Boundedness of Littlewood-Paley operators and their commutators on Herz-Morrey spaces with variable exponent

Abstract

The aim of this paper is to establish the vector-valued inequalities for Littlewood-Paley operators, including the Lusin area integrals, the Littlewood-Paley g-functions and $g_{\mu }^{\ast }$-functions, and their commutators on the Herz-Morrey spaces with variable exponent $M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)$. By applying the properties of $L^{p(\cdot )}({\mathbb{R}}^n)$ spaces and the vector-valued inequalities for Littlewood-Paley operators and their commutators generated by BMO function on $L^{p(\cdot )}({\mathbb{R}}^n)$, the boundedness of the vector-valued Littlewood-Paley operators and their commutators is obtained on $M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)$.

MSC:42B20, 42B25, 42B35.

1 Introduction

Let $\psi \in L^1({\mathbb{R}}^n)$ and satisfies

• (i) ${\int }_{{\mathbb{R}}^n}\psi (x)dx=0$,

• (ii) $|\psi (x)|⩽C{(1+|x|)}^{-n-\epsilon }$,

• (iii) $|\psi (x+y)-\psi (x)|⩽C{|y|}^{\gamma }{(1+|x|)}^{-n-\gamma -\epsilon }$, $|x|⩾2|y|$,

where C, ε, γ are all positive constants. Denote ${\psi }_t(x)=t^{-n}\psi (x/t)$ with $t>0$ and $x\in {\mathbb{R}}^n$. Given a function $f\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$, the Lusin area integral of f is defined by

$$S_{\psi ,a}(f)(x)={\left({\int }_{{\mathrm{\Gamma }}_a(x)}{|{\psi }_t\ast f(y)|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2},$$

where ${\mathrm{\Gamma }}_a(x)$ denote the usual cone of aperture one

$${\mathrm{\Gamma }}_a(x)=\{(t,y)\in {\mathbb{R}}_{+}^{n+1}:|y-x|<at,a\ge 1\}.$$

As $a=1$, we denote $S_{\psi ,a}(f)$ as $S_{\psi }(f)$.

Now let us turn to introduce the other two Littlewood-Paley operators. It is well known that the Littlewood-Paley operators include also the Littlewood-Paley g-functions and the Littlewood-Paley $g_{\mu }^{\ast }$-functions besides the Lusin area integrals. The Littlewood-Paley g-functions, which can be viewed as a ‘zero-aperture’ version of $S_{\psi }$, and $g_{\mu }^{\ast }$-functions, which can be viewed as a ‘infinite-aperture’ version of $S_{\psi }$, are defined, respectively, by

$$\begin{array}{c}g(f)(x)={\left({\int }_0^{\mathrm{\infty }}{|{\psi }_t\ast f(y)|}^2\frac{dt}{t}\right)}^{1/2},\hfill \\ g_{\mu }^{\ast }(f)(x)={\left({\int }_{{\mathbb{R}}_{+}^{n+1}}{\left(\frac{t}{t+|x-y|}\right)}^{\mu n}{|{\psi }_t\ast f(y)|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2},\mu >0.\hfill \end{array}$$

If we take ψ to be the poisson kernel, then the functions defined above are the classical Littlewood-Paley operators.

Let $b\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$, $m\ge 1$, the corresponding m-order commutators of Littlewood-Paley operators above generated by a function b are defined by

$$\begin{array}{c}[b^m,g_{\psi }](f)(x)={\left({\int }_0^{\mathrm{\infty }}\right|{\int }_{{\mathbb{R}}^n}{[b(x)-b(y)]}^m{\psi }_t(x-y)f(y)dy{|}^2\frac{dt}{t})}^{1/2},\hfill \\ [b^m,S_{\psi ,a}](f)(x)={\left({\int }_{{\mathrm{\Gamma }}_a(x)}\right|{\int }_{{\mathbb{R}}^n}{[b(x)-b(z)]}^m{\psi }_t(y-z)f(z)dz{|}^2\frac{dydt}{t^{n+1}})}^{1/2}\hfill \end{array}$$

and

$$\begin{array}{r}[b^m,g_{\mu }^{\ast }](f)(x)={\left({\int }_{{\mathbb{R}}_{+}^{n+1}}{\left(\frac{t}{t+|x-y|}\right)}^{\mu n}\right|{\int }_{{\mathbb{R}}^n}{[b(x)-b(z)]}^m{\psi }_t(y-z)f(z)dz{|}^2\frac{dydt}{t^{n+1}})}^{\frac{1}{2}},\\ \mu >0.\end{array}$$

The Littlewood-Paley operators are very important objects in the study of harmonic analysis. They play very important roles in harmonic analysis and PDE (see [1–3]), so it is natural and meaningful to consider the boundedness of Littlewood-Paley operators and their commutators. Lu and Yang investigated the behavior of Littlewood-Paley operators in the space ${CBMO}_p({\mathbb{R}}^n)$ in [4]. In 2005, Zhang and Liu proved the commutator $[b,g_{\psi }]$ is bounded on $L^p(\omega )$ (see [5]). In 2009, Xue and Ding gave the weighted estimate for Littlewood-Paley operators and their commutators (see [6]). There are some other results about Littlewood-Paley operators in [7–9].

On the other hand, Lebesgue spaces with variable exponent $L^{p(\cdot )}({\mathbb{R}}^n)$ become one of the important function spaces due to the fundamental paper [10] by Kováčik and Rákosník. In the past 20 years, the theory of these spaces has made progress rapidly, and the study of it has many applications in fluid dynamics, elasticity, calculus of variations and differential equations with non-standard growth conditions (see [11–15]). In [16], Cruz-Uribe et al. proved the extrapolation theorem which leads the boundedness of some classical operators including the commutator on $L^{p(\cdot )}({\mathbb{R}}^n)$. Karlovich and Lerner also independently obtained the boundedness of the singular integral commutators in [17]. Recently, Izuki considered the boundedness of vector-valued sub-linear operators and fractional integrals on Herz-Morrey spaces with variable exponent in [18] and [19], respectively.

Inspired by the above works, in this paper we will consider the vector-valued inequalities of the Littlewood-Paley operators and their m-order commutators on Herz-Morrey spaces with variable exponent. To do this, we need recall some definitions about the spaces with variable exponent.

Let E be a Lebesgue measurable set in ${\mathbb{R}}^n$ with measure $|E|>0$.

Definition 1.1 [10]

Let $p(\cdot ):E\to [1,\mathrm{\infty })$ be a measurable function.

The Lebesgue space with variable exponent $L^{p(\cdot )}(E)$ is defined by

$$L^{p(\cdot )}(E)=\{f\text{ is measurable}:{\int }_E{\left(\frac{|f(x)|}{\eta }\right)}^{p(x)}dx<\mathrm{\infty }\text{ for some constant }\eta >0\}.$$

The space $L_{\mathrm{loc}}^{p(\cdot )}(E)$ is defined by

$$L_{\mathrm{loc}}^{p(\cdot )}(E)=\{f\text{ is measurable}:f\in L^{p(\cdot )}(K)\text{ for all compact subsets }K\subset E\}.$$

The Lebesgue space $L^{p(\cdot )}(E)$ is a Banach space with the norm defined by

$${\parallel f\parallel }_{L^{p(\cdot )}(E)}=inf\{\eta >0:{\int }_E{\left(\frac{|f(x)|}{\eta }\right)}^{p(x)}dx\le 1\}.$$

We denote

$$p_{-}:=ess\hspace{0.17em}inf\{p(x):x\in E\},p_{+}:=ess\hspace{0.17em}sup\{p(x):x\in E\}.$$

Then $\mathcal{P}(E)$ consists of all $p(\cdot )$ satisfying $p_{-}>1$ and $p_{+}<\mathrm{\infty }$.

