})\) and \(r_{1}>0\). If α is log-Hölder continuous both at origin and at infinity, then
$$r_{1}^{\alpha(x)}\lesssim r_{2}^{\alpha(y)}\times \textstyle\begin{cases} (\frac{r_{1}}{r_{2}} )^{\alpha_{+}}, &0< r_{2}\leq{{r_{1}}/2},\\ 1, &{{r_{1}}/2}< r_{2}\leq2r_{1},\\ (\frac{r_{1}}{r_{2}} )^{\alpha_{-}}, &r_{2}>2r_{1}, \end{cases} $$
for any \(x\in B(0,r_{1})\backslash B(0,r_{1}/2)\) and \(y\in B(0,r_{2})\backslash B(0,r_{2}/2)\), with the implicit constant not depending on \(x,y,r_{1}\), and \(r_{2}\).

3 Main results and their proofs

In this section, we prove the boundedness of sublinear operators \(T_{\Omega}\) satisfying the size condition (1.2) and the commutators \([b,T_{\Omega}]\) defined as in (1.3) on the homogeneous Morrey-Herz space \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\), respectively.

Our main results in this paper can be stated as follows.

Theorem 3.1

Suppose that \(p(\cdot)\in{\mathscr{P}_{0}^{\log}({\Bbb{ R}}^{n})\cap\mathscr{P}_{\infty}^{\log}({\Bbb{ R}}^{n})}\) and \(\Omega\in L^{s}(S^{n-1})\), \(s>(p')_{+}\). Let \(0<\lambda<n\), \(0< q\leq\infty\), \(-1/s<\nu<\infty\) and let \(\alpha(\cdot)\in L^{\infty}({\Bbb{ R}}^{n})\) be log-Hölder continuous both at the origin and at infinity, such that
$$ \lambda-n\delta_{1}-\nu-n/s< \alpha_{-}\leq \alpha_{+}< n \delta_{2}-\nu-n/s, $$
(3.1)
where \(0<\delta_{1}, \delta_{2}<1\) are the constants appearing in Lemma  2.9. Then every sublinear operator \(T_{\Omega}\) satisfying (1.2) which is bounded on \(L^{p(\cdot)}({\Bbb{ R}}^{n})\) is also bounded on \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\).

Theorem 3.2

Suppose that \(b\in BMO({\Bbb{ R}}^{n})\) and let \([b,T_{\Omega}]\) be defined as in (1.3). Under the assumptions in Theorem  3.1, if \([b,T_{\Omega}]\) is bounded on \(L^{p(\cdot)}({\Bbb{ R}}^{n})\), then \([b,T_{\Omega}]\) is also bounded on \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\).

Remark 3.3

According to Remark 2.3, Theorem 3.1 and Theorem 3.2 extend the corresponding results in [29] to a more generalized function space. Moreover, comparing with [11, 28], our main results generalize the integral kernel to the case of \(\Omega\in L^{s}(S^{n-1})\).

Our proofs use partially some decomposition techniques already used in [30] where the constant exponent case was studied. We consider only \(0< q<\infty\), the arguments are similar in the case \(q=\infty\).

