1. Introduction
Suppose \(\mathbb{S}^{n-1}\) for \(n\geq 2\) is the unit sphere in \(\mathbb{R}^{n}\) equipped with the normalized Lebesgue measure \(\text{d}\sigma\). Further suppose that \(\Omega\) is a homogeneous function of degree zero on \(\mathbb{R}^{n}\) satisfying \(\Omega\in L^{1}(\mathbb{S}^{n-1})\) and
\begin{equation}\label{1.1}
\int_{\mathbb{S}^{n-1}}\Omega(x’)\text{d}\sigma(x’)=0,\mbox{where} x’=x/|x|(x\neq 0).
\end{equation}
(1)
For \(0< \rho0.\)
For \(m\in\mathbb{N},b\in \mbox{BMO}(\mathbb{R}^{n}),\) the higher-order commutator of parametrized Marcinkiewicz integral is defined as;
\begin{equation}\label{notag}[b^{m},\mu^{\rho}_{\Omega}](h)(x)=\left(\int^{\infty}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}\left[b(x)-b(y)\right]^{m}h(y)\text{d}y \right|^{2}\frac{dt}{t^{2\rho+1}}\right)^{1/2},t>0.
\end{equation}
(2)
It is easy to see that when \(\rho=1,\) and \(\mu^{\rho}(h)=\mu^{1}(h)\), then (2) is the classical Marcinkiewicz integral \(\mu(h)\) introduced by Stein in [1]. It has been proved in [1] that if \(\Omega\in Lip_{\gamma}(\mathbb{S}^{n-1})(0< \gamma\leq1)\) and \(\Omega\) is continuous, then the operator \(\mu(h)\) is of the type \((q,q)\mbox{for}1< q\leq2\) and of the weak type \((1,1)\). Benedek et al., [2] proved that if \(\Omega\in C^{1}(\mathbb{S}^{n-1})\), then \(\mu(h)\) it is of type \((q,q)\) for any \(1< q\leq \infty\). The \(L^{p}\) boundedness of the \(\mu(h)\) has been studied in [1, 3, 4, 5].
In 1960, Hörmander [4] introduced the parametrized Marcinkiewicz integral operators proved that if \(\Omega\in Lip_{\gamma}(\mathbb{S}^{n-1}),0< \gamma\leq1,\) then it is of strong type \((q,q)\) for \(1< q\leq2\). Sakamoto and Yabuta [6] proved the boundedness of the operator \(\mu^{\rho}(h)\) on \(L^{q}(\mathbb{R}^{n})\). Shi and Jiang [7] considered the weighted \(L^{q}-\)boundedness of parametrized Marcinkiewicz integral operator and its higher order commutator. Note that the Littlewood-paley \(g\)-function played very important roles in harmonic analysis and the parameterized Marcinkiewick integral is a special case of the Littlewood-paley \(g\)-function. Many authors studied properties of \(\mu^{\rho}(h)\) on different function spaces, for examples
[8, 9, 10, 11, 12, 13, 14].
In the last three decade, the generalized Orlicz-Lebesgue spaces and the corresponding generalized Orlicz-Sobolev spaces have been extensively studied by many researchers. The variable Lebesgue spaces are special cases of generalized orliz spaces which introduced by Nakano in [15] and developed in [16, 17]. In addition, for properties of \(L^{p(\cdot)}\) spaces we refer to [18, 19, 20], and the fundamental paper of Kováčik and Rákosník [21] appeared in 1990. By virtue of this works many function spaces appeared [22, 23, 24, 25]. Recently, in 2015, Lijuan and Tao established the Herz spaces with two variable exponents \(p(\cdot),q(\cdot)\) in the paper [26].
The main purpose of this work is to discuss the boundedness of parameterized Marcinkiewicz integral and it’s higher order commutators with rough kernels on Herz spaces with two variable exponents. The boundedness of higher order commutator generated by BOM function and parameterized Marcinkiewicz integral is also obtained.
Let \(\Upsilon\) be a measurable set in \(\mathbb{R}^{n}\) with \(|\Upsilon|> 0 \).
Definition 1.