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

Let $B_k=\{x\in {\mathbb{R}}^n:|x|\le 2^k\}$, $C_k=B_k\mathrm{\setminus }{B}_{k-1}$, $k\in \mathbb{Z}$, ${\chi }_k={\chi }_{C_k}$.

Definition 1.2 [18]

Let $\alpha \in {\mathbb{R}}^n$, $0\le \lambda <\mathrm{\infty }$, $0<p<\mathrm{\infty }$, and $q(\cdot )\in \mathcal{P}({\mathbb{R}}^n)$. The homogeneous Herz-Morrey spaces with variable exponent $M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)$ is defined by

$$M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }\left({\mathbb{R}}^n\right)=\{f\in L_{\mathrm{loc}}^{q(\cdot )}({\mathbb{R}}^n\setminus \{0\}):{\parallel f\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}<\mathrm{\infty }\},$$

where

$${\parallel f\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}=\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\parallel f{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\right\}}^{1/p}.$$

Remark 1.1 Comparing the homogeneous Herz-Morrey spaces with variable exponent $M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)$ with the homogeneous Herz spaces with variable exponent ${\dot{K}}_{q(\cdot )}^{\alpha ,p}({\mathbb{R}}^n)$ (see [20]), where ${\dot{K}}_{q(\cdot )}^{\alpha ,p}({\mathbb{R}}^n)$ is defined by

$${\dot{K}}_{q(\cdot )}^{\alpha ,p}\left({\mathbb{R}}^n\right)=\{f\in L_{\mathrm{loc}}^{q(\cdot )}({\mathbb{R}}^n\setminus \{0\}):{\parallel f\parallel }_{{\dot{K}}_{q(\cdot )}^{\alpha ,p}({\mathbb{R}}^n)}={\left(\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{2}^{\alpha kp}{\parallel f{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\right)}^{1/p}<\mathrm{\infty }\}.$$

Obviously, $M{\dot{K}}_{p,q(\cdot )}^{\alpha ,0}({\mathbb{R}}^n)={\dot{K}}_{q(\cdot )}^{\alpha ,p}({\mathbb{R}}^n)$.

We now make some conventions. Throughout this paper, given a function f, we denote the mean value of f on E by $f_E=:\frac{1}{|E|}{\int }_{E}f(x)dx$. $p^{\prime }(\cdot )$ means the conjugate exponent of $p(\cdot )$, namely $1/p(x)+1/p^{\prime }(x)=1$ holds. C always means a positive constant independent of the main parameters and may change from one occurrence to another.

2 Preliminary lemmas

In this section, we need some conclusions which will be used in the proofs of our main results.

Lemma 2.1 [10] (Generalized Hölder’s Inequality)

Let $p(\cdot ),p_1(\cdot ),p_2(\cdot )\in \mathcal{P}({\mathbb{R}}^n)$.

• (1) For any $f\in L^{p(\cdot )}({\mathbb{R}}^n)$, $g\in L^{p^{\prime }(\cdot )}({\mathbb{R}}^n)$,

  $${\int }_{{\mathbb{R}}^n}|f(x)g(x)|dx\le C_p{\parallel f\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}{\parallel g\parallel }_{L^{p^{\prime }(\cdot )}({\mathbb{R}}^n)},$$

where $C_p=1+1/p_{-}-1/p_{+}$.

• (2) For any $f\in L^{p_1(\cdot )}({\mathbb{R}}^n)$, $g\in L^{p_2(\cdot )}({\mathbb{R}}^n)$, when $1/p(x)=1/p_1(x)+1/p_2(x)$, we have

  $${\parallel f(x)g(x)\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}\le C_{p_1,p_2}{\parallel f\parallel }_{L^{p_1(\cdot )}({\mathbb{R}}^n)}{\parallel g\parallel }_{L^{p_2(\cdot )}({\mathbb{R}}^n)},$$

where $C_{p_1,p_2}={(1+1/p_{1_{-}}-1/p_{1_{+}})}^{1/p_{-}}$.

Lemma 2.2 [17]

If $p(\cdot )\in \mathcal{B}({\mathbb{R}}^n)$, then there exist constants ${\delta }_1,{\delta }_2,C>0$, such that for all balls $B\subset {\mathbb{R}}^n$ and all measurable subsets $S\subset B$,

$$\frac{{\parallel {\chi }_B\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}}{{\parallel {\chi }_S\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}}\le C\frac{|B|}{|S|},\frac{{\parallel {\chi }_S\parallel }_{L^{p^{\prime }(\cdot )}({\mathbb{R}}^n)}}{{\parallel {\chi }_B\parallel }_{L^{p^{\prime }(\cdot )}({\mathbb{R}}^n)}}\le C{\left(\frac{|S|}{|B|}\right)}^{{\delta }_1},\frac{{\parallel {\chi }_S\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}}{{\parallel {\chi }_B\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}}\le C{\left(\frac{|S|}{|B|}\right)}^{{\delta }_2}.$$

Lemma 2.3 [18]

If $p(\cdot )\in \mathcal{B}({\mathbb{R}}^n)$, then there exist constants $C>0$, such that for all balls $B\subset {\mathbb{R}}^n$,

$$\frac{1}{|B|}{\parallel {\chi }_B\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}{\parallel {\chi }_B\parallel }_{L^{p^{\prime }(\cdot )}({\mathbb{R}}^n)}\le C.$$

Lemma 2.4 [20]

Let $b\in BMO({\mathbb{R}}^n)$, m is a positive integer, there exist constants $C>0$, such that for any $k,j\in \mathbb{Z}$ with $k>j$,

• (1) $C^{-1}{\parallel b\parallel }_{\ast }^m\le {sup}_B\frac{1}{{\parallel {\chi }_B\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}}{\parallel {(b-b_B)}^m{\chi }_B\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}\le C{\parallel b\parallel }_{\ast }^m$;

• (2) ${\parallel {(b-b_{B_j})}^m{\chi }_{B_k}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}\le C{(k-j)}^m{\parallel b\parallel }_{\ast }^m{\parallel {\chi }_{B_k}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}$.

Lemma 2.5 [21]

Given an open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$, suppose that $p(\cdot )\in \mathcal{P}(\mathrm{\Omega })$ satisfies

$$|p(x)-p(y)|\le \frac{-C}{ln(|x-y|)},x,y\in \mathrm{\Omega },|x-y|\le 1/2,$$

(2.1)

$$|p(x)-p(y)|\le \frac{C}{ln(e+|x|)},x,y\in \mathrm{\Omega },|y|\ge |x|.$$

(2.2)

Then $p(\cdot )\in \mathcal{B}(\mathrm{\Omega })$, that is, the Hardy-Littlewood maximal operator is bounded on $L^{p(\cdot )}(\mathrm{\Omega })$.

Let $A_p$ be the classical Muckenhoupt weighted class.