Proof of Theorem 3.1

Let \(f\in M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\). We decompose
$$f(x)=\sum_{j=-\infty}^{\infty}f(x) \chi_{j}(x)=\sum_{j=-\infty}^{\infty} f_{j}(x). $$
Then we have
$$\begin{aligned} \bigl\| T_{\Omega}(f)\bigr\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})} ={}& \sup _{L\in\mathbb{Z}}2^{-L\lambda q}\sum_{k=-\infty}^{L} \bigl\| 2^{k\alpha(\cdot)}T_{\Omega}(f)\chi_{k}\bigr\| ^{q}_{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ \lesssim{}&\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L} \Biggl\| 2^{k\alpha(\cdot)} \Biggl(\sum _{j=-\infty}^{k-2} \bigl|T_{\Omega}(f_{j}) ( \cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \\ &{}+\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L} \Biggl\| 2^{k\alpha(\cdot)} \Biggl(\sum _{j=k-1}^{k+1} \bigl|T_{\Omega}(f_{j}) ( \cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \\ &{}+\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L} \Biggl\| 2^{k\alpha(\cdot)} \Biggl(\sum _{j=k+2}^{\infty} \bigl|T_{\Omega}(f_{j}) ( \cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \\ =:{}&U_{1} + U_{2} + U_{3}. \end{aligned}$$
First we estimate \(U_{1}\). Noting that if \(x\in R_{k}, y\in R_{j}\) and \(j\leq{k-2}\), then \(|x-y|\approx|x|\approx2^{k}\), by Lemma 2.11, we get
$$\begin{aligned} 2^{k\alpha(x)} \Biggl(\sum_{j=-\infty}^{k-2} \bigl\vert T_{\Omega}(f_{j}) (x)\bigr\vert \Biggr) \chi_{k}(x) &\lesssim\sum_{j=-\infty}^{k-2}2^{-kn} \int_{R_{j}}2^{k\alpha(x)}\bigl\vert \Omega (x-y)\bigr\vert \bigl\vert f_{j}(y)\bigr\vert \,dy\cdot\chi_{k}(x) \\ &\lesssim\sum_{j=-\infty}^{k-2}2^{-kn}2^{(k-j)\alpha_{+}} \int _{R_{j}}2^{j\alpha(y)}\bigl\vert \Omega(x-y)\bigr\vert \bigl\vert f_{j}(y)\bigr\vert \,dy\cdot\chi_{k}(x), \end{aligned}$$
which combining with Lemma 2.5 yields
$$\begin{aligned} & \Biggl\| 2^{k\alpha(\cdot)}\sum_{j=-\infty}^{k-2} \bigl|T_{\Omega}(f_{j}) (\cdot)\bigr|\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim\sum_{j=-\infty}^{k-2}2^{-kn}2^{(k-j)\alpha_{+}} \bigl\| 2^{j\alpha (\cdot)} f_{j}\bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}\bigl\| \Omega(x-\cdot) \chi_{j}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}\bigl\| \chi_{B_{k}}(\cdot) \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}. \end{aligned}$$
(3.2)
We define a variable exponent \(\widetilde{p}(\cdot)\) by \(\frac{1}{p'(x)}=\frac{1}{\widetilde{p}'(x)}+\frac{1}{s}\), since \(s>(p')_{+}\), by Lemma 2.7 and Lemma 2.6, we obtain
$$\begin{aligned} &\bigl\| \Omega(x-\cdot)\chi_{j}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim\bigl\| \Omega(x-\cdot)\chi_{j}(\cdot) \bigr\| _{{L^{s}}({\Bbb{R}}^{n})}\bigl\| \chi_{j}(\cdot) \bigr\| _{L^{\widetilde{p}'(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim 2^{-j\nu} \biggl( \int_{R_{j}}|y|^{s\nu}\bigl|\Omega(x-y)\bigr|^{s}\,dy \biggr)^{1/s}\bigl\| \chi _{B_{j}}(\cdot) \bigr\| _{L^{\widetilde{p}'(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim 2^{-j\nu}2^{k(\nu+n/s)}\|\Omega\|_{L^{s}(S^{n-1})}\bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}|B_{j}|^{-1/s}, \end{aligned}$$
(3.3)
where the last inequality is based on the fact that \(\|\chi_{B_{j}}(\cdot) \|_{L^{\widetilde{p}'(\cdot)}({\Bbb{R}}^{n})}\approx\|\chi_{B_{j}}(\cdot) \|_{L^{p'(\cdot)}({\Bbb{R}}^{n})}|B_{j}|^{-1/s}\); see [37], p.258, for the details.
From (3.2), (3.3), Lemma 2.8, and Lemma 2.9, we deduce that
$$\begin{aligned} & \Biggl\| 2^{k\alpha(\cdot)}\sum_{j=-\infty}^{k-2} \bigl|T_{\Omega}(f_{j}) (\cdot)\bigr|\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim\sum_{j=-\infty}^{k-2}2^{-kn}2^{(k-j)(\alpha_{+}+\nu+n/s)} \| \Omega\|_{L^{s}(S^{n-1})}\bigl\| 2^{j\alpha(\cdot)} f_{j}\bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}\bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}\bigl\| \chi_{B_{k}}(\cdot) \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim\sum_{j=-\infty}^{k-2}2^{(k-j)(\alpha_{+}+\nu+n/s)} \|\Omega\| _{L^{s}(S^{n-1})}\bigl\| 2^{j\alpha(\cdot)} f_{j}\bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \frac{\|\chi_{B_{j}}(\cdot) \|_{L^{p'(\cdot)}({\Bbb{R}}^{n})}}{\|\chi_{B_{k}}(\cdot) \|_{L^{p'(\cdot)}({\Bbb{R}}^{n})}} \\ &\quad\lesssim\sum_{j=-\infty}^{k-2}2^{(j-k)(n\delta_{2}-\alpha_{+}-\nu-n/s)} \bigl\| 2^{j\alpha(\cdot)} f_{j}\bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}. \end{aligned}$$
(3.4)

For convenience below we put \(\sigma=n\delta_{2}-\alpha_{+}-\nu-n/s\). It follows from the condition (3.1) that \(\sigma>0\). Now we can distinguish two cases as follows:

Case 1: If \(0< q\leq1\), using the well-known inequality
$$ \Biggl(\sum_{j=1}^{\infty}a_{j} \Biggr)^{q}\leq \sum_{j=1}^{\infty}a_{j}^{q} \quad (a_{j}>0, j=1,2,\ldots), $$
(3.5)
we have
$$\begin{aligned} U_{1} &\lesssim\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L}\sum _{j=-\infty}^{k-2} 2^{(j-k)\sigma q}\bigl\| 2^{j\alpha(\cdot)}f_{j} \bigr\| ^{q}_{L^{p(\cdot)}({\Bbb {R}}^{n})} \\ &\lesssim\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{j=-\infty}^{L-2}\bigl\| 2^{j\alpha(\cdot)}f_{j}\bigr\| ^{q}_{L^{p(\cdot)}({\Bbb {R}}^{n})}\sum_{k=j+2}^{L}2^{(j-k)\sigma q} \\ &\lesssim\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{j=-\infty}^{L-2}\bigl\| 2^{j\alpha(\cdot)}f_{j}\bigr\| ^{q}_{L^{p(\cdot)}({\Bbb {R}}^{n})} \\ &\lesssim\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})}. \end{aligned}$$
Case 2: If \(1< q<\infty\), by Hölder’s inequality, we have
$$\begin{aligned} U_{1} &\lesssim\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L} \Biggl(\sum _{j=-\infty}^{k-2} 2^{(j-k)\sigma q/2}\bigl\| 2^{j\alpha(\cdot)}f_{j} \bigr\| ^{q}_{L^{p(\cdot)}({\Bbb {R}}^{n})} \Biggr) \Biggl(\sum _{j=-\infty}^{k-2}2^{(j-k)\sigma q'/2} \Biggr)^{q/q'} \\ &\lesssim\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L} \Biggl(\sum_{j=-\infty}^{k-2} 2^{(j-k)\sigma q/2}\bigl\| 2^{j\alpha(\cdot)}f_{j}\bigr\| ^{q}_{L^{p(\cdot)}({\Bbb {R}}^{n})} \Biggr) \\ &\lesssim\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{j=-\infty}^{L-2} \bigl\| 2^{j\alpha(\cdot)}f_{j} \bigr\| ^{q}_{L^{p(\cdot)}({\Bbb {R}}^{n})}\sum_{k=j+2}^{L}2^{(j-k)\sigma q/2} \\ &\lesssim\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{j=-\infty}^{L-2} \bigl\| 2^{j\alpha(\cdot)}f_{j} \bigr\| ^{q}_{L^{p(\cdot)}({\Bbb {R}}^{n})} \\ &\lesssim\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})}. \end{aligned}$$
Next we estimate \(U_{2}\). By Proposition 2.4 and the hypothesis \(T_{\Omega}\) is bounded on \(L^{p(\cdot)}({\Bbb{ R}}^{n})\), so that
$$\begin{aligned} U_{2}\approx{}&\max \Biggl\{ \sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q} \sum_{k=-\infty}^{L} \Biggl\| 2^{k\alpha(0)} \Biggl( \sum_{j=k-1}^{k+1} \bigl|T_{\Omega}(f_{j}) (\cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q}, \\ &{}\sup_{L\geq 0,L\in\mathbb{Z}} \Biggl[2^{-L\lambda q}\sum _{k=-\infty}^{-1} \Biggl\| 2^{k\alpha(0)} \Biggl(\sum _{j=k-1}^{k+1} \bigl|T_{\Omega}(f_{j}) ( \cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \\ &{}+2^{-L\lambda q}\sum_{k=0}^{L} \Biggl\| 2^{k\alpha_{\infty}} \Biggl(\sum_{j=k-1}^{k+1} \bigl|T_{\Omega}(f_{j}) (\cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \Biggr] \Biggr\} \\ \lesssim{}&\max \Biggl\{ \sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L} \bigl\| 2^{k\alpha(0)} |f \chi_{k}| \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}^{q}, \\ &{}\sup_{L\geq 0,L\in\mathbb{Z}} \Biggl[2^{-L\lambda q}\sum _{k=-\infty}^{-1} \bigl\| 2^{k\alpha(0)}|f\chi_{k}| \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}^{q} +2^{-L\lambda q}\sum _{k=0}^{L} \bigl\| 2^{k\alpha_{\infty}}|f\chi_{k}| \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}^{q} \Biggr] \Biggr\} \\ \lesssim{}&\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})}. \end{aligned}$$
For \(U_{3}\), once again by Proposition 2.4, we have
$$\begin{aligned} U_{3}\approx{}&\max \Biggl\{ \sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q} \sum_{k=-\infty}^{L} \Biggl\| 2^{k\alpha(0)} \Biggl( \sum_{j=k+2}^{\infty} \bigl|T_{\Omega}(f_{j}) (\cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q}, \\ &{}\sup_{L\geq 0,L\in\mathbb{Z}} \Biggl[2^{-L\lambda q}\sum _{k=-\infty}^{-1} \Biggl\| 2^{k\alpha(0)} \Biggl(\sum _{j=k+2}^{\infty} \bigl|T_{\Omega}(f_{j}) ( \cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \\ &{}+2^{-L\lambda q}\sum_{k=0}^{L} \Biggl\| 2^{k\alpha_{\infty}} \Biggl(\sum_{j=k+2}^{\infty} \bigl|T_{\Omega}(f_{j}) (\cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \Biggr] \Biggr\} \\ =:{}& \max\{I,J\}. \end{aligned}$$
(3.6)
To estimate I, observe that if \(x\in R_{k}, y\in R_{j}\) and \(j\geq {k+2}\), then \(|x-y|\approx|y|\approx2^{j}\). We apply Lemma 2.5 and obtain
$$\begin{aligned} \bigl|T_{\Omega}(f_{j}) (x)\bigr|&\lesssim2^{-jn} \int_{R_{j}}\bigl|\Omega(x-y)\bigr|\bigl|f_{j}(y)\bigr|\,dy \\ &\lesssim 2^{-jn}\|f_{j}\|_{L^{p(\cdot)}({\Bbb{R}}^{n})}\bigl\| \Omega(x-\cdot) \chi_{j}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}. \end{aligned}$$
(3.7)
An application of (3.7), (3.3), and Lemma 2.9 gives
$$\begin{aligned} & \Biggl\| \sum_{j=k+2}^{\infty} \bigl|T_{\Omega}(f_{j}) (\cdot)\bigr|\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim\sum_{j=k+2}^{\infty}2^{-jn} \|f_{j}\|_{L^{p(\cdot)}({\Bbb {R}}^{n})}\bigl\| \Omega(x-\cdot)\chi_{j}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}\bigl\| \chi_{B_{k}}(\cdot) \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim\sum_{j=k+2}^{\infty}2^{(k-j)(\nu+n/s)} \|\Omega\| _{L^{s}(S^{n-1})}\|f_{j}\|_{L^{p(\cdot)}({\Bbb{R}}^{n})} \frac{\|\chi_{B_{k}}(\cdot) \|_{L^{p(\cdot)}({\Bbb{R}}^{n})}}{\|\chi_{B_{j}}(\cdot) \|_{L^{p(\cdot)}({\Bbb{R}}^{n})}} \\ &\quad\lesssim\sum_{j=k+2}^{\infty}2^{(k-j)(n\delta_{1}+\nu+n/s)} \|f_{j}\|_{L^{p(\cdot)}({\Bbb{R}}^{n})}. \end{aligned}$$