Let \(p(\cdot): \Upsilon \rightarrow {[1,\infty)}\) be a measurable function. The Lebesgue space with variable exponent \(L^{p(\cdot)}(\Upsilon)\) is defined by
$$L^{p(\cdot)}(\Upsilon)= \left\{{ h \mbox{is measurable} : \int_{\Omega}\left(\frac{|h(x)|}{\eta}\right)^{p(x)} dx 0\right\}$$
The space \(L _{loc}^{p(\cdot)} {(\Upsilon)}\) is defined by
$$L_{loc}^{p(\cdot)} {(\Upsilon)}= \{ \mbox {h is measurable} : h\in {L^{p(\cdot)} {(K)}}\mbox{for all compact}K\subset{\Upsilon}\}$$
The Lebesgue spaces \(L^{p(\cdot)} {(\Upsilon)}\) is a Banach spaces with the norm defined by
\begin{equation}\label{eq1.1}\|h\|_{L^{p(\cdot)}(\Upsilon)}= \inf\left\{\eta> 0 : \int_{\Omega}\left(\frac{|h(x)|}{\eta}\right)^{p(x)}dx \leq 1\right\},\end{equation}
(3)
We denote
$$p_{-}= \text{essinf} \{p(x): x \in \Upsilon\},p_{+}=\text{ess}\sup \{p(x): x \in \Upsilon\},$$
then \(\mathcal{P}(\Upsilon)\) consists of all \(p(\cdot)\) satisfying \(p_{-} > 1\) and \(p_{+} < \infty\).
Let \(M\) be the Hardy-Littlewood maximal operator. We denote \(\mathcal{B}(\Upsilon)\) to be the set of all function \(p(\cdot)\in \mathcal{P}(\Upsilon)\) such that \(M\) is bounded on \(L^{p(\cdot)}(\Upsilon)\).
Now, let us recall the definition of Herz spaces with variable exponents.
Definition 2.[26]
Let \(\alpha \in\mathbb{R}^{n} ,q (\cdot),p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\). The homogeneous Herz space with variable exponent \(\dot{K}_{p(\cdot)}^{\alpha,q(\cdot)}(\mathbb{R}^{n})\) is defined by
$$ \dot{K}_{p(\cdot)}^{\alpha, q(\cdot)}(\mathbb{R}^{n})= \{h\in {L_{loc}^{p(\cdot)}}(\mathbb{R}^{n}\backslash\{0\}) : \|h\|_{\dot{K}_{p(\cdot)}^{\alpha, q(\cdot)}(\mathbb{R}^{n})} 0 : \sum\limits_{k=-\infty}^{\infty} \left\| \left( \frac{2^{k\alpha}|h\chi_{k}|}{\eta}\right)^{q(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q(\cdot)}}}\leq 1\right\}.
\end{eqnarray*}
Remark 1.Let \(v\in \mathbb{N},a_{v}\geq 0,1\leq p_{v} 1.
\end{array}\right.$$
Remark 2.[26]
- If \( q_{1} (\cdot),q_{2}(\cdot)\in \mathcal{P} (\mathbb{R}^{n})\) satisfying \( (q_{1})_{+}\leq (q_{2})_{+}\), then
\({K}_{p(\cdot)}^{\alpha, q_{1}(\cdot)}(\mathbb{R}^{n}) \subset {K}_{p(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n}),
\dot{K}_{p(\cdot)}^{\alpha, q_{1}(\cdot)}(\mathbb{R}^{n}) \subset \dot{K}_{p(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n}).\)
- If \( q_{1} (\cdot),q_{2}(\cdot)\in \mathcal{P} (\mathbb{R}^{n})\) and \( (q_{1})_{+}\leq (q_{2})_{-}\), then \(\frac{q_{2}(\cdot)}{q_{1}(\cdot)}\in \mathcal{P}(\mathbb{R}^{n})\)
and \(\frac{q_{2}(\cdot)}{q_{1}(\cdot)}\geq 1 \).
By Remark 1, for any \( h\in \dot{K}_{p(\cdot)}^{\alpha, q(\cdot)}(\mathbb{R}^{n}) \), we have
\begin{eqnarray*}
\sum\limits_{k=-\infty}^{\infty} \left\| \left( \frac{2^{k\alpha}|h\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}
&&\leq \sum\limits_{k=-\infty}^{\infty} \left\| \left( \frac{2^{k\alpha}|h\chi_{k}|}{\eta}\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}^{p_{v}}\leq \left\{ \sum\limits_{k=-\infty}^{\infty} \left\| \left( \frac{2^{k\alpha}|h\chi_{k}|}{\eta}\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}^{p_{h}}
\right\}^{p_{*}}\leq 1;
\end{eqnarray*}
where
\begin{eqnarray*}
&p_{v}&= \left\{\begin{array}{ll}
(\frac{q_{2}(\cdot)}{q_{1}(\cdot)})_{-},\quad\quad\quad\frac{2^{k\alpha}|f\chi_{k}|}{\eta} \leq 1,\\
(\frac{q_{2}(\cdot)}{q_{1}(\cdot)})_{+},\quad\quad\quad\frac{2^{k\alpha}|f\chi_{k}|}{\eta}>1,
\end{array}\right .