Lemma 2.6 [16]

• (1) Given a family ℱ and open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$, assume that for some $p_0:1<p_0<\mathrm{\infty }$, and for every $\omega \in A_{p_0}$,

  $${\int }_{\mathrm{\Omega }}f{(x)}^{p_0}\omega (x)dx\le C{\int }_{\mathrm{\Omega }}g{(x)}^{p_0}\omega (x)dx,(f_j,g_j)\in \mathcal{F}.$$

Let $p(\cdot )\in \mathcal{P}(\mathrm{\Omega })$ be such that there exists $1<p_1<p_{-}$, with ${(p(\cdot )/p_1)}^{\prime }\in \mathcal{B}(\mathrm{\Omega })$. Then for every $1<q<\mathrm{\infty }$ and sequence ${\{(f_j,g_j)\}}_j\in \mathcal{F}$,

$${\parallel {\left(\sum _j{(f_j)}^q\right)}^{1/q}\parallel }_{L^{p(\cdot )}(\mathrm{\Omega })}\le C{\parallel {\left(\sum _j{(g_j)}^q\right)}^{1/q}\parallel }_{L^{p(\cdot )}(\mathrm{\Omega })}.$$

• (2) If $p(\cdot )\in \mathcal{B}({\mathbb{R}}^n)$, then for all $1<q<\mathrm{\infty }$,

  $${\parallel {\left(\sum _j{(M{f}_j)}^q\right)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}\le C{\parallel {\left(\sum _j{(|f_j|)}^q\right)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}.$$

Lemma 2.7 [6]

Let $\psi \in L^1({\mathbb{R}}^n)$ satisfies (i)-(iii), $\mu >2$, $0<\gamma <min\{(\mu -2)n/2,\epsilon \}$, there exists a constant C and for all bounded functions f with compact support:

• (1) If $0<p<\mathrm{\infty }$, $\omega \in A_{\mathrm{\infty }}$, then

  $${\int }_{{\mathbb{R}}^n}{[g_{\mu }^{\ast }(f)(x)]}^p\omega (x)dx\le C{[\omega ]}_{A_{\mathrm{\infty }}}^p{\int }_{{\mathbb{R}}^n}{[M(f)(x)]}^p\omega (x)dx.$$

• (2) If $1<p<\mathrm{\infty }$, $\omega \in A_p$, $b\in BMO$, then

  $${\int }_{{\mathbb{R}}^n}{([b^m,g_{\mu }^{\ast }](f)(x))}^p\omega (x)dx\le C{\int }_{{\mathbb{R}}^n}{|(f)(x)|}^p\omega (x)dx.$$

Remark 2.1

• (1) If we replace $g_{\mu }^{\ast }$ with $S_{\psi ,a}$, then the results of Lemma 2.7 also hold. That is due to for any $x\in {\mathbb{R}}^n$, $\mu \ge 1$, the inequalities $S_{\psi ,a}(f)(x)\le C{g}_{\mu }^{\ast }(f)(x)$ and $[b^m,S_{\psi ,a}](f)(x)\le C[b^m,g_{\mu }^{\ast }](f)(x)$ hold.

• (2) According to the argument in [6], we may deduce that Lemma 2.7 is also suit to $g_{\psi }$.

Combining Lemmas 2.6-2.7 and Remark 2.1, we get the following conclusions.

Lemma 2.8 Let $\psi \in L^1({\mathbb{R}}^n)$ satisfies (i)-(iii). If $p(\cdot )\in \mathcal{B}({\mathbb{R}}^n)$, $1<q<\mathrm{\infty }$, then for all bounded compactly support functions $f_j$ such that ${\{f_j\}}_{j=1}^{\mathrm{\infty }}\in L^{p(\cdot )}(l^q)$, that is ${\parallel {({\sum }_j{(|f_j|)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}<\mathrm{\infty }$, the below vector-valued inequalities hold:

• (1) ${\parallel {({\sum }_j{(S_{\psi ,a}{(f)}_j)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}\le C{\parallel {({\sum }_j{(|f_j|)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}$,

• (2) ${\parallel {({\sum }_j{(g_{\psi }{(f)}_j)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}\le C{\parallel {({\sum }_j{(|f_j|)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}$,

• (3) ${\parallel {({\sum }_j{(g_{\mu }^{\ast }{(f)}_j)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}\le C{\parallel {({\sum }_j{(|f_j|)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}$,

where $\mu >2$, $0<\gamma <min\{(\mu -2)n/2,\epsilon \}$.

Lemma 2.9 Let $b\in BMO$, $\psi \in L^1({\mathbb{R}}^n)$ satisfies (i)-(iii), $m\in \mathbb{N}\mathrm{\setminus }\{0\}$. If $p(\cdot )\in \mathcal{B}({\mathbb{R}}^n)$, $1<q<\mathrm{\infty }$, then for all bounded compactly support functions $f_j$ such that ${\{f_j\}}_{j=1}^{\mathrm{\infty }}\in L^{p(\cdot )}(l^q)$, that is ${\parallel {({\sum }_j{(|f_j|)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}<\mathrm{\infty }$, the following vector-valued inequalities hold:

• (1) ${\parallel {({\sum }_j{([b^m,S_{\psi ,a}]{(f)}_j)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}\le C{\parallel {({\sum }_j{(|f_j|)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}$,

• (2) ${\parallel {({\sum }_j{([b^m,g_{\psi }]{(f)}_j)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}\le C{\parallel {({\sum }_j{(|f_j|)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}$,

• (3) ${\parallel {({\sum }_j{([b^m,g_{\mu }^{\ast }]{(f)}_j)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}\le C{\parallel {({\sum }_j{(|f_j|)}^q)}^{1/q}\parallel }_{L^{p(\cdot )}({\mathbb{R}}^n)}$,

where $\mu >2$, $0<\gamma <min\{(\mu -2)n/2,\epsilon \}$.

3 Main results and their proofs

In this section, we will establish the vector-valued inequalities of the Littlewood-Paley operators, $S_{\psi ,a}$, $g_{\psi }$, $g_{\mu }^{\ast }$, and their m-order commutators on Herz-Morrey spaces with variable exponent, respectively.

We begin with the study of the vector-valued inequalities of the Littlewood-Paley operators on $M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)$.

Theorem 3.1 Suppose that $\psi \in L^1({\mathbb{R}}^n)$ satisfies (i)-(iii), $q(\cdot )\in \mathcal{B}({\mathbb{R}}^n)$. Let $0<p<\mathrm{\infty }$, $1<r<\mathrm{\infty }$, $\lambda -n{\delta }_2<\alpha <\lambda +n{\delta }_1$, where ${\delta }_1$, ${\delta }_2$ is the constant in Lemma  2.2. Then for all function sequences ${\{f_h\}}_{h=1}^{\mathrm{\infty }}:{\parallel {\{{\sum }_h{|f_h|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}<\mathrm{\infty }$, the following vector-valued inequalities hold:

• (1) ${\parallel {\{{\sum }_h{|S_{\psi ,a}(f_h)|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\le C{\parallel {\{{\sum }_h{|f_h|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}$,

• (2) ${\parallel {\{{\sum }_h{|g_{\psi }(f_h)|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\le C{\parallel {\{{\sum }_h{|f_h|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}$,

• (3) ${\parallel {\{{\sum }_h{|g_{\mu }^{\ast }(f_h)|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\le C{\parallel {\{{\sum }_h{|f_h|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}$,

where $\mu >3+2(\epsilon +\gamma )/n$, $0<\gamma <min\{(\mu -2)n/2,\epsilon \}$, and the constant C is independent of ${\{f_h\}}_{h=1}^{\mathrm{\infty }}$.

