Boundedness of Littlewood-Paley Operators with Variable Kernel on Weighted Herz Spaces with Variable Exponent

1. Introduction

The boundedness of Littlewood-Paley operators on function spaces are one of the very important tools, not only in harmonic analysis, but also in potential theory and in partial differential equations (see [1, 2, 3, 4, 5, 6], for details). In 2004, Ding, Lin and Shao [7] investigated the \({L}^{2}\)-boundedness for a class of Marcinkiewicz integral operators with variable kernels \(\mu_{\Omega}\) and \(\mu_{\Omega,s}\) related to the Littlewood-Paley function \(\mu^{*}_{\Omega,\lambda}\) and the area integral \(g^{*}_{\lambda}\). In 2006, the authors [8] proved the \(L^{p}\)-boundedness of the Littlewood-Paley operators with variable kernels. In 2009, Xue and Ding [9] established the weighted estimate for Littlewood-Paley operators and their commutators.

In 1960, Hörmander [10] introduced the parameterized Littlewood-Paley operators for the first time. Now, let us recall the definitions of the parameterized Lusin area integral and Littlewood-Paley \(g^{*}_{\lambda}\) function. Let \(S^{n-1} (n\geq 2)\) be the unit sphere in \(\mathbb{R}^{n}\) with normalized Lebesgue measure \(\mathrm{d}\sigma(x')\). Take \(\Omega(x,z)\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r\geq{1})\) to be a homogeneous function of degree zero and where \(\Omega\) satisfies the following conditions:

1. For any \(x,z \in{\mathbb{R}^{n}}\) and any \(\lambda > 0\), we have \(\Omega{(x,\lambda{z})}=\Omega(x,z)\);
2. \(\|\Omega\|_{L^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}}:=\sup\limits_{r\geq0, y\in\mathbb{R}^{n}}\left(\int_{s^{n-1}}|{\Omega(rz'+y ,z')}|^{r}\mathrm{d}{\sigma}(z')\right)^{\frac{1}{r}}{< \infty}.\)

The parameterized Littlewood-Paley operators \(\mu^{\rho}_{\Omega,s}\) and \(\mu^{*,\rho}_{\Omega,\lambda}\) with variable kernels, which are related to the Lusin area integral and the Littlewood-Paley \(g_{\lambda}^{*}\) function are defined by $$ \mu^{\rho}_{\Omega,s}(f)(x)=\left(\int \int_{\Gamma(x)}\left|\frac{1}{t^{\rho}}\int_{|y-z|\leq t}\frac{\Omega(y,y-z)}{|y-z|^{n-\rho}}f(z)\mathrm{d}z\right|^{2}\frac{\mathrm{d}y\mathrm{d}t}{t^{n+1}}\right) ^{\frac{1}{2}} $$ and \begin{eqnarray*} &&\mu^{*,\rho}_{\Omega,\lambda}(f)(x)=\\&&\left(\int \int_{\mathbb{R}^{n+1}_{+}}\left(\frac{t}{t + |x-y|}\right)^{\lambda n}\left|\frac{1}{t^{\rho}}\int_{|y-z|\leq t}\frac{\Omega(y,y-z)}{|y-z|^{n-\rho}}f(z)\mathrm{d}z\right|^{2}\frac{\mathrm{d}y\mathrm{d}t}{t^{n+1}}\right)^{\frac{1}{2}}, \end{eqnarray*} where \(\Gamma(x)= \{ (y,t)\in {\mathbb{R}^{n+1}_{+}}: |x-y|< t\) and \(\lambda >1 \}\).

In 2013, Wei and Tao [11] investigated the boundedness of parameterized Littlewood Paley operators on weighted weak Hardy spaces. Lin and Xuan [12] established the boundedness for commutators of parameterized Littlewood-Paley operators and area integrals on weighted Lebesgue spaces \(L^{p}(w)\).

The theory of the variable exponent function spaces has been rapidly developed after the work [13], where Kováčik and Rákosník have clarified fundamental properties of Lebesgue spaces with variable exponent. After that, many researchers have been interested in the theory of the variable exponent spaces (see [14, 15, 16, 17, 18, 19, 20]).

The generalization of the Muckenhoupt weights with variable exponent \(A_{p(\cdot)}\) has been considered in [21, 22, 23, 24]. The equivalence between the Muckenhoupt condition and the boundedness of the Hardy-Littlewood maximal operator on weighted Lebesgue spaces with variable exponent were discussed in [21, 22]. After that, Cruz-Uribe and Wang [25] proved the boundedness of some classical operators on weighted Lebesgue spaces with variable exponent \(L^{p(\cdot)}(w)\).