\end{eqnarray*}
and
\begin{eqnarray*} &p_{*}&= \left\{\begin{array}{ll}
\min\limits_{v\in \mathbb{N} }p_{v},\quad\quad\quad\sum\limits_{v=0}^{\infty} a_{v}\leq 1,\\
\max\limits_{v\in \mathbb{N} }p_{v},\quad\quad\quad\sum\limits_{v=0}^{\infty} a_{v}>1.
\end{array}\right .
\end{eqnarray*}
This implies that \(\dot{K}_{p(\cdot)}^{\alpha, q_{1}(\cdot)}(\mathbb{R}^{n}) \subset \dot{K}_{p(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n})\).Similarly, we get \({K}_{p(\cdot)}^{\alpha, q_{1}(\cdot)}(\mathbb{R}^{n}) \subset {K}_{p(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n}).\)
Definition 3.
For all \(0< \gamma \leq 1,\) the Lipschitz space \(\dot{\Lambda}_{\gamma}(\mathbb{R}^{n})\) is defined by
$$\dot{\Lambda}_{\gamma}(\mathbb{R}^{n})=\left\{h:\|h\|_{\dot{\Lambda}_{\gamma}(\mathbb{R}^{n})}= \sup\limits_{x,y\in \mathbb{R}^{n};x\neq y}\frac{|h(x)-h(y)|}{|x-y|^{\gamma}}< \infty\right\}.$$
Definition 4.
The BMO function and BMO norm are defined by
\begin{align*}
\mathrm{BMO}(\mathbb{R}^{n})&:=\left\{b\in L^{1}_{loc}(\mathbb{R}^{n}):\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}< 0\right\},\\
\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}&:=\sup\limits_{Q:\text{cube}}|Q|^{-1}\int_{Q}|b(x)-b_{Q}|\text{d}x.
\end{align*}
From here, we suppose that \(B_{k}=\{ x\in\mathbb{R}^{n} : |x|\leq 2^{k}\},\) and \( C_{k}= B_{k}\backslash B_{k-1} , \chi_{k}= \chi_{C_{k}} , \) \; \( k \in{\mathbb{Z}}.\)
2. Preliminary Lemmas
Proposition 1.
[27]
Let a function \(p(\cdot): \mathbb{R}^{n} \rightarrow [ 1 , \infty).\) If \(p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) satisfies
\begin{equation}\label{eq2.1}| p(x) – p(y)|\leq \frac{ -C}{Log( |x – y|)}; | x – y| \leq 1/ 2 ,\end{equation}
(1)
and
\begin{equation}\label{eq2.2}| p(x) – p(y)|\leq \frac{ C}{Log( e +|x|)}; |y|\geq|x|,\end{equation}
(1)
then \(p(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\).
Lemma 1. [21]
(Generalized Hölder Inequality) Let \(p(\cdot),p_{1}(\cdot),p_{2}(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\), then
- for every \(h\in L^{p_{1}(\cdot)}(\mathbb{R}^{n})\text{and}g\in L^{p_{2}(\cdot)}(\mathbb{R}^{n})\), we have
\(\int_{\mathbb{R}^{n}}|h(x) g(x)| dx \leq C\|h\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{p'(\cdot)}(\mathbb{R}^{n})},\)
where \(C_{p}=1+\frac{1}{p_{-}}-\frac{1}{p_{+}}\);
- for every \(h\in L^{p_{1}(\cdot)}(\mathbb{R}^{n}),g\in L^{p_{2}(\cdot)}(\mathbb{R}^{n})\),when \(\frac{1}{p(\cdot)}=\frac{1}{p_{2}(\cdot)}+\frac{1}{p_{1}(\cdot)}\),we have
\(\|h(x)g(x)\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq C \|g(x)\|_{L^{p_{2}}(\mathbb{R}^{n})}\|h(x)\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})},\)
where \(C_{p_{1},p_{2}}=[1+\frac{1}{p_{1-}}-\frac{1}{p_{1+}}]^{\frac{1}{p_{-}}}\).