Proof Firstly, we consider the inequality (1). For any function sequence ${\{f_h\}}_h$ satisfies ${\parallel {\parallel {\{f_h\}}_h\parallel }_{l^r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}<\mathrm{\infty }$, we write

$$f_h(x)=\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_h(x){\chi }_j(x)\triangleq \sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_h^j(x).$$

Thus

$$\begin{array}{r}{\parallel {\left\{\sum _h{|S_{\psi ,a}(f_h)|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\\ =\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\parallel {\left\{\sum _h{\left|\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{S}_{\psi ,a}\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\right\}}^{1/p}\\ \le \underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left(\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{\parallel {\left\{\sum _h{\left|S_{\psi ,a}\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ \le C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left(\sum _{j=-\mathrm{\infty }}^{k-3}{\parallel {\left\{\sum _h{\left|S_{\psi ,a}\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ +C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left(\sum _{j=k-2}^{k+2}{\parallel {\left\{\sum _h{\left|S_{\psi ,a}\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ +C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left(\sum _{j=k+3}^{\mathrm{\infty }}{\parallel {\left\{\sum _h{\left|S_{\psi ,a}\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ \triangleq D_1+D_2+D_3.\end{array}$$

For the term $D_2$, noting that $supp{f}_h^j\subset C_j$, we can easily obtain, by Lemma 2.8,

$$\begin{array}{rl}{D}_2& \le C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left(\sum _{j=k-2}^{k+2}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ \le C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\right\}}^{1/p}\\ =C{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}.\end{array}$$

We now turn to estimate $D_1$. Let ${\mathrm{\Gamma }}^{\prime }=\{y\in {\mathbb{R}}^n:|y-x|<at,y\le 2^{j+1}\}$, ${\mathrm{\Gamma }}^{″}=\{y\in {\mathbb{R}}^n:|y-x|<at,y>2^{j+1}\}$. We write

$$\begin{array}{rl}{S}_{\psi ,a}\left(f_h^j\right)(x)=& {\left({\int }_{{\mathrm{\Gamma }}_a(x)}{|{\int }_{{\mathbb{R}}^n}{t}^{-n}\psi \left(\frac{y-z}{t}\right)f_h^j(z)dz|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2}\\ \le & {\left({\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{\prime }}{|{\int }_{{\mathbb{R}}^n}{t}^{-n}\psi \left(\frac{y-z}{t}\right)f_h^j(z)dz|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2}\\ +{\left({\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{″}}{|{\int }_{{\mathbb{R}}^n}{t}^{-n}\psi \left(\frac{y-z}{t}\right)f_h^j(z)dz|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2}\\ \triangleq & I+II.\end{array}$$

For I, observe that as $a\ge 1$, $x\in C_k$, $j\le k-3$, $z\in C_j$, then we have $t>\frac{|x-y|}{a}\ge \frac{|x|-|y|}{a}>\frac{2^{k-1}-2^{j+1}}{a}\ge \frac{2^{j+1}}{a}$, and $t+|y-z|\ge \frac{|x-y|}{a}+|y-z|\ge \frac{|x-y|+|y-z|}{a}\ge \frac{|x|-|z|}{a}\ge \frac{3|x|}{4a}$. Hence, it follows from the condition (ii) that

$$\begin{array}{rl}I& \le C{\left\{{\int }_{\frac{2^{j+1}}{a}}^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{\prime }}{\left({\int }_{{\mathbb{R}}^n}|f_h^j(z)|t^{-n}{(1+\frac{|y-z|}{t})}^{-n-\epsilon }dz\right)}^2\frac{dydt}{t^{n+1}}\right\}}^{1/2}\\ \le C{\left\{{\int }_{\frac{2^{j+1}}{a}}^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{\prime }}{\left({\int }_{{\mathbb{R}}^n}|f_h^j(z)|\frac{t^{\epsilon }}{{(t+|y-z|)}^{n+\epsilon }}dz\right)}^2\frac{dydt}{t^{n+1}}\right\}}^{1/2}\\ \le C{\left(\frac{4a}{3|x|}\right)}^n{\left\{{\int }_{\frac{2^{j+1}}{a}}^{\mathrm{\infty }}{t}^{-n-1}dt{\int }_{|y|<2^{j+1}}dy\right\}}^{1/2}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}\\ \le C{a}^{3n/2}{|x|}^{-n}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.\end{array}$$

For the estimate of II, denote

$$\begin{array}{rl}II\le & C{\left\{{\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{″}}{\left({\int }_{{\mathbb{R}}^n}|f_h^j(z)|t^{-n}|\psi \left(\frac{y-z}{t}\right)-\psi \left(\frac{y}{t}\right)|dz\right)}^2\frac{dydt}{t^{n+1}}\right\}}^{1/2}\\ +C{\left\{{\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{″}}{\left({\int }_{{\mathbb{R}}^n}|f_h^j(z)|t^{-n}\left|\psi \left(\frac{y}{t}\right)\right|dz\right)}^2\frac{dydt}{t^{n+1}}\right\}}^{1/2}\\ \triangleq & I{I}_1+I{I}_2.\end{array}$$

Noting that $y>2^{j+1}\ge 2|z|$, by the condition (iii), we have

$$\begin{array}{rl}I{I}_1\le & C{\left\{{\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{″}}{\left({\int }_{{\mathbb{R}}^n}|f_h^j(z)|t^{-n}{\left(\frac{|z|}{t}\right)}^{\gamma }{(1+\frac{|y|}{t})}^{-n-\gamma -\epsilon }dz\right)}^2\frac{dydt}{t^{n+1}}\right\}}^{1/2}\\ \le & C{\left\{{\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{″}}\frac{t^{2\epsilon -n-1}}{{(t+|y|)}^{2(n+\gamma +\epsilon )}}dydt\right\}}^{1/2}{\int }_{{\mathbb{R}}^n}|f_h^j(z)|\cdot {|z|}^{\gamma }dz\\ \le & C{2}^{j\gamma }\{{\int }_0^{|x|}{\int }_{{\mathrm{\Gamma }}^{″}}\frac{t^{2\epsilon -n-1}}{{(t+|y|)}^{2(n+\gamma +\epsilon )}}dydt\\ {+{\int }_{|x|}^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{″}}\frac{t^{2\epsilon -n-1}}{{(t+|y|)}^{2(n+\gamma +\epsilon )}}dydt\}}^{1/2}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.\end{array}$$

Since $t+|y|>\frac{|x-y|}{a}+|y|\ge \frac{|x-y|+|y|}{a}\ge \frac{|x|}{a}$, then we obtain

$$\begin{array}{r}{\int }_0^{|x|}{\int }_{{\mathrm{\Gamma }}^{″}}\frac{t^{2\epsilon -n-1}}{{(t+|y|)}^{2(n+\gamma +\delta )}}dydt\\ \le C{\left(\frac{a}{|x|}\right)}^{2(n+\gamma +\epsilon )}{\int }_0^{|x|}{\int }_{|y-x|<at}{t}^{2\epsilon -n-1}dydt\\ \le C{\left(\frac{a}{|x|}\right)}^{2(n+\gamma +\epsilon )}{\int }_0^{|x|}{t}^{2\delta -n-1}{a}^n{t}^{n}dt\\ \le C{a}^{3n+2\gamma +2\epsilon }{|x|}^{-2(n+\gamma )}\end{array}$$

and

$$\begin{array}{r}{\int }_{|x|}^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{″}}\frac{t^{2\epsilon -n-1}}{{(t+|y|)}^{2(n+\gamma +\delta )}}dydt\\ \le C{\int }_{|x|}^{\mathrm{\infty }}{\int }_{|y-x|<at}{t}^{-3n-2\gamma -1}dydt\\ \le C{a}^n{\int }_{|x|}^{\mathrm{\infty }}{t}^{-2n-2\gamma -1}dydt\le C{a}^n{|x|}^{-2(n+\gamma )}.\end{array}$$