Recently, Izuki and Noi [26] introduced the weighted Herz spaces with variable exponent, and also studied the boundedness of fractional integrals on those spaces.

In this paper, we establish the boundedness of parameterized Littlewood-Paley operators with variable kernels on weighted Herz spaces with variable exponent \(\dot{K}_{p(\cdot)}^{\alpha,q}(w)\). Let \(E\) be a Lebesgue measurable set in \(\mathbb{R}^{n}\) with measure \(|E|>0\), \(\chi_{E}\) means its characteristic function. We shall recall some definitions.

Definition 1.1. Let \(p(\cdot): E \rightarrow {[1,\infty)}\) be a measurable function. The variable exponent Lebesgue space is defined as $$ L^{p(\cdot)}(E)= \{{ f~\mbox{is measurable}: \int_{E}\left(\frac{|f(x)|}{\eta}\right)^{p(x)} \mathrm{d}x <\infty}~ \mbox{for some constant } \eta > 0\}. $$ The space \(L _{\mathrm{loc}}^{p(\cdot)} {(E)}\) is defined as $$ L_{\mathrm{loc}}^{p(\cdot)} {(E)}= \{f \mbox { is measurable}: f\in {L^{p(\cdot)} {(K)}}~\mbox{for all compact} ~K\subset E\}. $$ The Lebesgue spaces \(L^{p(\cdot)} {(E)}\) is a Banach spaces with the norm defined as $$ \|f\|_{L^{p(\cdot)}(E)}= \inf\left\{\eta> 0: \int_{E}\left(\frac{|f(x)|}{\eta}\right)^{p(x)}\mathrm{d}x \leq 1\right\}. $$

We denote \(p_{-}= \mbox{ess inf} \{p(x): x \in E\},\) \(p_{+}= \mbox{ ess sup} \{p(x): x \in E\}\), then \(\mathcal{P}(E)\) consists of all \(p(\cdot)\) satisfying \(p_{-} > 1\) and \(p_{+} < \infty\).

Definition 1.2.[27] Let \(p(\cdot):\mathbb{R}^{n}\rightarrow [1,\infty)\). A measurable function \(p(\cdot)\) is said to be globally log-Höder continuous if it satisfies

1. \(|p(x)-p(y)|\leq \frac{1}{\mathrm{-log}(|x-y|)}, ~~~~~~x,y\in \mathbb{R}^{n}, |x-y|\leq 1/2;\)
2. \(|p(x)-p_{\infty}|\leq \frac{1}{\mathrm{log}(e+|x|)}, ~~~~~~~~~x\in \mathbb{R}^{n},\)

for some \(p_{\infty}\geq 1\). The set of \(p(\cdot)\) satisfying conditions (1) and (2) is denoted by \(LH(\mathbb{R}^{n})\).

We know that, if \(p(\cdot)\in LH(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n})\), the Hardy-Littlewood maximal operator \(M\) is bounded on \(L^{p(\cdot)}(\mathbb{R}^{n})\) (see [28]).

Definition 1.3.[29] Suppose that \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) and \(w\) is a weight function. The weighted Lebesgue spaces with variable exponent \(L^{p(\cdot)}(w)\) is the set of all complex-valued measurable function \(f\) such that \(fw^{1/p(\cdot)}\in L^{p(\cdot)}(\mathbb{R}^{n})\). The space \(L^{p(\cdot)}(w)\) is a Banach space equipped with the norm $$ \|f\|_{L^{p(\cdot)}(w)}= \|fw^{1/p(\cdot)}\|_{L^{p(\cdot)}}. $$ \(p'(\cdot)\) is the conjugate of \(p(\cdot)\) such that \(\frac{1}{ p'(\cdot) }+\frac{1}{ p(\cdot) }=1\). Next, we introduce the classical Muckenhoupt \(A_{p}\) weight.

Definition 1.4.[30] Let \(1< p< \infty\), then \(w\in A_{p}\) for every cube \(Q\), $$ \left( \frac{1}{|Q|}\int_{Q}w(x)\mathrm{d}x\right)\left( \frac{1}{|Q|}\int_{Q}w(x)^{1-p'}\mathrm{d}x\right)^{p-1}\leq C< \infty. $$ We say that \(w\in A_{1}\) if it satisfies \(Mw(x)\leq w(x)\) for all \(x\in \mathbb{R}^{n}\). The set \(A_{1}\) consists of all Muckenhoupt \(A_{1}\) weights.