Lemma 2. [18, [19] Let \(p(\cdot) \in \mathcal{B}(\mathbb{R}^{n})\). If there exists a positive constants \(C,\) \(\delta_{1},\) \(\delta_{2}\) such that \(\delta_{1},\delta_{2}< 1\), then, for all balls \(B\subset\mathbb{R}^{n}\) and all measurable subset \(R\subset B,\) we have
$$\frac{\|\chi_{R}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\leq C \frac{|R|}{|B|},
\frac{\|\chi_{R}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p_{u}^{\prime}(\cdot)}(\mathbb{R}^{n})}}\leq C\left(\frac{|R|}{|B|}\right)^{\delta_{2}},\frac{\|\chi_{R}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}{
\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\leq C\left(\frac{|R|}{|B|}\right)^{\delta_{1}}.$$
Lemma 3.[28]
Let \(p(\cdot) \in \mathcal{B}(\mathbb{R}^{n}),\) then there exists a constant \(C > 0\) such that for any balls B in \(\mathbb{R}^{n}\), we have
$$
\frac{1}{|B|}\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \|\chi_{B}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}\leq C.
$$
Lemma 4. [29] Let \(p(\cdot) \in \mathcal{B}(\mathbb{R}^{n}),\) and \(b\in \mathrm{BMO}(\mathbb{R}^{n})\). If \(i,j\in\mathbb{Z}\) with \(i< j\), then we have
- \(C^{-1}\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}
\leq\sup\limits_{B}\frac{1}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\|(b-b_{B})\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}
\leq C\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})};\)
- \(\|(b-b_{B_{i}})\chi_{B_{j}}\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C( j-i)\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{q(\cdot)}(\mathbb{R}^{n})}.\)
Lemma 5.[26] Let \(p(\cdot),q(\cdot) \in\mathcal{P}(\mathbb{R}^{n}).\) If \(h\in L^{p(\cdot)q(\cdot)}\), then
$$
\min ( \|h\|_{L^{p(\cdot)q(\cdot)}}^{q_{+}}, \|h\|_{L^{p(\cdot)q(\cdot)}}^{q_{-}} )\leq\||h|^{q(\cdot)}\|_{L^{p(\cdot)}}\leq\max
( \|h\|_{L^{p(\cdot)q(\cdot)}}^{q_{+}}, \|h\|_{L^{p(\cdot)q(\cdot)}}^{q_{-}}).
$$
Lemma 6.[30] Let \(a>0,0< d \leq s,1\leq s\leq\infty\) and \(\frac{-sn+(n-1)d}{s}< v< \infty,\) then
$$\left(\int_{|y|\leq a|x|}|y|^{v}|\Omega(x-y)|^{d}\mbox{d}y\right)^{1/d}\leq C |x|^{(v+n)/d}\|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}.$$
Lemma 7.[31] Let the variable exponent \(\tilde{q}(\cdot)\) is defined by \(\frac{1}{p(x)}=\frac{1}{\tilde{q}(x)}+\frac{1}{q}(x\in\mathbb{R}^{n})\), then we have
$$\|hg\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq C \|g\|_{L^{q}(\mathbb{R}^{n})} \|h\|_{L^{\tilde{q}(\cdot)}(\mathbb{R}^{n})}.$$
Lemma 8. Let \(p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}),\Omega\in L^{s}(\mathbf{\mathbb{S}}^{n-1})\) and \(0<\rho0\) independent of \(h\), then \(\mu_{\Omega}^{\rho}\) is bounded from \(L^{p(\cdot)}\) to it self.
Lemma 9. Let \(b\in\mathrm{BMO}(\mathbb{R}^{n})\) and \(m\in\mathbb{N}\). Further let that \(p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}),\Omega\in L^{s}(\mathbf{\mathbb{S}}^{n-1})\) and \(0< \rho0\) independent of \(h\), then \([b^{m},\mu_{\Omega}^{\rho}]\) is bounded from \(L^{p(\cdot)}\) to itself.
Lemma 10. Let \(b\in\dot{\Lambda}_{\gamma}(\mathbb{R}^{n}),0< \gamma\leq1,m\in\mathbb{N}\) and \(0< \rho< n.\) If \(q_{1}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) satisfies (4) and (5) in Proposition 1 with \(q^{+}_{1}q^{+}_{2})\) with \(1\leq r'< q^{-}_{2}\). Then the commutator \([b^{m},\mu^{\rho}_{\Omega}]\) is bounded from \(L^{q_{1}(\cdot)}(\mathbb{R}^{n})\) to \(L^{q_{2}(\cdot)}(\mathbb{R}^{n}).\)
Lemma 11.