Thus, we have

$$I{I}_1\le C{a}^{3n/2+\gamma +\epsilon }{|x|}^{-(n+\gamma )}{2}^{j\gamma }{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}\le C{a}^{3n/2+\gamma +\epsilon }{|x|}^{-n}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.$$

On the other hand, by the condition (ii) and $t+|y|>\frac{|x|}{a}$, we deduce

$$\begin{array}{rl}I{I}_2\le & C{\left\{{\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{″}}{\left({\int }_{{\mathbb{R}}^n}|f_h^j(z)|t^{-n}{(1+\frac{|y|}{t})}^{-n-\epsilon }dz\right)}^2\frac{dydt}{t^{n+1}}\right\}}^{1/2}\\ \le & C{\left\{{\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{″}}\frac{t^{2\epsilon -n-1}}{{(t+|y|)}^{2(n+\epsilon )}}dydt\right\}}^{1/2}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}\\ \le & C\{{\int }_0^{|x|}{\int }_{{\mathrm{\Gamma }}^{″}}\frac{t^{2\epsilon -n-1}}{{(t+|y|)}^{2(n+\epsilon )}}dydt\\ {+{\int }_{|x|}^{|\mathrm{\infty }|}{\int }_{{\mathrm{\Gamma }}^{″}}\frac{t^{2\epsilon -n-1}}{{(t+|y|)}^{2(n+\epsilon )}}dydt\}}^{1/2}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.\end{array}$$

Thus, similar to $I{I}_1$, we get

$$I{I}_2\le C{a}^{3n/2+\epsilon }{|x|}^{-n}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.$$

Combining the estimates of I, II, we obtain

$$S_{\psi ,a}\left(f_h^j\right)(x)\le C{a}^{3n/2+\gamma +\epsilon }{|x|}^{-n}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.$$

Therefore, by using the above inequality and Minkowski’s inequality, we have

$$\begin{array}{rl}{D}_1\le & C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}\\ {\times {\left(\sum _{j=-\mathrm{\infty }}^{k-3}{\parallel {\left\{\sum _h{\left(a^{3n/2+\gamma +\epsilon }{|x|}^{-n}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}\right)}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\}}^{1/p}\\ \le & C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{(\alpha -n)kp}{\parallel {\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p{\left(\sum _{j=-\mathrm{\infty }}^{k-3}{\left\{\sum _h{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}^r\right\}}^{1/r}\right)}^p\right\}}^{1/p}\\ \le & C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{(\alpha -n)kp}{\parallel {\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p{\left(\sum _{j=-\mathrm{\infty }}^{k-3}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^1({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}.\end{array}$$

It follows from Hölder’s inequality and Lemmas 2.2-2.3 that

$$\begin{array}{rl}{D}_1\le & C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{(\alpha -n)kp}{\parallel {\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\\ {\times {\left(\sum _{j=-\mathrm{\infty }}^{k-3}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\chi }_{B_j}\parallel }_{L^{q^{\prime }(\cdot )}({\mathbb{R}}^n)}\right)}^p\}}^{1/p}\\ \le & C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{(\alpha -n)kp}\\ {\times {\{\sum _{j=-\mathrm{\infty }}^{k-3}\frac{{\parallel {\chi }_{B_j}\parallel }_{L^{q^{\prime }(\cdot )}({\mathbb{R}}^n)}}{{\parallel {\chi }_{B_k}\parallel }_{L^{q^{\prime }(\cdot )}({\mathbb{R}}^n)}}|B_k|{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\}}^p\}}^{1/p}\\ \le & C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{\alpha kp}{\left\{\sum _{j=-\mathrm{\infty }}^{k-3}{\left(\frac{|B_j|}{|B_k|}\right)}^{{\delta }_1}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right\}}^p\right\}}^{1/p}.\end{array}$$

Noting that $2^{j\alpha }{\parallel {\{{\sum }_h{|f_h^j|}^r\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\le {\{{\sum }_{l=-\mathrm{\infty }}^j{2}^{\alpha lp}{\parallel {\{{\sum }_h{|f_h^l|}^r\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\}}^{1/p}$, thus, when $\alpha <\lambda +n{\delta }_1$, we get

$$\begin{array}{rl}{D}_1\le & C\underset{k_0\in \mathbb{Z}}{sup}\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{\lambda (k-k_0)p}\right\{\sum _{j=-\mathrm{\infty }}^{k-3}{2}^{(j-k)(n{\delta }_1+\lambda -\alpha )}\\ {{\times 2^{-j\lambda }{\left(\sum _{l=-\mathrm{\infty }}^j{2}^{\alpha lp}{\parallel {\left\{\sum _h{\left|f_h^l\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\right)}^{1/p}\}}^p\}}^{1/p}\\ \le & C\underset{k_0\in \mathbb{Z}}{sup}{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{\lambda (k-k_0)p}{\left(\sum _{j=-\mathrm{\infty }}^{k-3}{2}^{(j-k)(n{\delta }_1+\lambda -\alpha )}\right)}^p\right\}}^{1/p}{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\\ \le & C{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}.\end{array}$$

Finally, let us to estimate $D_3$. By using Minkowski’s inequality, we obtain

$$\begin{array}{rl}{S}_{\psi ,a}\left(f_h^j\right)(x)=& {\left({\int }_{{\mathrm{\Gamma }}_a(x)}{|{\int }_{{\mathbb{R}}^n}{t}^{-n}\psi \left(\frac{y-z}{t}\right)f_h^j(z)dz|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2}\\ \le & {\int }_{{\mathbb{R}}^n}|f_h^j(z)|{\left({\int }_{{\mathrm{\Gamma }}_a(x)}{t}^{-3n-1}{\left|\psi \left(\frac{y-z}{t}\right)\right|}^{2}dydt\right)}^{1/2}dz\\ \le & {\int }_{{\mathbb{R}}^n}|f_h^j(z)|{\left({\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{\prime }}{t}^{-3n-1}{\left|\psi \left(\frac{y-z}{t}\right)\right|}^{2}dydt\right)}^{1/2}dz\\ +{\int }_{{\mathbb{R}}^n}|f_h^j(z)|{\left({\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{″}}{t}^{-3n-1}{\left|\psi \left(\frac{y-z}{t}\right)\right|}^{2}dydt\right)}^{1/2}dz.\end{array}$$

Let $a\ge 1$, $x\in C_k$, $j\ge k+3$, $z\in C_j\subset B_j$. Then we have $t+|y-z|\ge \frac{|x-y|}{a}+|y-z|\ge \frac{|x-y|+|y-z|}{a}\ge \frac{|z|-|x|}{a}\ge \frac{3|z|}{4a}$. Hence, it follows from the condition (ii) that

$$\begin{array}{r}{\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{\prime }}{t}^{-3n-1}{\left|\psi \left(\frac{y-z}{t}\right)\right|}^{2}dydt\\ \le {\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{\prime }}\frac{t^{2\epsilon -n-1}}{{(t+|y-z|)}^{2(n+\epsilon )}}dydt\\ \le {\int }_0^{2^{j+1}}{\int }_{{\mathrm{\Gamma }}^{\prime }}\frac{t^{2\epsilon -n-1}}{{(t+|y-z|)}^{2(n+\epsilon )}}dydt+{\int }_{2^{j+1}}^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{\prime }}\frac{t^{2\epsilon -n-1}}{{(t+|y-z|)}^{2(n+\epsilon )}}dydt\\ \le C\frac{a^{2(n+\epsilon )}}{{|z|}^{2(n+\epsilon )}}{\int }_0^{2^{j+1}}{\int }_{|x-y|<at}{t}^{2\epsilon -n-1}dydt+{\int }_{2^{j+1}}^{\mathrm{\infty }}{\int }_{y\le 2^{j+1}}{t}^{-3n-1}dydt\\ \le C{a}^{3n+2\epsilon }{2}^{-2jn}.\end{array}$$