(3.8)
In the sequel, we put \(\eta=n\delta_{1}+\nu+n/s\) for short. If \(0< q\leq1\), in view of \(\eta>0\), from (3.6), (3.8), and (3.5), we conclude that
$$\begin{aligned} I\lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L}2^{k\alpha(0)q}\sum _{j=k+2}^{\infty}2^{(k-j)\eta q}\|f_{j} \|^{q}_{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ \lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L}2^{k\alpha(0)q}\sum _{j=k+2}^{L-1}2^{(k-j)\eta q}\|f_{j} \|^{q}_{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ &{} +\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L}2^{k\alpha(0)q}\sum _{j=L}^{\infty }2^{(k-j)\eta q}\|f_{j} \|^{q}_{L^{p(\cdot)}({\Bbb{R}}^{n})} =:I_{1}+I_{2}. \end{aligned}$$
For \(I_{1}\), noting that \(\eta+\alpha(0)>\eta+\alpha_{-}>0\), hence we have
$$\begin{aligned} I_{1}&\lesssim\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{j=-\infty}^{L-1}2^{j\alpha(0)q}\|f_{j} \|^{q}_{L^{p(\cdot )}({\Bbb{R}}^{n})} \sum_{k=-\infty}^{j-2}2^{(k-j)(\eta+\alpha(0))q} \\ &\lesssim\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{j=-\infty}^{L-1}2^{j\alpha(0)q}\|f_{j} \|^{q}_{L^{p(\cdot )}({\Bbb{R}}^{n})} \lesssim\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})}. \end{aligned}$$
For \(I_{2}\), since \(\lambda-\eta-\alpha(0)<\lambda-\eta-\alpha_{-}<0\), we get
$$\begin{aligned} I_{2}\lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L}2^{k\alpha(0)q}\sum _{j=L}^{\infty }2^{(k-j)\eta q}2^{-j\alpha(0)q}2^{j\lambda q} \\ &{}\times2^{-j\lambda q}\sum_{l=-\infty}^{j}2^{l\alpha(0)q} \|f_{l}\|^{q}_{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ \lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q} \Biggl(\sum _{k=-\infty}^{L}2^{k(\alpha(0)+\eta)q} \Biggr) \Biggl(\sum _{j=L}^{\infty}2^{j(\lambda-\eta-\alpha(0))q} \Biggr)\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})} \\ \lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q} 2^{L(\alpha(0)+\eta)q} 2^{L(\lambda-\eta-\alpha(0))q}\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})} \\ \lesssim{}&\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})}. \end{aligned}$$
If \(1< q<\infty\), we have
$$\begin{aligned} I\lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L}2^{k\alpha(0)q} \Biggl(\sum _{j=k+2}^{L}2^{(k-j)\eta }\|f_{j} \|_{L^{p(\cdot)}({\Bbb{R}}^{n})} \Biggr)^{q} \\ &{} +\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L}2^{k\alpha(0)q} \Biggl(\sum _{j=L+1}^{\infty}2^{(k-j)\eta }\|f_{j} \|_{L^{p(\cdot)}({\Bbb{R}}^{n})} \Biggr)^{q} \\ =:{}&I_{a}+I_{b}. \end{aligned}$$
For \(I_{a}\), by Hölder’s inequality, we get
$$\begin{aligned} I_{a}\lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L}\sum _{j=k+2}^{L}2^{j\alpha(0)q }\|f_{j} \|^{q}_{L^{p(\cdot)}({\Bbb{R}}^{n})}2^{(k-j)(\eta+\alpha(0))q/2} \\ &{} \times \Biggl(\sum_{j=k+2}^{L}2^{(k-j)(\eta+\alpha (0))q'/2} \Biggr)^{q/q'} \\ \lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{j=-\infty}^{L}2^{j\alpha(0)q }\|f_{j} \|^{q}_{L^{p(\cdot)}({\Bbb{R}}^{n})}\sum_{k=-\infty}^{j-2} 2^{(k-j)(\eta+\alpha(0))q/2} \\ \lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{j=-\infty}^{L}2^{j\alpha(0)q }\|f_{j} \|^{q}_{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ \lesssim{}&\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})}. \end{aligned}$$
For \(I_{b}\), as argued in \(I_{2}\), we obtain
$$\begin{aligned} I_{b}\lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L} \Biggl(\sum _{j=L+1}^{\infty }2^{j\alpha(0) }\|f_{j} \|_{L^{p(\cdot)}({\Bbb{R}}^{n})}2^{(k-j)(\eta+\alpha(0)+\lambda )/2} 2^{(k-j)(\eta+\alpha(0)-\lambda)/2} \Biggr)^{q} \\ \lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L}\sum_{j=L+1}^{\infty}2^{j\alpha(0)q } \|f_{j}\|^{q}_{L^{p(\cdot)}({\Bbb{R}}^{n})}2^{(k-j)(\eta+\alpha(0)+\lambda )q/2} \\ &{} \times \Biggl(\sum_{j=L+1}^{\infty}2^{(k-j)(\eta+\alpha (0)-\lambda)q'/2} \Biggr)^{q/q'} \\ \lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L}\sum_{j=L+1}^{\infty}2^{(k-j)(\eta +\alpha(0)+\lambda)q/2} 2^{j\lambda q} \\ &{} \times2^{-j\lambda q}\sum_{l=-\infty}^{j}2^{l\alpha(0)q} \|f_{l}\|^{q}_{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ \lesssim{}&\sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L}2^{k\lambda q}\sum _{j=L+1}^{\infty}2^{(k-j)(\eta+\alpha(0)-\lambda)q/2}\| f\|^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})} \\ \lesssim{}&\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})}. \end{aligned}$$
Hence, we arrive at the desired inequality,
$$I\lesssim I_{1}+I_{2}\lesssim\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})}. $$