Definition 1.5.[21, 25] Given \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) and a weight \(w\), then \(w\in A_{p(\cdot)}\) if $$ \sup\limits_{\mathrm{B:ball}}|B|^{-1}\|w^{1/p(\cdot)}\chi_{B}\|_{L^{p(\cdot)}(w)}\|w^{-1/p(\cdot)}\chi_{B}\|_{L^{p'(\cdot)} (w)}< \infty .$$

Definition 1.6. [ 25] Given \(p_{1}(\cdot),~p_{2}(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) and \(1/p_{1}(x)-1/p_{2}(x)=\mu/n\) such that \(0< \mu< n\). Then \(w\in A_{{p_{1}(\cdot)}, {p_{2}(\cdot)}}\) if $$ \|w\chi_{B}\|_{L^{p_{2}(\cdot)}}\|w^{-1}\chi_{B}\|_{L^{p'_{1}(\cdot)}}\leq|B|^{\frac{n-\mu}{n}} $$ holds for all balls \(B\in\mathbb{R}^{n}\).

Definition 1.7. [ 25] Let \(p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) and \(w\) be a weight. We say that \((p(\cdot),w)\) is an \(M\)-pair if the maximal operator \(M\) is bounded on \(L^{p(\cdot)}(w)\) and \(L^{p'(\cdot)}(w^{-1})\).

Now, we to need to give the definition of weighted Herz space with variable exponent. For all \(k \in{\mathbb{Z}}\), we denote \(B_{k}=\{ x\in\mathbb{R}^{n}: |x|\leq 2^{k}\},\) \(C_{k}= B_{k}\backslash B_{k-1},\) \(\chi_{k}= \chi_{C_{k}}.\)

Definition 1.8. [ 26] Suppose that \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\), \(0< q < \infty\), \(\alpha \in \mathbb{R}.\) The homogeneous weighted Herz space with variable exponent \(\dot{K}_{p(\cdot)}^{\alpha,q}(w)\) is the collection of \(f \in L_{\mathrm{loc}}^{p(\cdot)}(\mathbb{R}^{n} \backslash \{0\},w)\) such that $$ \|f\|_{\dot{K}^{\alpha,q}_{p(\cdot)}(w)} :=\left(\sum^{\infty}_{k=-\infty}2^{\alpha q k}\|f\chi_{k}\|^{q}_{L^{p(\cdot)}(w)}\right)^{1 /{q}}< \infty. $$ It is easy to see that if \(w = 1\), then \(\dot{K}_{p(\cdot)}^{\alpha,q}(w)= \dot{K}_{p(\cdot)}^{\alpha,q}(\mathbb{R}^{n})\) is the Herz space with variable exponen [ 17]. If \(w = 1\) and \(p(\cdot)=p\), then \( \dot{K}_{p(\cdot)}^{\alpha,q}(w)= \dot{K}_{p}^{\alpha,q}(\mathbb{R}^{n})\) is the classical Herz space introduced in [31]. If \(p(\cdot)=p\), then \( \dot{K}_{p(\cdot)}^{\alpha,q}(w)= \dot{K}_{p}^{\alpha,q}(w)\) is the weighted Herz space [ 32].

Definition 1.9. We say a kernel function \(\Omega(x , z )\) satisfies the \(L^{r}\)-Dini condition \((r\geq{1})\), if $$ \int_{0}^{1} \frac{\omega_{r}(\delta)}{\delta}(1+|\log\delta|^{\sigma})\mathrm{d}\delta < \infty, $$ where \({\omega_{r}(\delta)}\) denotes the integral modulus of continuity of order \(r\) of \(\Omega\) defined by $$ {\omega_{r}(\delta)}= \sup_{x\in{\mathbb{R}^{n}} , |\rho|< \delta } \left(\int\limits_{s^{n - 1}}|\Omega(x , \rho{z'}) - \Omega (x , z')|^{r}\mathrm{d}\sigma(z')\right)^{\frac{1}{r}}, $$ where \(\rho\) is the rotation in \(\mathbb{R}^{n}\), \(\|\rho\|=\sup\limits_{z'\in S^{n - 1}}\|\rho{z'}- {z'}\|.\)

Preliminaries and notations

In order to prove our main theorems, we need the following Lemmas.