If $y>2^{j+1}$, then $t>\frac{|x-y|}{a}\ge \frac{|x|-|y|}{a}>\frac{2^{j+1}-2^k}{a}\ge \frac{2^j}{a}$. Applying the condition (ii), we obtain

$$\begin{array}{r}{\int }_0^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{″}}{t}^{-3n-1}{\left|\psi \left(\frac{y-z}{t}\right)\right|}^{2}dydt\\ \le C{\int }_{\frac{2^j}{a}}^{\mathrm{\infty }}{\int }_{{\mathrm{\Gamma }}^{″}}{t}^{-3n-1}{(1+\frac{|y-z|}{t})}^{-2(n+\epsilon )}dydt\\ \le C{\int }_{\frac{2^j}{a}}^{\mathrm{\infty }}{\int }_{|x-y|<at}\frac{t^{2\epsilon -n-1}}{{(t+|y-z|)}^{-2(n+\epsilon )}}dydt\\ \le C{a}^{3n}{2}^{-2jn}.\end{array}$$

Thus

$$S_{\psi ,a}\left(f_h^j\right)(x)\le {\int }_{{\mathbb{R}}^n}|f_h^j(z)|{(C{a}^{3n+2\epsilon }{2}^{-2jn}+C{a}^{3n}{2}^{-2jn})}^{1/2}dz\le C{a}^{3n/2+\epsilon }{2}^{-jn}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.$$

Therefore, similar to $D_1$, as $\lambda -n{\delta }_2<\alpha$, by applying Minkowski’s inequality, Lemmas 2.1-2.3, and $2^{j\alpha }{\parallel {\{{\sum }_h{|f_h^j|}^r\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\le {\{{\sum }_{l=-\mathrm{\infty }}^j{2}^{\alpha lp}{\parallel {\{{\sum }_h{|f_h^l|}^r\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\}}^{1/p}$, we have

$$\begin{array}{rl}{D}_3\le & C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left(\sum _{j=k+3}^{\mathrm{\infty }}{\parallel {\left\{\sum _h{\left(a^{3n/2+\epsilon }{2}^{-jn}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}\right)}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ \le & C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{\alpha kp}{\parallel {\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p{\left(\sum _{j=k+3}^{\mathrm{\infty }}{2}^{-jn}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^1({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ \le & C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{\alpha kp}{\{\sum _{j=k+3}^{\mathrm{\infty }}{2}^{-jn}|B_j|\frac{{\parallel {\chi }_{B_k}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}}{{\parallel {\chi }_{B_j}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\}}^p\right\}}^{1/p}\\ \le & C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{\alpha kp}{\left\{\sum _{j=k+3}^{\mathrm{\infty }}{\left(\frac{|B_k|}{|B_j|}\right)}^{{\delta }_2}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right\}}^p\right\}}^{1/p}\\ \le & C\underset{k_0\in \mathbb{Z}}{sup}\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{\lambda (k-k_0)p}\right\{\sum _{j=k+3}^{\mathrm{\infty }}{2}^{(j-k)(-n{\delta }_2+\lambda -\alpha )}\\ {{\times 2^{-j\lambda }{\left(\sum _{l=-\mathrm{\infty }}^j{2}^{\alpha lp}{\parallel {\left\{\sum _h{\left|f_h^l\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\right)}^{1/p}\}}^p\}}^{1/p}\\ \le & C{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}.\end{array}$$

Adding up the results of $D_1$, $D_2$, $D_3$, we have

$${\parallel {\left\{\sum _h{|S_{\psi ,a}(f_h)|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\le C{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}.$$

That is, the inequality (1) in Theorem 3.1 holds.

Next we show the other two vector-valued inequalities also hold.

We consider $g_{\psi }$ first. Similar to $S_{\psi ,a}$, via calculation, we shall get the following conclusions (also see [22]):

• (1) If $x\in C_k$, $j\le k-3$, $supp{f}_h^j\subset C_j$, then $g_{\psi }(f_h^j)(x)\le C{|x|}^{-n}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}$;

• (2) If $x\in C_k$, $j\ge k+3$, $supp{f}_h^j\subset C_j$, then $g_{\psi }(f_h^j)(x)\le C{2}^{-jn}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}$.

Thus, with a similar argument in the proof of the inequality (1) in Theorem 3.1, by Lemma 2.8 and the above two conclusions, we shall get the following inequality immediately:

$${\parallel {\left\{\sum _h{|g_{\psi }(f_h)|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\le C{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}.$$

For $g_{\mu }^{\ast }$, by the definition of $S_{\psi ,a}$ and $g_{\mu }^{\ast }$, we have

$$\begin{array}{rl}{g}_{\mu }^{\ast }(f)(x)=& {\left({\int }_0^{\mathrm{\infty }}{\int }_{{\mathbb{R}}^n}{\left(\frac{t}{t+|x-y|}\right)}^{\mu n}{|{\psi }_t\ast f(y)|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2}\\ \le & {\left({\int }_0^{\mathrm{\infty }}{\int }_{|x-y|<t}{\left(\frac{t}{t+|x-y|}\right)}^{\mu n}{|{\psi }_t\ast f(y)|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2}\\ +\sum _{l=1}^{\mathrm{\infty }}{\left({\int }_0^{\mathrm{\infty }}{\int }_{2^{l-1}t\le |x-y|<2^{l}t}{\left(\frac{t}{t+|x-y|}\right)}^{\mu n}{|{\psi }_t\ast f(y)|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2}\\ \le & S_{\psi ,a}\left(f_h^j\right)(x)+\sum _{l=1}^{\mathrm{\infty }}{(1+2^{l-1})}^{-\mu n/2}{S}_{\psi ,2^{l}a}\left(f_h^j\right)(x).\end{array}$$

By the above estimate of $S_{\psi ,a}$, we know that, as $x\in C_k$, $j\le k-3$, $supp{f}_h^j\subset C_j$,

$$S_{\psi ,a}\left(f_h^j\right)(x)\le C{a}^{3n/2+\epsilon +\gamma }{|x|}^{-n}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.$$

Thus, when $\mu >3+2(\epsilon +\gamma )/n$, we have

$$\begin{array}{rl}{g}_{\mu }^{\ast }\left(f_h^j\right)(x)& \le C{a}^{3n/2+\epsilon +\gamma }(1+\sum _{l=1}^{\mathrm{\infty }}{2}^{(3n/2+\epsilon +\gamma -\mu n/2)}){|x|}^{-n}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}\\ \le C{a}^{3n/2+\epsilon +\gamma }{|x|}^{-n}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.\end{array}$$

On the other hand, as $x\in C_k$, $j\ge k+3$, $supp{f}_h^j\subset C_j$, we have

$$S_{\psi ,a}\left(f_h^j\right)(x)\le C{a}^{3n/2+\epsilon }{2}^{-jn}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.$$

Furthermore, when $\mu >3+2(\epsilon +\gamma )/n$, we obtain

$$\begin{array}{rl}{g}_{\mu }^{\ast }\left(f_h^j\right)(x)& \le C{a}^{3n/2+\epsilon }(1+\sum _{l=1}^{\mathrm{\infty }}{2}^{(3n/2+\epsilon -\mu n/2)})2^{-jn}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}\\ \le C{a}^{3n/2+\epsilon }{2}^{-jn}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.\end{array}$$

Hence, similar to the proof of the inequality (1) too, by Lemma 2.8 and the above two conclusions about $g_{\mu }^{\ast }$, we get the following inequality immediately:

$${\parallel {\left\{\sum _h{|g_{\mu }^{\ast }(f_h)|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\le C{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}.$$

This completes the proof of Theorem 3.1. □

Now, let us to establish the vector-valued inequalities of the commutators generated by the Littlewood-Paley operators with BMO functions on $M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)$.