We omit the estimation of J since it is essentially similar to that of I. Consequently, the proof of Theorem 3.1 is complete. □

Proof of Theorem 3.2

Let \(f\in M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\). We decompose
$$f(x)=\sum_{j=-\infty}^{\infty}f(x) \chi_{j}(x)=\sum_{j=-\infty}^{\infty} f_{j}(x). $$
Then we have
$$\begin{aligned} \bigl\| [b,T_{\Omega}](f)\bigr\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda }({\Bbb{R}}^{n})}={}& \sup _{L\in\mathbb{Z}}2^{-L\lambda q}\sum_{k=-\infty}^{L} \bigl\| 2^{k\alpha(\cdot)}[b,T_{\Omega}](f)\chi _{k}\bigr\| ^{q}_{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ \lesssim{}&\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L} \Biggl\| 2^{k\alpha(\cdot)} \Biggl(\sum _{j=-\infty}^{k-2} \bigl|[b,T_{\Omega}](f_{j}) ( \cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \\ &{}+\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L} \Biggl\| 2^{k\alpha(\cdot)} \Biggl(\sum _{j=k-1}^{k+1} \bigl|[b,T_{\Omega}](f_{j}) ( \cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \\ &{}+\sup_{L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L} \Biggl\| 2^{k\alpha(\cdot)} \Biggl(\sum _{j=k+2}^{\infty} \bigl|[b,T_{\Omega}](f_{j}) ( \cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \\ =:{}&V_{1} + V_{2} + V_{3}. \end{aligned}$$
First we estimate \(V_{1}\). Noting that \(|x-y|\approx|x|\approx2^{k}\) for \(x\in R_{k}, y\in R_{j}\) and \(j\leq{k-2}\), then, from Lemma 2.11 and Lemma 2.5, it follows that
$$\begin{aligned} &2^{k\alpha(x)}\sum_{j=-\infty}^{k-2} \bigl|[b,T_{\Omega}](f_{j}) (x)\bigr|\chi_{k}(x) \\ &\quad\lesssim\sum_{j=-\infty}^{k-2}2^{-kn} \int_{R_{j}}2^{k\alpha (x)}\bigl|b(x)-b(y)\bigr|\bigl|\Omega(x-y)\bigr|\bigl|f_{j}(y)\bigr|\,dy \cdot\chi_{k}(x) \\ &\quad\lesssim\sum_{j=-\infty}^{k-2}2^{-kn}2^{(k-j)\alpha_{+}} \int _{R_{j}}2^{j\alpha(y)}\bigl|b(x)-b(y)\bigr|\bigl|\Omega(x-y)\bigr|\bigl|f_{j}(y)\bigr|\,dy \cdot\chi_{k}(x) \\ &\quad\lesssim\sum_{j=-\infty}^{k-2}2^{-kn}2^{(k-j)\alpha_{+}} \biggl(\bigl|b(x)-b_{B_{j}}\bigr| \int_{R_{j}}2^{j\alpha(y)}\bigl|\Omega(x-y)\bigr|\bigl|f_{j}(y)\bigr|\,dy \\ &\qquad{} + \int_{R_{j}}2^{j\alpha(y)}\bigl|b_{B_{j}}-b(y)\bigr|\bigl|\Omega (x-y)\bigr|\bigl|f_{j}(y)\bigr|\,dy \biggr). \end{aligned}$$
(3.9)
Similar to (3.3), we define a variable exponent \(\widetilde{p}(\cdot)\) by \(\frac{1}{p'(x)}=\frac{1}{\widetilde{p}'(x)}+\frac{1}{s}\), since \(s>(p')_{+}\), by Lemma 2.7, Lemma 2.6, and Lemma 2.10, we get
$$\begin{aligned} &\bigl\| \bigl(b_{B_{j}}-b(\cdot)\bigr)\Omega(x-\cdot) \chi_{j}(\cdot) \bigr\| _{L^{p'(\cdot )}({\Bbb{R}}^{n})} \\ &\quad\lesssim\bigl\| \Omega(x-\cdot)\chi_{j}(\cdot) \bigr\| _{{L^{s}}({\Bbb{R}}^{n})}\bigl\| \bigl(b_{B_{j}}-b(\cdot)\bigr)\chi_{j}(\cdot) \bigr\| _{L^{\widetilde{p}'(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim\|b\|_{BMO}\bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{\widetilde{p}'(\cdot)}({\Bbb{R}}^{n})}\bigl\| \Omega(x-\cdot)\chi_{j}(\cdot) \bigr\| _{{L^{s}}({\Bbb{R}}^{n})} \\ &\quad\lesssim\|b\|_{BMO} 2^{(k-j)(\nu+n/s)}\|\Omega\|_{L^{s}(S^{n-1})}\bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim2^{(k-j)(\nu+n/s)}\bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}. \end{aligned}$$
(3.10)
Then by (3.9), Lemma 2.5, (3.3), (3.10), Lemma 2.10, and Lemma 2.9, we obtain
$$\begin{aligned} & \Biggl\| 2^{k\alpha(\cdot)}\sum_{j=-\infty}^{k-2} \bigl|[b,T_{\Omega}](f_{j}) (\cdot)\bigr|\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})} \\ &\quad\lesssim\sum_{j=-\infty}^{k-2}2^{-kn}2^{(k-j)\alpha_{+}} \bigl\| 2^{j\alpha (\cdot)} f_{j}\bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \bigl(\bigl\| \bigl(b_{B_{j}}-b( \cdot)\bigr)\chi_{k}(\cdot) \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}\bigl\| \Omega(x-\cdot) \chi_{j}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})} \\ &\qquad{} +\bigl\| \bigl(b_{B_{j}}-b(\cdot)\bigr)\Omega(x-\cdot) \chi_{j}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}\bigl\| \chi_{k}(\cdot) \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \bigr) \\ &\quad\lesssim\sum_{j=-\infty}^{k-2}2^{-kn}2^{(k-j)\alpha_{+}} \bigl\| 2^{j\alpha (\cdot)} f_{j}\bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ &\qquad{} \times \bigl((k-j)\|b\|_{BMO} 2^{(k-j)(\nu+n/s)}\|\Omega \|_{L^{s}(S^{n-1})}\bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}\bigl\| \chi_{B_{k}}(\cdot) \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ &\qquad{} +\|b\|_{BMO} 2^{(k-j)(\nu+n/s)}\|\Omega\|_{L^{s}(S^{n-1})}\bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}\bigl\| \chi_{B_{k}}(\cdot) \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \bigr) \\ &\quad\lesssim\sum_{j=-\infty}^{k-2}(k-j)2^{-kn}2^{(k-j)\alpha_{+}} \bigl\| 2^{j\alpha(\cdot)} f_{j}\bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}2^{(k-j)(\nu+n/s)}\bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}\bigl\| \chi_{B_{k}}(\cdot) \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim\sum_{j=-\infty}^{k-2}(k-j)2^{(k-j)(\alpha_{+}+\nu+n/s)} \bigl\| 2^{j\alpha(\cdot)} f_{j}\bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}\frac{\|\chi_{B_{j}}(\cdot) \|_{L^{p'(\cdot)}({\Bbb{R}}^{n})}}{\|\chi_{B_{k}}(\cdot) \|_{L^{p'(\cdot)}({\Bbb{R}}^{n})}} \\ &\quad\lesssim\sum_{j=-\infty}^{k-2}(k-j)2^{(j-k)(n\delta_{2}-\alpha_{+}-\nu-n/s)} \bigl\| 2^{j\alpha(\cdot)} f_{j}\bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}. \end{aligned}$$
(3.11)
By comparing (3.11) with (3.4), after applying the same arguments used in the estimation of \(U_{1}\) in Theorem 3.1, we can easily get, for all \(0< q<\infty\),
$$V_{1}\lesssim\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})}. $$
Next we estimate \(V_{2}\). Using Proposition 2.4 and the boundedness of \([b,T_{\Omega}]\) on \(L^{p(\cdot)}({\Bbb{ R}}^{n})\), we derive the estimate
$$\begin{aligned} V_{2}\approx{}&\max \Biggl\{ \sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q} \sum_{k=-\infty}^{L} \Biggl\| 2^{k\alpha(0)} \Biggl( \sum_{j=k-1}^{k+1} \bigl|[b,T_{\Omega}](f_{j}) (\cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q}, \\ &{}\sup_{L\geq 0,L\in\mathbb{Z}} \Biggl[2^{-L\lambda q}\sum _{k=-\infty}^{-1} \Biggl\| 2^{k\alpha(0)} \Biggl(\sum _{j=k-1}^{k+1} \bigl|[b,T_{\Omega}](f_{j}) ( \cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \\ &{} +2^{-L\lambda q}\sum_{k=0}^{L} \Biggl\| 2^{k\alpha_{\infty}} \Biggl(\sum_{j=k-1}^{k+1} \bigl|[b,T_{\Omega}](f_{j}) (\cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \Biggr] \Biggr\} \\ \lesssim{}&\max \Biggl\{ \sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q}\sum _{k=-\infty}^{L} \bigl\| 2^{k\alpha(0)} |f \chi_{k}| \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}^{q}, \\ &{}\sup_{L\geq 0,L\in\mathbb{Z}} \Biggl[2^{-L\lambda q}\sum _{k=-\infty}^{-1} \bigl\| 2^{k\alpha(0)}|f\chi_{k}| \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}^{q}+2^{-L\lambda q}\sum _{k=0}^{L} \bigl\| 2^{k\alpha_{\infty}}|f\chi_{k}| \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}^{q} \Biggr] \Biggr\} \\ \lesssim{}&\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})}. \end{aligned}$$
For \(V_{3}\), we apply Proposition 2.4 again and obtain
$$\begin{aligned} V_{3}\approx{}&\max \Biggl\{ \sup_{L< 0,L\in\mathbb{Z}}2^{-L\lambda q} \sum_{k=-\infty}^{L} \Biggl\| 2^{k\alpha(0)} \Biggl( \sum_{j=k+2}^{\infty} \bigl|[b,T_{\Omega}](f_{j}) (\cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q}, \\ &{} \sup_{L\geq 0,L\in\mathbb{Z}} \Biggl[2^{-L\lambda q}\sum _{k=-\infty}^{-1} \Biggl\| 2^{k\alpha(0)} \Biggl(\sum _{j=k+2}^{\infty} \bigl|[b,T_{\Omega}](f_{j}) ( \cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \\ &{} +2^{-L\lambda q}\sum_{k=0}^{L} \Biggl\| 2^{k\alpha_{\infty}} \Biggl(\sum_{j=k+2}^{\infty} \bigl|[b,T_{\Omega}](f_{j}) (\cdot)\bigr| \Biggr)\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb {R}}^{n})}^{q} \Biggr] \Biggr\} \\ =:{}& \max\{G,H\}. \end{aligned}$$