Lemma 2.1.[3] Suppose that \(X\subset \mathcal{M}\) is a Banach function space.

1. (The generalized Hölder inequality) For all \(f\in X\) and \(g\in X'\), we have $$\int_{\mathbb{R}^{n}}|f(x)g(x)|\mathrm{d}x \leq \|f\|_{X}\|g\|_{X}.$$
2. For all \(f\in X\), we have $$ \sup\left\{\left|\int_{\mathbb{R}^{n}}f(x)g(x)\mathrm{d}x\right|:\|g\|_{X'} \leq 1 \right\}=\|f\|_{X}. $$

In particular, space \((X')'= X\).

As an application of the generalized Hölder inequality above, we have the following Lemma.

Lemma 2.2. Let \(X\) be a Banach function space, we have $$ 1\leq \frac{1}{|B|}\|\chi_{B}\|_{X}\|\chi_{B}\|_{X'} $$ hold for all balls \(B\).

Lemma 2.3.[24] Let \(X\) be a Banach function space. If the Hardy-Littlewood maximal operator \(M\) is weakly bounded on \(X\), that is $$ \|\chi_{\{Mf>\lambda\}}\|_{X}\leq \lambda^{-1}\|f\|_{X}, $$ holds for all \(f\in X\) and \(\lambda>0\)l, then we get $$ \sup\limits_{\mathrm{B: Ball}} \frac{1}{|B|}\|\chi_{B}\|_{X}\|\chi_{B}\|_{X'}< \infty. $$

Remark 2.1.[29] The weighted Banach function space \(X( \mathbb{R}^{n},W)\) is a Banach function space equipped the norm \(\|f\|_{X(\mathbb{R}^{n},W)}:=\|fW\|_{X}.\) The associated space of \(X( \mathbb{R}^{n},W)\) is a Banach function space and equals \(X'(\mathbb{R}^{n},W^{-1} )\).

Remark 2.2. If \(p_{1}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\), by comparing the definition of the weighted Banach function space with weighted variable Lebesgue space, we have

1. If \(X= L^{p_{1}(\cdot)}(\mathbb{R}^{n})\) and \(W = w\), then we obtain $$L^{p_{1}(\cdot)}(\mathbb{R}^{n}, w)= L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}).$$
2. If \(X= L^{p'_{1}(\cdot)}(\mathbb{R}^{n})\) and \(W = w^{-1}.\) Using Lemma 2.4, we obtain $$ L^{p'_{1}(\cdot)}(\mathbb{R}^{n}, w^{-1})= L^{p'_{1}(\cdot)}(w^{-p'_{1}(\cdot)})=(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'. $$

Lemma 2.4.[33] Suppose that \(X\) is a Banach space. Let \(M\) be bounded on the associated space \(X'\). Then there exists a constant \(0< \delta< 1 \) such that $$ \frac{\|\chi_{E}\|_{X}}{\|\chi_{B}\|_{X}}\leq \left(\frac{|E|}{|B|}\right)^{\delta} $$ holds for all balls \(B\) and all measurable sets \(E\subset B\).

Lemma 2.5.[26] Suppose that \(p_{1}(\cdot)\in LH(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n})\) and \(w^{p_{1}(\cdot)}\in A_{1}\). Let \(M\) be a bounded on \(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\) and \(L^{p'_{1}(\cdot)}(w^{-p'_{1}(\cdot)})\), then there exist constants \(\delta_{1},\delta_{2}\in (0,1) \) such that $$ \frac{\|\chi_{S}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}} {\|\chi_{B}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}} \leq C \left(\frac{|S|}{|B|}\right)^{\delta_{1}}, \frac{\|\chi_{S}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}} {\|\chi_{B}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}} \leq C \left(\frac{|S|}{|B|}\right)^{\delta_{2}}, $$ hold for all balls \(B\) and all measurable sets \(S\subset B\).