[32] Let \(p(\cdot)\in \mathcal{P}(\Omega)\) abd \(h:\Omega\times \Omega\rightarrow \mathbb{R}\) is a measurable function (with respect to product measure) such that, \(y\in \Omega,h(\cdot,y)\in L^{p(\cdot)}(\Omega)\), then we have
$$\left\|\int_{\Omega}h(\cdot,y)dy\right\|_{L^{p(\cdot)}(\Omega)}\leq C \int_{\Omega}\left\|h(\cdot,y)\right\|_{L^{p(\cdot)}(\Omega)}dy.$$
3. Main Results
Theorem 1.
Let \(0< \rho< n,0(p_{1}’)_{+}\) and \(q_{1}(\cdot),q_{2}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) with \((q_{2})_{-}\geq(q_{1})_{+}\). If \(-n\delta_{1}-v-n/s< \alpha< n\delta_{2}-v-n/s\) with \(\delta_{1},\delta_{2}\) as defined in Lemma 2, then the operator \(\mu^{\rho}_{\Omega}\) is bounded from
\( \dot{K}_{p_{1}(\cdot)}^{\alpha,q_{1}(\cdot)}(\mathbb{R}^{n})\)
to
\(\dot{K}_{p_{1}(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n})\) and from \(\left({K}_{p_{1}(\cdot)}^{\alpha,q_{1}(\cdot)}(\mathbb{R}^{n})\right)\)
to \(\left({K}_{p_{1}(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n})\right)\).
Proof.
Let \(h(x)\in \dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})\). Rewrite
\(h(x)=\sum_{j=-\infty}^{\infty}h(x)\chi_{j}=\sum_{j=-\infty}^{\infty}h_{j}(x).\) From Definition 2, we have
\begin{equation*}
\|\mu^{\rho}_{\Omega}(h)\|_{\dot{K}^{\alpha,q_{2}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}=\inf\left\{\eta>0 :
\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }|\mu^{\rho}_{\Omega}(h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}
\leq1\right\}.
\end{equation*}
Since
\(
\left\|\left(\frac{2^{k\alpha}|\mu^{\rho}_{\Omega}(h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}
\leq\left\|\left(
\frac{2^{k\alpha}|\sum^{\infty}_{j=-\infty}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\sum^{3}_{i=1}\eta_{1i}}
\right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\)\\
\(\leq\left\|\left(
\frac{2^{k\alpha}|\sum^{k-2}_{j=-\infty}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{11}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}+\left\|\left(
\frac{2^{k\alpha}|\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{12}}
\right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}
+\left\|\left(
\frac{2^{k\alpha}|\sum^{\infty}_{j=k+2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{13}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}},
\)
where
\begin{eqnarray*}
\quad\quad\quad\quad\quad&\eta_{11}&=\left\|\left\{2^{k\alpha }|\sum^{k-2}_{j=-\infty}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|
\right\}^{\infty}_{k=-\infty}\right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})},\\
&\eta_{12}&=\left\|\left\{2^{k\alpha }|\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|\right\}^{\infty}_{k=-\infty}
\right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})},\\
&\eta_{13}&=\left\|\left\{2^{k\alpha }|\sum^{\infty}_{j=k+2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|\right\}^{\infty}_{k=-\infty}
\right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})},
\end{eqnarray*}
and
$$\eta=\eta_{11}+\eta_{12}+\eta_{13}=\sum^{3}_{i=1}\eta_{1i}.$$
Thus,
$$\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }|\mu^{\rho}_{\Omega}(h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C.$$
Meanwhile,
$$\|\mu^{\rho}_{\Omega}(h)\|_{\dot{K}^{\alpha,q_{2}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}\leq C\eta= C \sum^{3}_{i=1}\eta_{1i}.$$
To show Theorem 1, we only need to estimate \(\eta_{11},\eta_{12}\text{and}\eta_{13}\leq C \|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}\). To do this, denote \(\eta_{10}=\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}.\)
Step 1.For \(\eta_{12}\). From Lemma 5, we get
\begin{eqnarray}\label{4.1}
\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }
|\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}
&\leq&\sum^{\infty}_{k=-\infty}\left\|\frac{2^{k\alpha }
|\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{10}}\right\|^{(q^{1}_{2})_{k}}_{L^{p_{1}(\cdot)}}\nonumber\\
&\leq&\sum^{\infty}_{k=-\infty}\left(\left\|\frac{2^{k\alpha }
|\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}
\right)^{(q^{1}_{2})_{k}},
\end{eqnarray}
(6)
where
$${(q^{1}_{2})_{k}}= \left\{\begin{array}{ll}
(q_{2})_{-},\quad\quad\quad\left\|\left(\frac{2^{k\alpha }
|\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\
(q_{2})_{+},\quad\quad\quad\left\|\left(\frac{2^{k\alpha }
|\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1.