Theorem 3.2 Suppose that $\psi \in L^1({\mathbb{R}}^n)$ satisfies (i)-(iii), $b\in BMO({\mathbb{R}}^n)$, $q(\cdot )\in \mathcal{B}({\mathbb{R}}^n)$, $m\in \mathbb{N}\mathrm{\setminus }\{0\}$. Let $0<p<\mathrm{\infty }$, $1<r<\mathrm{\infty }$, $\lambda -n{\delta }_2<\alpha <\lambda +n{\delta }_1$, where ${\delta }_1$, ${\delta }_2$ is the constant in Lemma  2.2. Then for all function sequence ${\{f_h\}}_{h=1}^{\mathrm{\infty }}:{\parallel {\{{\sum }_h{|f_h|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}<\mathrm{\infty }$, the following vector-valued inequalities hold:

• (1) ${\parallel {\{{\sum }_h{|[b^m{S}_{\psi ,a}](f_h)|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\le C{\parallel b\parallel }_{\ast }^m{\parallel {\{{\sum }_h{|f_h|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}$,

• (2) ${\parallel {\{{\sum }_h{|[b^m,g_{\psi }](f_h)|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\le C{\parallel b\parallel }_{\ast }{\parallel {\{{\sum }_h{|f_h|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}$,

• (3) ${\parallel {\{{\sum }_h{|[b^m,g_{\mu }^{\ast }](f_h)|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\le C{\parallel b\parallel }_{\ast }^m{\parallel {\{{\sum }_h{|f_h|}^r\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}$,

where $\mu >3+2(\epsilon +\gamma )/n$, $0<\gamma <min\{(\mu -2)n/2,\epsilon \}$, and the constant C is independent of ${\{f_h\}}_{h=1}^{\mathrm{\infty }}$.

Proof Firstly, we consider the inequality (1). Let $b\in BMO({\mathbb{R}}^n)$. For any function sequence ${\{f_h\}}_h$ satisfies ${\parallel {\parallel {\{f_h\}}_h\parallel }_{l^r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}<\mathrm{\infty }$. We write

$$f_h(x)=\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_h(x){\chi }_j(x)\triangleq \sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_h^j(x).$$

Thus,

$$\begin{array}{r}{\parallel {\left\{\sum _h{|[b^m,S_{\psi ,a}](f_h)|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\\ =\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\parallel {\left\{\sum _h{|[b^m,S_{\psi ,a}](f_h)|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\right\}}^{1/p}\\ \le \underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left(\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{\parallel {\left\{\sum _h{\left|[b^m,S_{\psi ,a}]\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ \le C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left(\sum _{j=-\mathrm{\infty }}^{k-3}{\parallel {\left\{\sum _h{\left|[b^m,S_{\psi ,a}]\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ +C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left(\sum _{j=k-2}^{k+2}{\parallel {\left\{\sum _h{\left|[b^m,S_{\psi ,a}]\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ +C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left(\sum _{j=k+3}^{\mathrm{\infty }}{\parallel {\left\{\sum _h{\left|[b^m,S_{\psi ,a}]\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ \triangleq E_1+E_2+E_3.\end{array}$$

We are now going to estimate each term, respectively. For the term $E_2$, noting that $supp{f}_h^j\subset C_j$, we can easily obtain by Lemma 2.9

$$\begin{array}{rl}{E}_2& \le C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\left(\sum _{j=k-2}^{k+2}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ \le C\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{k\alpha p}{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\right\}}^{1/p}\\ =C{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}.\end{array}$$

For the term $E_1$, observe that as $x\in C_k$, $j\le k-3$, $supp{f}_h^j\subset C_j$. With the same argument as in the estimate of $D_1$, we have

$$S_{\psi ,a}\left(f_h^j\right)(x)\le C{a}^{3n/2+\epsilon +\gamma }{|x|}^{-n}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.$$

Thus,

$$\begin{array}{rl}|[b^m,S_{\psi ,a}]\left(f_h^j\right)(x)|& =|S_{\psi ,a}\left[{(b(x)-b)}^{m}f\right](x)|\\ \le C{a}^{3n/2+\epsilon +\gamma }{|x|}^{-n}{\parallel {(b(\cdot )-b)}^m{f}_h^j\parallel }_{L^1({\mathbb{R}}^n)}.\end{array}$$

Hence, by Minkowski’s inequality, we get

$$\begin{array}{r}{\parallel {\left\{\sum _h{\left|[b^m,S_{\psi ,a}]\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\\ \le C{\parallel {|x|}^{-n}{\left\{\sum _h{\parallel {(b(\cdot )-b)}^m{f}_h^j\parallel }_{L^1({\mathbb{R}}^n)}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\\ \le C{\parallel {|x|}^{-n}{\parallel {\left\{\sum _h{\left|{(b(\cdot )-b)}^m{f}_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^1({\mathbb{R}}^n)}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\\ \le C{2}^{-nk}{\parallel {(b-b_j)}^m{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^1({\mathbb{R}}^n)}\\ +2^{-nk}{\parallel {\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\left\{\sum _h{\left|{(b-b_j)}^m{f}_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^1({\mathbb{R}}^n)}.\end{array}$$

Furthermore, by Hölder’s inequality and Lemmas 2.2-2.4, we obtain

$$\begin{array}{r}{\parallel {\left\{\sum _h{\left|[b^m,S_{\psi ,a}]\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\\ \le C{2}^{-nk}{(k-j)}^m{\parallel b\parallel }_{\ast }^m{\parallel {\chi }_{B_k}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\chi }_j\parallel }_{L^{q^{\prime }(\cdot )}({\mathbb{R}}^n)}\\ +2^{-nk}{\parallel {\chi }_{B_k}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {(b-b_j)}^m{\chi }_j\parallel }_{L^{q^{\prime }(\cdot )}({\mathbb{R}}^n)}\\ \le C{2}^{-nk}[{(k-j)}^m+1]{\parallel b\parallel }_{\ast }^m{\parallel {\chi }_{B_k}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\chi }_{B_j}\parallel }_{L^{q^{\prime }(\cdot )}({\mathbb{R}}^n)}\\ \le C{2}^{-nk}{(k-j)}^m{\parallel b\parallel }_{\ast }^m|B_k|\frac{{\parallel {\chi }_{B_j}\parallel }_{L^{q^{\prime }(\cdot )}({\mathbb{R}}^n)}}{{\parallel {\chi }_{B_k}\parallel }_{L^{q^{\prime }(\cdot )}({\mathbb{R}}^n)}}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\\ \le C{(k-j)}^m{\parallel b\parallel }_{\ast }^m\frac{|B_j|}{|B_k|}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\\ \le C{2}^{(j-k)n{\delta }_1}{(k-j)}^m{\parallel b\parallel }_{\ast }^m{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}.\end{array}$$