Lemma 2.6.[34] Suppose that \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}\) satisfies equation (1) and definition (1.9), \(\lambda>2\), \(2\rho -n>0\), \(1 < p < \infty\). Then for all \(f\in L^{q}(w)\) there exists \(C>0\) independent of \(f\) such that $$ \|\mu^{\rho}_{\Omega,s}f\|_{L^{q}(w)}\leq C \|f\|_{L^{q}(w)} $$ and $$ \|\mu^{*,\rho}_{\Omega,\lambda}f\|_{L^{q}(w)}\leq C \|f\|_{L^{q}(w)}. $$

Lemma 2.7.[25] Assume that for \(p_{0}\), \(1< p_{0}< \infty\) and every \(w_{0}\in A_{p_{0}}\), $$ \int\limits_{\mathbb{R}^{n}} f(x)^{p_{0}}w_{0}(x)\mathrm{d}x \leq \int\limits_{\mathbb{R}^{n}} g(x)^{p_{0}}w_{0}(x)\mathrm{d}x,\qquad\qquad\qquad(f,g)\in\mathcal{F}. $$ Then for any \(M\)-pair \((p(\cdot),w) \), $$ \|f\|_{L^{p(\cdot)}(w)}\leq C \|g\|_{L^{p(\cdot)}(w)},\qquad\qquad\qquad\qquad\qquad(f,g)\in\mathcal{F}, $$

Lemma 2.7 holds for \(p_{0}=1\) and the maximal operator is bounded on \(L^{p'(\cdot)}(w^{-1})\). We know the Hardy-Littlewood maximal operator is bounded on \(L^{p(\cdot)}(w)\) and \(L^{p'(\cdot)}(w^{-p'(\cdot)})\) (see [33]).
Combining Lemma 2.6 with Lemma 2.7, we obtain the following conclusion.

Corollary 2.8. Let \(p(\cdot)\in LH(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n})\), \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r\geq1)\) and \(w\in A_{p(\cdot)}\). Then the parameterized Littlewood-Paley operators \(\mu^{\rho}_{\Omega,s}\) and \(\mu^{*,\rho}_{\Omega,\lambda}\) with variable kernels are bounded on \(L^{p(\cdot)}(w)\).

3. Main Theorems and their proofs

In this section, we will prove the boundedness of the parameterized Littlewood-Paley operators with variable kernels on variable weighted Herz spaces.

Theorem 3.1. Let \(p(\cdot)\in LH(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n})\), \(0< q_{1} \leq q_{2}< \infty\), \(\lambda>2\), \(2\rho -n>0\) and \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}\) satisfies (1.1) and (1.2). If \(w^{p(\cdot)}\in A_{1}\) and \(-n\delta_{1}< \alpha < n \delta_{2} \), where \(\delta_{1},\delta_{2}\) are the constants in Lemma 2.5, then the operator \(\mu_{\Omega ,s}\) is bounded from \(\dot{K}^{\alpha,q_{2}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\) to \(\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\).

Theorem 3.2. Let \(p(\cdot)\in LH(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n}),\) \( 0< q_{1}\leq q_{2}< \infty\), \(\lambda>2\), \(2\rho -n>0\) and \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}\) satisfies (1.1) and (1.2). If \(w^{p(\cdot)}\in A_{1}\) and \(-n\delta_{1}< \alpha< n\delta_{2}\), where \(\delta_{1},\delta_{2}\) are the constants in Lemma 2.5, then the operator \(\mu^{*}_{\Omega,\lambda}\) is bounded from \(\dot{K}^{\alpha,q_{2}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\) to \(\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\).

Remark 3.1. As it is well known that, \(\mu^{\rho}_{\Omega,s}f(x)\leq 2^{n\lambda}\mu^{*,\rho}_{\Omega,\lambda}f(x)\) (see [6], p.89). Therefore, we give only the proof of Theorem 3.2.