\end{array}\right.$$
So, by using the Lemma 6, Remark 2 and \(h(x)\in \dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})\), we have \(\left\|\frac{2^{k\alpha}|h\chi_{k}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}\leq1\) and \(\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }
|h\chi_{k}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^\frac{{p_{1}(\cdot)}}{{q_{1}(\cdot)}}}\leq1\). Hence
\begin{eqnarray}\label{4.2}
&&\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }
|\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}
\leq C\sum^{\infty}_{k=-\infty}\left(\sum^{k+2}_{j=k-2}\left\|\frac{2^{k\alpha }
|h_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}\right)^{(q^{1}_{2})_{k}}\notag
\\
&&\leq C \sum^{\infty}_{k=-\infty}\left\|\frac{2^{k\alpha }
|h\chi_{k}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}^{(q^{1}_{2})_{k}}\leq C \sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }
|h\chi_{k}|}{\eta_{10}}\right)^{q_{1}(\cdot)}
\right\|^{\frac{(q^{1}_{2})_{k}}{(q_{1})_{+}}}_{L^{{\frac{p_{1}(\cdot)}{q_{1}(\cdot)}}}}\leq C \left\{\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }
|h\chi_{k}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{1}(\cdot)}}}}\right\}^{q_{*}}\leq C.\notag\\&&
\end{eqnarray}
(7)
Which, together with \((p_{1})_{+}\leq(p_{2})_{-}\leq(q^{1}_{2})_{k}\) and \(q_{*}= \min\limits_{k\in N}\frac{(q^{1}_{2})_{k}}{(q_{1})_{+}}\) gives;
\begin{equation}\label{4.3}\eta_{12}\leq C\eta_{10}\leq C\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}.\end{equation}
(8)
Step 2.
Now, let us deal with \(\eta_{11}\). Since
\begin{eqnarray*}
&\qquad|\mu_{\Omega}^{\rho}(h_{j})(x)|&
:= \left(\int^{\infty}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\mathrm{d}y
\right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\
&&\leq \left(\int^{|x|}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y
\right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\
&&\quad+\left(\int_{|x|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y
\right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\
&&:=\eta_{11}’+\eta_{11}”.
\end{eqnarray*}
Now we estimate \(\eta_{11}’\text{and}\eta_{11}”\).
For \(\eta_{11}’\), note that \(x\in A_{k},y\in A_{j}\) and \(j\leq k-2.\) Since \(|x-y|\sim|x|\) so by virtue of the Mean Value Theorem, we have
\begin{equation}\label{4.4}\left|\frac{1}{|x-y|^{2\rho}}-\frac{1}{|x|^{2\rho}}\right|\leq C \frac{|y|}{|x-y|^{2\rho+1}}.\end{equation}
(9)
Substituting the inequality (9) into \(\eta_{11}’\) and by virtue of Minkowski’s inequality, we deduced that
\begin{eqnarray}\label{4.5}
\eta_{11}’&\leq& C\int_{\mathbb{R}^{n}}\frac{\left|\Omega(x-y)\right|}{|x-y|^{n-\rho}}
|h_{j}(y)|\left(\int^{|x|}_{|x-y|}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\text{d}y\leq C\int_{\mathbb{R}^{n}}\frac{\left|\Omega(x-y)\right|}{|x-y|^{n-\rho}}
|h_{j}(y)|\left|\frac{1}{|x-y|^{2\rho}}-\frac{1}{|x|^{2\rho}}\right|^{1/2}\text{d}y\notag\\
&\leq& C\int_{\mathbb{R}^{n}}\frac{\left|\Omega(x-y)\right|}{|x-y|^{n-\rho}}
|h_{j}(y)|\frac{|y|^{1/2}}{|x-y|^{\rho+1/2}}\text{d}y\leq C\frac{2^{j/2}}{|x|^{n+1/2}}\int_{A_{j}}\left|\Omega(x-y)\right||h_{j}(y)|\text{d}y\notag\\
&\leq& C2^{j/2}2^{-k(n+1/2)}\int_{A_{j}}\left|\Omega(x-y)\right||h_{j}(y)|\text{d}y\leq C2^{(j-k)/2}2^{-nk}\int_{A_{j}}\left|\Omega(x-y)\right||h_{j}(y)|\text{d}y.