(3.12)
To estimate G, we note that if \(x\in R_{k}, y\in R_{j}\) and \(j\geq {k+2}\), then \(|x-y|\approx|y|\approx2^{j}\). By Lemma 2.5, we have
$$\begin{aligned} \bigl|[b,T_{\Omega}](f_{j}) (x)\bigr|\lesssim{}&2^{-jn} \int_{R_{j}}\bigl|b(x)-b(y)\bigr|\bigl|\Omega (x-y)\bigr|\bigl|f_{j}(y)\bigr|\,dy \\ \lesssim{}& 2^{-jn} \biggl(\bigl|b(x)-b_{B_{k}}\bigr| \int_{R_{j}}\bigl|\Omega(x-y)\bigr|\bigl|f_{j}(y)\bigr|\,dy \\ &{} + \int_{R_{j}}\bigl|b(y)-b_{B_{k}}\bigr|\bigl|\Omega(x-y)\bigr|\bigl|f_{j}(y)\bigr|\,dy \biggr) \\ \lesssim{}&2^{-jn} \|f_{j}\|_{L^{p(\cdot)}({\Bbb{R}}^{n})} \bigl(\bigl|b(x)-b_{B_{k}}\bigr|\bigl\| \Omega(x-\cdot )\chi_{j}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})} \\ &{} +\bigl\| \Omega(x-\cdot) \bigl(b(\cdot)-b_{B_{k}}\bigr)\chi_{j}( \cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})} \bigr). \end{aligned}$$
(3.13)
As argued for (3.10), we get actually
$$\begin{aligned} &\bigl\| \Omega(x-\cdot) \bigl(b(\cdot)-b_{B_{k}}\bigr) \chi_{j}(\cdot) \bigr\| _{L^{p'(\cdot )}({\Bbb{R}}^{n})} \\ &\quad\lesssim\bigl\| \Omega(x-\cdot)\chi_{j}(\cdot) \bigr\| _{{L^{s}}({\Bbb{R}}^{n})}\bigl\| \bigl(b(\cdot)-b_{B_{k}}\bigr)\chi_{j}(\cdot) \bigr\| _{L^{\widetilde{p}'(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim(j-k)\|b\|_{BMO}\bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{\widetilde{p}'(\cdot)}({\Bbb{R}}^{n})} \bigl\| \Omega(x-\cdot)\chi_{j}(\cdot) \bigr\| _{{L^{s}}({\Bbb{R}}^{n})} \\ &\quad\lesssim (j-k)\|b\|_{BMO}2^{-j\nu}2^{k(\nu+n/s)}\| \Omega\|_{L^{s}(S^{n-1})}\bigl\| \chi _{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)} ({\Bbb{R}}^{n})}|B_{j}|^{-1/s} \\ &\quad\lesssim(j-k)2^{(k-j)(\nu+n/s)}\bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}. \end{aligned}$$
(3.14)
From (3.13) and (3.14), it follows that
$$\begin{aligned} &\bigl\| [b,T_{\Omega}](f_{j}) (\cdot) \chi_{k}(\cdot)\bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ &\quad\lesssim 2^{-jn}\|f_{j}\|_{L^{p(\cdot)}({\Bbb{R}}^{n})} \bigl(2^{(k-j)(\nu+n/s)}\| \Omega\|_{L^{s}(S^{n-1})} \bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}\bigl\| \bigl(b(\cdot)-b_{B_{k}}\bigr)\chi_{k}( \cdot)\bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ &\qquad{} +(j-k)\|b\|_{BMO} 2^{(k-j)(\nu+n/s)}\|\Omega\|_{L^{s}(S^{n-1})} \bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}\bigl\| \chi_{B_{k}}(\cdot) \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \bigr) \\ &\quad\lesssim2^{-jn}\|f_{j}\|_{L^{p(\cdot)}({\Bbb{R}}^{n})} \bigl(2^{(k-j)(\nu +n/s)}\|\Omega\|_{L^{s}(S^{n-1})} \bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}\|b\|_{BMO}\bigl\| \chi_{B_{k}}(\cdot)\bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \\ &\qquad{} +(j-k)\|b\|_{BMO} 2^{(k-j)(\nu+n/s)}\|\Omega\|_{L^{s}(S^{n-1})} \bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}\bigl\| \chi_{B_{k}}(\cdot) \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \bigr) \\ &\quad\lesssim(j-k)\|f_{j}\|_{L^{p(\cdot)}({\Bbb{R}}^{n})}2^{-jn} 2^{(k-j)(\nu+n/s)}\bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p'(\cdot)}({\Bbb{R}}^{n})}\bigl\| \chi_{B_{j}}(\cdot) \bigr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})}\frac{\|\chi_{B_{k}}(\cdot) \|_{L^{p(\cdot)}({\Bbb{R}}^{n})}}{\|\chi_{B_{j}}(\cdot) \|_{L^{p(\cdot)}({\Bbb{R}}^{n})}} \\ &\quad\lesssim (j-k)2^{(k-j)(n\delta_{1}+\nu+n/s)} \|f_{j}\|_{L^{p(\cdot)}({\Bbb{R}}^{n})}. \end{aligned}$$
(3.15)
Hence we have
$$ \Biggl\| \sum_{j=k+2}^{\infty} \bigl|[b,T_{\Omega}](f_{j}) (\cdot)\bigr|\chi_{k}(\cdot) \Biggr\| _{L^{p(\cdot)}({\Bbb{R}}^{n})} \lesssim\sum _{j=k+2}^{\infty}(j-k)2^{(k-j)(n\delta_{1}+\nu+n/s)} \|f_{j} \|_{L^{p(\cdot)}({\Bbb{R}}^{n})}. $$
(3.16)
By comparing (3.16) with (3.8), as long as we repeat the same procedure as the estimation of I in Theorem 3.1, we can immediately get for all \(0< q<\infty\),
$$G\lesssim\| f\| ^{q}_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{R}}^{n})}. $$