Therefore, as $\alpha <\lambda +n{\delta }_1$, noting that

$$2^{j\alpha }{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\le {\left\{\sum _{l=-\mathrm{\infty }}^j{2}^{\alpha lp}{\parallel {\left\{\sum _h{\left|f_h^l\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\right\}}^{1/p},$$

then we have

$$\begin{array}{rl}{E}_1\le & C{\parallel b\parallel }_{\ast }^m\underset{k_0\in \mathbb{Z}}{sup}{2}^{-\lambda k_0}{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{\alpha kp}{\left(\sum _{j=-\mathrm{\infty }}^{k-3}{2}^{(j-k)n{\delta }_1}{(k-j)}^m{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ \le & C{\parallel b\parallel }_{\ast }^m\underset{k_0\in \mathbb{Z}}{sup}\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{\lambda (k-k_0)p}\right(\sum _{j=-\mathrm{\infty }}^{k-3}{(k-j)}^m{2}^{(j-k)(n{\delta }_1+\lambda -\alpha )}\\ {{\times 2^{-j\lambda }{\left\{\sum _{l=-\mathrm{\infty }}^j{2}^{\alpha lp}{\parallel {\left\{\sum _h{\left|f_h^l\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\right\}}^{1/p})}^p\}}^{1/p}\\ \le & C{\parallel b\parallel }_{\ast }^m{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}.\end{array}$$

Now we turn to estimate $E_3$. Observe that as $x\in C_k$, $j\ge k+3$, $supp{f}_h^j\subset C_j$. With the same argument as in the estimate of $D_3$, then we have

$$S_{\psi ,a}\left(f_h^j\right)(x)\le C{a}^{3n/2+\epsilon }{2}^{-jn}{\parallel f_h^j\parallel }_{L^1({\mathbb{R}}^n)}.$$

Thus,

$$|[b^m,S_{\psi ,a}]\left(f_h^j\right)(x)|\le C{a}^{3n/2+\epsilon }{2}^{-jn}{\parallel {(b(\cdot )-b)}^m{f}_h^j\parallel }_{L^1({\mathbb{R}}^n)}.$$

Hence, by Minkowski’s inequality, we get

$$\begin{array}{r}{\parallel {\left\{\sum _h{\left|[b^m,S_{\psi ,a}]\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\\ \le C{\parallel 2^{-jn}{\left\{\sum _h{\parallel {(b(\cdot )-b)}^m{f}_h^j\parallel }_{L^1({\mathbb{R}}^n)}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\\ \le C{\parallel 2^{-jn}{\parallel {\left\{\sum _h{\left|{(b(\cdot )-b)}^m{f}_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^1({\mathbb{R}}^n)}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\\ \le C{2}^{-jn}{\parallel {(b-b_k)}^m{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^1({\mathbb{R}}^n)}\\ +2^{-jn}{\parallel {\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\left\{\sum _h{\left|{(b-b_k)}^m{f}_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^1({\mathbb{R}}^n)}.\end{array}$$

Furthermore, by Hölder’s inequality and Lemmas 2.2-2.4, we obtain

$$\begin{array}{r}{\parallel {\left\{\sum _h{\left|[b^m,S_{\psi ,a}]\left(f_h^j\right)\right|}^r\right\}}^{1/r}{\chi }_k\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\\ \le C{2}^{-jn}{\parallel b\parallel }_{\ast }^m{\parallel {\chi }_{B_k}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\chi }_j\parallel }_{L^{q^{\prime }(\cdot )}({\mathbb{R}}^n)}\\ +2^{-jn}{\parallel {\chi }_{B_k}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {(b-b_k)}^m{\chi }_j\parallel }_{L^{q^{\prime }(\cdot )}({\mathbb{R}}^n)}\\ \le C{2}^{-jn}[{(k-j)}^m+1]{\parallel b\parallel }_{\ast }^m{\parallel {\chi }_{B_k}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}{\parallel {\chi }_{B_j}\parallel }_{L^{q^{\prime }(\cdot )}({\mathbb{R}}^n)}\\ \le C{2}^{-jn}{(k-j)}^m{\parallel b\parallel }_{\ast }^m|B_j|\frac{{\parallel {\chi }_{B_k}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}}{{\parallel {\chi }_{B_j}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\\ \le C{(k-j)}^m{\parallel b\parallel }_{\ast }^m\frac{|B_k|}{|B_j|}{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\\ \le C{2}^{(k-j)n{\delta }_2}{(j-k)}^m{\parallel b\parallel }_{\ast }^m{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}.\end{array}$$

Therefore, similar to $E_1$, as $\alpha >\lambda -n{\delta }_2$, noting that

$$2^{j\alpha }{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\le {\left\{\sum _{l=-\mathrm{\infty }}^j{2}^{\alpha lp}{\parallel {\left\{\sum _h{\left|f_h^l\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\right\}}^{1/p},$$

then we have

$$\begin{array}{rl}{E}_3\le & C{\parallel b\parallel }_{\ast }^m\underset{k_0\in \mathbb{Z}}{sup}{2}^{-\lambda k_0}{\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{\alpha kp}{\left(\sum _{j=k+3}^{\mathrm{\infty }}{2}^{(k-j)n{\delta }_2}{(j-k)}^m{\parallel {\left\{\sum _h{\left|f_h^j\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}\right)}^p\right\}}^{1/p}\\ \le & C{\parallel b\parallel }_{\ast }^m\underset{k_0\in \mathbb{Z}}{sup}\left\{\sum _{k=-\mathrm{\infty }}^{k_0}{2}^{\lambda (k-k_0)p}\right(\sum _{j=k+3}^{\mathrm{\infty }}{(j-k)}^m{2}^{(j-k)(-n{\delta }_2+\lambda -\alpha )}\\ {{\times 2^{-j\lambda }{\left\{\sum _{l=-\mathrm{\infty }}^j{2}^{\alpha lp}{\parallel {\left\{\sum _h{\left|f_h^l\right|}^r\right\}}^{1/r}\parallel }_{L^{q(\cdot )}({\mathbb{R}}^n)}^p\right\}}^{1/p})}^p\}}^{1/p}\\ \le & C{\parallel b\parallel }_{\ast }^m{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}.\end{array}$$

Adding up the results of $E_1$, $E_2$, $E_3$, we have

$${\parallel {\left\{\sum _h{|\left[b^m{S}_{\psi ,a}\right](f_h)|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\le C{\parallel b\parallel }_{\ast }^m{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}.$$

That is, the inequality (1) in Theorem 3.2 holds.

For the proofs about the vector-valued inequalities of $[b^m,g_{\psi }]$ and $[b^m,g_{\mu }^{\ast }]$, with a similar arguments in the proof of the vector-valued inequality (1) in Theorem 3.2, by Lemma 2.9 and the estimates of $g_{\psi }$, and $g_{\mu }^{\ast }$ in Theorem 3.1, it is not difficult to deduce

$${\parallel {\left\{\sum _h{|[b^m,g_{\psi }](f_h)|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\le C{\parallel b\parallel }_{\ast }{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}$$

and, if $\mu >3+2(\epsilon +\gamma )/n$ and $0<\gamma <min\{(\mu -2)n/2,\epsilon \}$,

$${\parallel {\left\{\sum _h{|[b^m,g_{\mu }^{\ast }](f_h)|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}\le C{\parallel b\parallel }_{\ast }^m{\parallel {\left\{\sum _h{|f_h|}^r\right\}}^{1/r}\parallel }_{M{\dot{K}}_{p,q(\cdot )}^{\alpha ,\lambda }({\mathbb{R}}^n)}.$$

We complete the proof of Theorem 3.2. □

Remark 3.1 By Remark 1.1, we can easily to see that the results in Theorems 3.1-3.2 are also suitable for Herz spaces with variable exponent ${\dot{K}}_{p(\cdot )}^{\alpha ,q}({\mathbb{R}}^n)$.