\end{eqnarray}
(10)
Similarly, we obtain
\begin{eqnarray}\label{4.6}
\eta_{11}”&\leq& C\int_{\mathbb{R}^{n}}\frac{\left|\Omega(x-y)\right|}{|x-y|^{n-\rho}}
|h_{j}(y)|\left(\int^{\infty}_{|x|}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\text{d}y\leq C\int_{\mathbb{R}^{n}}\frac{\left|\Omega(x-y)\right|}{|x-y|^{n-\rho}}
|h_{j}(y)|\left(\frac{1}{|x|^{2\rho}}\right)^{1/2}\text{d}y\notag\\
&\leq& C\int_{\mathbb{R}^{n}}\frac{\left|\Omega(x-y)\right|}{|x-y|^{n}}|h_{j}(y)|\text{d}y\leq C2^{-nk}\int_{A_{j}}\left|\Omega(x-y)\right||h_{j}(y)|\text{d}y.
\end{eqnarray}
(11)
Combining the inequality (11) with Lemma 1, we get
\begin{eqnarray}\label{4.7}
|\mu_{\Omega}^{\rho}(h_{j})(x)| &\leq& C 2^{-nk}\int_{A_{j}}\left|\Omega(x-y)\right||h_{j}(y)|\text{d}y\leq C 2^{-nk}\|\left(\Omega(x-\cdot)\right).\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}}\|h_{j}\|_{L^{p_{1}(\cdot)}}.
\end{eqnarray}
(12)
Now, consider \(\tilde{p}_{1}'(\cdot)>1\) and \(1/p_{1}'(x)=1/\tilde{p}’_{1}(x)+1/s\). Since \(s>(p_{1}’)_{+}\), so by virtue of Lemma 1 and Lemma 8, we get
\begin{eqnarray}\label{4.8}
&&\notag\|\left(\Omega(x-\cdot)\right).\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} \leq \|\Omega(x-\cdot)\|_{L^{s}}\|\chi_{B_{j}}\|_{L^{\tilde{p}'(\cdot)}}
\leq 2^{-jv}\left(\int_{A_{j}}|y|^{sv}|\Omega(x-y)|^{s}\text{d}y\right)^{1/s}
\|\chi_{B_{j}}\|_{L^{\tilde{p}_{1}'(\cdot)}}\\
&&\notag \leq 2^{-jv}2^{k(v+n/s)}\|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}\|\chi_{B_{j}}\|_{L^{\tilde{p}_{1}'(\cdot)}}\leq 2^{-jv}2^{k(v+n/s)}\|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}
\|\chi_{B_{j}}\|_{L^{{p}_{1}'(\cdot)}}/|B_{j}|^{1/s}\\
&&\leq 2^{(k-j)(v+n/s)}\|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}
\|\chi_{B_{j}}\|_{L^{{p}_{1}'(\cdot)}}.
\end{eqnarray}
(13)
By using (12), (13), Lemmas 1, 2, 3, 5 and \(\left\|\frac{2^{j\alpha }
|h\chi_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)q_{1}}}\leq1\), we get
\begin{eqnarray}\label{4.9}
&&\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }
|\sum^{k-2}_{j=-\infty}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C \sum^{\infty}_{k=-\infty}\left(\left\|\frac{2^{k\alpha }
|\sum^{\infty}_{j=}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}
\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{(q^{2}_{2})_{k}}\notag\\
&&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{k-2}_{j=-\infty}2^{-kn}2^{(k-j)(v+n/s)}\left\|\frac{h_{j}}{\eta_{10}}
\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}
\|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{(q^{2}_{2})_{k}}\notag\\
&&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{k-2}_{j=-\infty}2^{(k-j)(v+n/s)}\left\|\frac{h\chi_{j}}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}
\frac{\|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B_{k}}
\|_{L^{p^{\prime}_{1}(\cdot)}(\mathbb{R}^{n})}}
\right)^{(q^{2}_{2})_{k}}\notag\\
&&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{k-2}_{j=-\infty}2^{(k-j)(v+n/s)}2^{-j\alpha}\left\|\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}
\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}
\frac{\|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B_{k}}
\|_{L^{p^{\prime}_{1}(\cdot)}(\mathbb{R}^{n})}}\right)^{(q^{2}_{2})_{k}}\notag\\
&&\leq C \sum^{\infty}_{k=-\infty}\left\{
\sum^{k-2}_{j=-\infty}2^{(k-j)(\alpha+v+n/s-n\delta_{2})}\left\|\left(\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}
\right)^{q_{1}(\cdot)}\right\|^{\frac{1}{(q_{1})+}}_{L^{p_{1}(\cdot)q_{1}(\cdot)}(\mathbb{R}^{n})}
\right\}^{(q^{2}_{2})_{k}},
\end{eqnarray}
(14)
where
$${(q^{2}_{2})_{k}}= \left\{\begin{array}{ll}
(q_{2})_{-},\quad\quad\quad\left\|\left(\frac{2^{k\alpha }
|\sum^{k-2}_{j=-\infty}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\
(q_{2})_{+},\quad\quad\quad\left\|\left(\frac{2^{k\alpha }
|\sum^{k-2}_{j=-\infty}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1.