Proof of Theorem 3.2 Let \(f\in\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\). By the Jensen inequality, we have \begin{align*} \|\mu^{*,\rho}_{\Omega ,\lambda}f\|^{q_{1}}_{\dot{K}^{\alpha,q_{2}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} &\leq \left(\sum^{\infty}_{k=-\infty}2^{\alpha q_{2} k}\|(\mu^{*,\rho}_{\Omega ,\lambda}f)\chi_{k}\|^{q_{2}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{\frac{q_{1}}{q_{2}}}\\ &\leq \sum^{\infty}_{k=-\infty}2^{\alpha q_{1} k}\|(\mu^{*,\rho}_{\Omega ,\lambda}f)\chi_{k}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{align*} Denote \(f_{j}= f\chi_{j}\) for each \(j\in z\), then \(f = \sum_{j=-\infty}^{\infty}f_{j}\), so we have \begin{align*} \|\mu^{*,\rho}_{\Omega ,\lambda}f\|^{q_{1}}_{\dot{K}^{\alpha,q_{2}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} &\leq \sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\|(\mu^{*,\rho}_{\Omega ,\lambda}f)\chi_{k}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &\leq \sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\left(\sum^{k-2}_{j=-\infty}\|(\mu^{*,\rho}_{\Omega ,\lambda}f_{j})\chi_{k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq \sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\left(\sum^{ k+2}_{j=k-2}\|(\mu^{*,\rho}_{\Omega ,\lambda}f_{j})\chi_{k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq \sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\left(\sum^{\infty}_{j=k+2}\|(\mu^{*,\rho}_{\Omega ,\lambda}f_{j})\chi_{k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &=: \mathrm{L_{1}}+\mathrm{ L_{2}}+ \mathrm{L_{3}}. \end{align*} First, we consider \(\mathrm{L_{2}}\). Using Lemma 2.1 and \(-2\leq k-j\leq 2\), it is easy to get \begin{align*} \mathrm{L_{2}}&=\sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\left(\sum^{ k+2}_{j=k-2}\|(\mu^{*,\rho}_{\Omega ,\lambda}f_{j})\chi_{k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &=\sum^{\infty}_{k=-\infty}\left(\sum^{ k+2}_{j=k-2}2^{\alpha(k-j)}2^{\alpha j} \|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\qquad\leq C \|f\|^{q_{1}}_{\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}.\\ \end{align*} Now we need to consider \(\mu^{*,\rho}_{\Omega ,\lambda}f_{j}\). Applying the Minkowski inequality, we conclude that \begin{eqnarray*} &&|\mu^{*,\rho}_{\Omega,\lambda}(f_{j})(x)|=\\&&\left(\int^{\infty}_{0} \int_{\mathbb{R}^{n}}\left(\frac{t}{t + |x-y|}\right)^{\lambda n}\left|\frac{1}{t^{\rho}}\int_{|y-z|\leq t}\frac{\Omega(y,y-z)}{|y-z|^{n-\rho}}f_{j}(z)\mathrm{d}z\right|^{2}\frac{\mathrm{d}y\mathrm{d}t}{t^{n+1}}\right)^{\frac{1}{2}}\\ &&\qquad\leq \int_{\mathbb{R}^{n}}f_{j}(z)\left(\int^{\infty}_{0}\int_{|y-z|\leq t} \left(\frac{t}{t + |x-y|}\right)^{\lambda n}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho+n+1}}\right)^{\frac{1}{2}} \mathrm{d}z\\ &&\qquad\leq \int_{\mathbb{R}^{n}}f_{j}(z)\left(\int^{|x-z|}_{0}\int_{|y-z|\leq t} \left(\frac{t}{t + |x-y|}\right)^{\lambda n}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho+n+1}}\right)^{\frac{1}{2}} \mathrm{d}z\\ &&\qquad+\int_{\mathbb{R}^{n}}f_{j}(z)\left(\int^{\infty}_{|x-z|}\int_{|y-z|\leq t} \left(\frac{t}{t + |x-y|}\right)^{\lambda n}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho+n+1}}\right)^{\frac{1}{2}} \mathrm{d}z. \end{eqnarray*} For \(\Omega\in{L^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}}\) and \(2\rho-n>0\), the following inequality holds \begin{align*} \int_{|y-z|\leq t}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\mathrm{d}y &\leq\int_{s^{n-1}}\int^{t}_{0}\frac{|\Omega(sy'+z,y')|^{2}}{s^{2n-2\rho}}s^{n-1}\mathrm{d}s\mathrm{d}\sigma(y')\\ &\leq \|\Omega\|^{2}_{{L^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}}} t^{2\rho-n}. \end{align*} Since \(|x-z|\leq|x-y|+|y-z|\leq|x-y|+t\). For \(\lambda > 2\), taking \(0< \delta < (\lambda -2)n \), we have \begin{eqnarray*} &&\int^{|x-z|}_{0}\int_{|y-z|\leq t} \left(\frac{t}{t + |x-y|}\right)^{\lambda n}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho+n+1}}\\ &&\leq {\int^{|x-z|}_{0} \int_{|y-z|\leq t}} \left(\frac{t}{t + |x-y|}\right)^{\lambda n-2n-\delta} \frac{1}{|x-z|^{2n+\delta}} \\&&\times\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho-n-\delta+1}}\\ &&\leq \frac{1}{|x-z|^{2n+\delta}} {\int^{|x-z|}_{0} \int_{|y-z|\leq t}}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho-n-\delta+1}}\\ &&\leq \frac{\|\Omega\|^{2}_{{L^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}}}}{|x-z|^{2n+\delta}}\int^{|x-z|}_{0} t^{\delta-1}\mathrm{d}t\\ &&\leq C |x-z|^{-2n}.