We omit the estimation of H since it is essentially similar to that of G. Consequently, the proof of Theorem 3.2 is complete. □

Declarations

Acknowledgements

The authors would like to thank the referee for his very valuable comments and suggestions. Meng Qu is supported by National Natual Science Foundation of China (11471033) and University NSR Project of Anhui Province (KJ2014A087).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Applied Mathematics, School of Mathematics and Physics, Anhui Polytechnic University
(2)
Department of Applied Mathematics, School of Mathematics and Computer Science, Anhui Normal University

References

  1. Almeida, A, Hästö, P: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258, 1628-1655 (2010) MathSciNetView ArticleMATHGoogle Scholar
  2. Cruz-Uribe, D, Fiorenza, A: Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis. Birkhäuser, Basel (2013) View ArticleMATHGoogle Scholar
  3. Diening, L, Hästö, P, Roudenko, S: Function spaces of variable smoothness and integrability. J. Funct. Anal. 256, 1731-1768 (2009) MathSciNetView ArticleMATHGoogle Scholar
  4. Yan, X, Yang, D, Yuan, W, Zhuo, C: Variable weak Hardy spaces and their applications. J. Funct. Anal. (2016). doi:10.1016/j.jfa.2016.07.006 Google Scholar
  5. Yang, D, Zhuo, C, Yuan, W: Triebel-Lizorkin type spaces with variable exponents. Banach J. Math. Anal. 9, 146-202 (2015) MathSciNetView ArticleMATHGoogle Scholar
  6. Yang, D, Zhuo, C, Yuan, W: Besov-type spaces with variable smoothness and integrability. J. Funct. Anal. 269, 1840-1898 (2015) MathSciNetView ArticleMATHGoogle Scholar
  7. Chen, Y, Levine, S, Rao, R: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383-1406 (2006) MathSciNetView ArticleMATHGoogle Scholar
  8. Diening, L, Ru̇žička, M: Calderón-Zygmund operators on generalized Lebesgue spaces \(L^{p(\cdot)}\) and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197-220 (2003) MathSciNetMATHGoogle Scholar
  9. Harjulehto, P, Hästö, P, Lê, ÚV, Nuortio, M: Overview of differential equations with non-standard growth. Nonlinear Anal. 72, 4551-4574 (2010) MathSciNetView ArticleMATHGoogle Scholar
  10. Kováčik, O, Rákosník, J: On spaces \(L^{p(x)}\) and \(W^{k,p(x)}\). Czechoslov. Math. J. 41, 592-618 (1991) MATHGoogle Scholar
  11. Almeida, A, Drihem, D: Maximal, potential and singular type operators on Herz spaces with variable exponents. J. Math. Anal. Appl. 394, 781-795 (2012) MathSciNetView ArticleMATHGoogle Scholar
  12. Cruz-Uribe, D, SFO, Fiorenza, A, Martell, J, Pérez, C: The boundedness of classical operators on variable \(L^{p}\) spaces. Ann. Acad. Sci. Fenn., Math. 31, 239-264 (2006) MathSciNetMATHGoogle Scholar
  13. Diening, L, Harjulehto, P, Hästö, P, Ru̇žička, M: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011) MATHGoogle Scholar
  14. Izuki, M: Herz and amalgam spaces with variable exponent, the Haar wavelets and greediness of the wavelet system. East J. Approx. 15, 87-109 (2009) MathSciNetMATHGoogle Scholar
  15. Kokilashvili, V, Meskhi, A, Sarwar, M: Potential operators in variable exponent Lebesgue spaces: two-weight estimates. J. Inequal. Appl. 2010, Article ID 329571 (2010) MathSciNetView ArticleMATHGoogle Scholar
  16. Qu, M, Wang, J: A note on fractional integral operators on Herz spaces with variable exponent. J. Inequal. Appl. 2016, Article ID 2 (2016) MathSciNetView ArticleMATHGoogle Scholar
  17. Samko, S: Variable exponent Herz spaces. Mediterr. J. Math. 10, 2007-2025 (2013) MathSciNetView ArticleMATHGoogle Scholar
  18. Xu, J: Variable Besov and Triebel-Lizorkin spaces. Ann. Acad. Sci. Fenn., Math. 33, 511-522 (2008) MathSciNetMATHGoogle Scholar
  19. Zhang, P, Wu, J: Commutators of the fractional maximal function on variable exponent Lebesgue spaces. Czechoslov. Math. J. 64, 183-197 (2014) MathSciNetView ArticleMATHGoogle Scholar
  20. Herz, C: Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms. J. Math. Mech. 18, 283-324 (1968) MathSciNetMATHGoogle Scholar
  21. Baernstein, A, Sawyer, E: Embedding and multiplier theorems for \(H^{p}({\Bbb{R}}^{n})\). Mem. Am. Math. Soc. 53, 318 (1985) MathSciNetMATHGoogle Scholar
  22. Lu, S, Yang, D, Hu, G: Herz Type Spaces and Their Applications. Science Press, Beijing (2008) Google Scholar
  23. Ragusa, M: Homogeneous Herz spaces and regularity results. Nonlinear Anal. 71, 1909-1914 (2009) MathSciNetView ArticleMATHGoogle Scholar
  24. Soria, F, Weiss, G: A remark on singular integrals and power weights. Indiana Univ. Math. J. 43, 187-204 (1994) MathSciNetView ArticleMATHGoogle Scholar
  25. Hernández, E, Yang, D: Interpolation of Herz spaces and applications. Math. Nachr. 205, 69-87 (1999) MathSciNetView ArticleMATHGoogle Scholar
  26. Li, X, Yang, D: Boundedness of some sublinear operators on Herz spaces. Ill. J. Math. 40, 484-501 (1996) MathSciNetMATHGoogle Scholar
  27. Lu, S, Yang, D: The decomposition of the weighted Herz spaces and its application. Sci. China Ser. A 38, 147-158 (1995) MathSciNetMATHGoogle Scholar
  28. Izuki, M: Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Anal. Math. 36, 33-50 (2010) MathSciNetView ArticleMATHGoogle Scholar
  29. Lu, S, Tang, L, Yang, D: Boundedness of commutators on homogeneous Herz spaces. Sci. China Ser. A 41, 1023-1033 (1998) MathSciNetView ArticleMATHGoogle Scholar
  30. Lu, S, Xu, L: Boundedness of rough singular integral operatorson the homogeneous Morrey-Herz spaces. Hokkaido Math. J. 34, 299-314 (2005) MathSciNetView ArticleMATHGoogle Scholar
  31. Izuki, M: Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent. Rend. Circ. Mat. Palermo 59, 461-472 (2010) MathSciNetView ArticleMATHGoogle Scholar
  32. Lu, Y, Zhu, Y: Boundedness of some sublinear operators and commutators on Morrey-Herz spaces with variable exponents. Czechoslov. Math. J. 64, 969-987 (2014) MathSciNetView ArticleMATHGoogle Scholar
  33. Izuki, M: Fractional integrals on Herz-Morrey spaces with variable exponent. Hiroshima Math. J. 40, 343-355 (2010) MathSciNetMATHGoogle Scholar
  34. Lu, Y, Zhu, Y: Boundedness of multilinear Calderón-Zygmund singular operators on Morrey-Herz spaces with variable exponents. Acta Math. Sin. Engl. Ser. 30, 1180-1194 (2014) MathSciNetView ArticleMATHGoogle Scholar
  35. Muckenhoupt, B, Wheeden, R: Weighted norm inequalities for singular and fractional integrals. Trans. Am. Math. Soc. 161, 249-258 (1971) MathSciNetView ArticleMATHGoogle Scholar
  36. Nakai, E, Sawano, Y: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665-3748 (2012) MathSciNetView ArticleMATHGoogle Scholar
  37. Wang, H: Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent. Czechoslov. Math. J. 66, 251-269 (2016) MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Wang et al. 2016