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad(1.3) \end{eqnarray*} If we take \(1< \lambda_{1} < 2\), then \(\lambda_{1} n-n> 0\) and \(\lambda_{1} n-2n< 0\), so we have \begin{eqnarray*} &&\int^{\infty}_{|x-z|}\int_{|y-z|\leq t} \left(\frac{t}{t + |x-y|}\right)^{\lambda n}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho+n+1}}\\ &&\leq \int^{\infty}_{|x-z|}\int_{|y-z|\leq t} |x-z|^{-\lambda_{1} n} \frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho-\lambda_{1}n+n+1}}\\ &&\leq \int^{\infty}_{|x-z|} |x-z|^{-\lambda_{1} n} \int_{|y-z|\leq t} \frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-\lambda_{1} n}}\frac{\mathrm{d}y\mathrm{d}t}{t^{n+1}}\\ &&\leq \int^{\infty}_{|x-z|} |x-z|^{-\lambda_{1} n} \int_{s^{n-1}} \int^{t}_{0}\frac{|\Omega(y',(y-z)')|^{2}}{s^{2n-\lambda_{1}n}}s^{n-1}\mathrm{d}s\mathrm{d}\sigma(y') \frac{\mathrm{d}t}{t^{n+1}}\\ &&\leq C \|\Omega\|^{2}_{{L^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}}} |x-z|^{-\lambda_{1} n} \int^{\infty}_{|x-z|} t^{\lambda_{1} n-2n-1}\mathrm{d}t\\ &&\leq C |x-z|^{-2n}.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad(1.4) \end{eqnarray*} Combining the above two estimates, we obtain $$ \mu^{*}_{\Omega,\lambda}(f)(x)\leq \int_{\mathbb{R}^{n}}\frac{|f(z)|}{|x-z|^{n}} \mathrm{d}z. \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (1.5) $$ Next, we consider \(\mathrm{L_{1}}\). Noting that for \(x\in A_{k}\), \(z\in A_{j}\) and \(j\leq k-2\), then \(|x-z|\sim |x|\). By the virtue of the generalized Hölder's inequality, we have $$ \mu^{*,\rho}_{\Omega,\lambda}(f_{j})(x)\leq C 2^{-kn}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \|\chi_{j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}. $$ Applying Lemma 2.3 and Lemma 2.5, we take \(\|.\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\) for each side, we have \begin{eqnarray*} &&\|\mu^{*,\rho}_{\Omega,\lambda}(f_{j})(x)\chi_{k}\|_{{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}}\\ &&\leq C 2^{-kn}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \|\chi_{j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'} \|\chi_{k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &&\leq C 2^{-kn}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \|\chi_{B_j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'} \|\chi_{B_k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &&\leq C \|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\|\chi_{B_j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'} \|\chi_{B_k}\|^{-1}_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}\\ &&\leq C\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\frac{\|\chi_{B_j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}} {\|\chi_{B_k}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}}\\ &&\leq C 2^{(j-k)n\delta_{2}}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{eqnarray*} Thus, we have \begin{align*} \mathrm{L_{1}} &\leq C \sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\left(\sum^{k-2}_{j=-\infty}2^{(j-k)n\delta_{2}}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{k-2}_{j=-\infty}2^{j\alpha } 2^{(k-j)(\alpha-n\delta_{2} )}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}. \end{align*} Now we have two cases: \(1< q_{1}< \infty\) and \(0< q_{1}\leq1\). When \(1< q_{1}< \infty\), by using the Hölder's inequality, we have \begin{align*} \mathrm{L_{1}} &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{k-2}_{j=-\infty}2^{j\alpha } 2^{(k-j)(\alpha-n\delta_{2})}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{k-2}_{j=-\infty}2^{j\alpha q_{1} } 2^{(k-j)( \alpha-n\delta_{2} )q_{1}/2}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right) \\&\times\left(\sum^{k-2}_{j=-\infty}2^{(k-j)(\alpha-n\delta_{2})q'_{1}/2}\right)^{q_{1}/q'_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\sum^{k-2}_{j=-\infty}2^{j\alpha q_{1} } 2^{(k-j)( \alpha-n\delta_{2} )q_{1}/2}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &\leq C \sum^{\infty}_{j=-\infty}2^{j\alpha q_{1} }\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \sum^{\infty}_{k=j+2} 2^{(k-j)( \alpha-n\delta_{2} )q_{1}/2}\\ &\leq C \|f\|^{q_{1}}_{\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{align*} When \(0< q_{1}\leq1\), again by the Jensen inequality, we obtain \begin{align*} \mathrm{L_{1}} &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{k-2}_{j=-\infty}2^{j\alpha } 2^{(k-j)(\alpha-n\delta_{2})}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\sum^{k-2}_{j=-\infty}2^{j\alpha q_{1} } 2^{(k-j)( \alpha-n\delta_{2} )q_{1}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &\leq C \sum^{\infty}_{j=-\infty}2^{j\alpha q_{1} }\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \sum^{\infty}_{k=j+2} 2^{(k-j)( \alpha-n\delta_{2} )q_{1}}\\ &\leq C \|f\|^{q_{1}}_{\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{align*} Finally, we estimate \(\mathrm{L_{3}}\). Noting that for \(x\in A_{k}\), \(y\in A_{j}\) and \(j\geq k+2\), then \(|y-x|\sim |y|\). By (1.5) and the virtue of the generalized Hölder's inequality, we have $$ \mu^{*,\rho}_{\Omega,\lambda}(f_{j})(x)\leq C 2^{-jn}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \|\chi_{j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}. $$ Applying Lemma 2.3 and Lemma 2.5, we can take \(\|.\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\) for each side, we have \begin{eqnarray*} &&\|\mu^{*,\rho}_{\Omega,\lambda}(f_{j})(x)\chi_{k}\|_{{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}}\\ &&\leq C 2^{-jn}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \|\chi_{j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'} \|\chi_{k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &&\leq C 2^{-jn}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \|\chi_{B_j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'} \|\chi_{B_k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &&\leq C \|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\|\chi_{B_k}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))} \|\chi_{B_j}\|^{-1}_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))}\\ &&\leq C \|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\frac{\|\chi_{B_k}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))}} {\|\chi_{B_j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))}}\\ &&\leq C 2^{(k-j)n\delta_{1}}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{eqnarray*} Thus, we have \begin{align*} \mathrm{L_{3}} &\leq C \sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\left(\sum^{\infty}_{j=k+2}2^{(k-j)n\delta_{1}}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{\infty}_{j=k+2}2^{j\alpha } 2^{(k-j)(\alpha+n\delta_{1} )}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}. \end{align*} Now we also have two cases: \(1< q_{1}< \infty\) and \(0< q_{1} \leq1\). When \(1< q_{1}< \infty\), by using the Hölder's inequality, we have \begin{align*} \mathrm{L_{3}} &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{\infty}_{j=k+2}2^{j\alpha } 2^{(k-j)(\alpha+n\delta_{1})}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{\infty}_{j=k+2}2^{j\alpha q_{1} } 2^{(k-j)( \alpha+n\delta_{1} )q_{1}/2}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)\\&\times \left(\sum^{\infty}_{j=k+2}2^{(k-j)(\alpha+n\delta_{1})q'_{1}/2}\right)^{q_{1}/q'_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\sum^{\infty}_{j=k+2}2^{j\alpha q_{1} } 2^{(k-j)( \alpha+n\delta_{1} )q_{1}/2}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &\leq C \sum^{\infty}_{j=-\infty}2^{j\alpha q_{1} }\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \sum^{j-2}_{k=-\infty} 2^{(k-j)( \alpha+n\delta_{1} )q_{1}/2}\\ &\leq C \|f\|^{q_{1}}_{\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{align*} When \(0< q_{1}\leq1\), applying the Jensen inequality, we obtain \begin{align*} \mathrm{L_{3}} &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{\infty}_{j=k+2}2^{j\alpha } 2^{(k-j)(\alpha+n\delta_{1})}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\sum^{\infty}_{j=k+2}2^{j\alpha q_{1} } 2^{(k-j)( \alpha+n\delta_{1} )q_{1}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &\leq C \sum^{\infty}_{j=-\infty}2^{j\alpha q_{1} }\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \sum^{j-2}_{k=-\infty} 2^{(k-j)( \alpha-n\delta_{2} )q_{1}}\\ &\leq C \|f\|^{q_{1}}_{\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{align*} This completes the proof of Theorem 3.2.

Acknowledgments

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