\end{array}\right.$$
Which, together with \((q_{1})_{+}< 1\) and \((p_{1})_{+}\leq(p_{2})_{-}\leq(q^{2}_{2})_{k}\) gives;\\
\(
\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }
|\sum^{k-2}_{j=-\infty}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C \left\{\sum^{\infty}_{j=-\infty}
\left\|\left(\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}\right)^{q_{1}(\cdot)}
\right\|_{L^{p_{1}(\cdot)q_{1}(\cdot)}}\sum^{\infty}_{k=j+2}2^{(k-j)(\alpha+v+n/s-n\delta_{2})}
\right\}^{q_{*}}\)
\begin{eqnarray}\label{4.10}
\leq C,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\end{eqnarray}
(15)
where \(q_{*}= \min\limits_{k\in\mathbb{N}}\frac{(q^{1}_{2})_{k}}{(q_{1})_{+}}.\)
Since \(\alpha< n\delta_{2}-(v+n/s)\), so if \((q_{1})_{+}\geq1\) and \((q^{2}_{2})_{k}\geq(q_{2})_{-}\geq(q_{1})_{+}\geq1\) then by using Remark 2% correct remark number
and applying the generalized Hölder's inequality, we get
\begin{align}\label{4.11}
\quad\quad&\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }
|\sum^{k-2}_{j=-\infty}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\notag\\
&\leq C \sum^{\infty}_{k=-\infty}\left\{\sum^{k-2}_{j=-\infty}
2^{(k-j)(\alpha+v+n/s-n\delta_{2})(q_{1})_{+}/2}
\left\|\left(\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^{p_{1}(\cdot)q_{1}(\cdot)}}
\right\}^{\frac{(q^{2}_{2})_{k}}{(q_{1})+}}\notag\\
&\quad\times \left(\sum^{k-2}_{j=-\infty}
2^{(k-j)(\alpha+v+n/s-n\delta_{2})((q_{1})_{+})^{\prime}/2}
\right)^{\frac{(q^{2}_{2})_{k}}{((q_{1})+)^{\prime}}}\notag\\
&\leq C \left\{\sum^{\infty}_{j=-\infty}
\left\|\left(\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^{p_{1}(\cdot)q_{1}(\cdot)}}
\sum^{\infty}_{k=j+2}2^{(k-j)(\alpha+v+n/s-n\delta_{2})(q_{1})_{+}/2}
\right\}^{q_{*}}\notag\\
&\leq C,\end{align}
(16)
where \(q_{*}= \min\limits_{k\in\mathbb{N}}\frac{(q^{2}_{2})_{k}}{(q_{1})_{+}}.\) Hence we have
\begin{equation}\label{4.12}
\eta_{11}\leq C\eta_{10}\leq C\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}.\end{equation}
(17)
Step 3.
Finally, we estimate \(\eta_{13}\). For each \(x\in A_{j}\) and \(j\geq k+2\), we have
\begin{align*}
|\mu_{\Omega}^{\rho}(h_{j})(x)|&:= \left(\int^{\infty}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y
\right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\
&\leq \left(\int^{|y|}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y
\right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\
&\quad+\left(\int_{|y|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y
\right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\
&:=\eta_{13}’+\eta_{13}”.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\end{align*}
The estimates of \(\eta_{13}’\) and \(\eta_{13}”\) can be obtained similarly as that of \(\eta_{11}’\) and \(\eta_{11}”\) in Step 2 and we get
\begin{eqnarray}
\eta_{13}’&
\leq& C2^{(j-k)/2}2^{-jn}\int_{A_{j}}\left|\Omega(x-y)\right||h_{j}(y)|\text{d}y,
\label{4.13}\end{eqnarray}
(18)
and
\begin{eqnarray}\label{4.14}
\eta_{13}”&
\leq& C2^{-jn}\int_{A_{j}}\left|\Omega(x-y)\right||h_{j}(y)|\text{d}y.
\end{eqnarray}
(19)
Thus, we have
\begin{eqnarray}\label{4.15}
|\mu_{\Omega}^{\rho}(h_{j})(x)|
\leq C 2^{-jn}\int_{A_{j}}\left|\Omega(x-y)\right||h_{j}(y)|\text{d}y\leq C 2^{-jn}\|\left(\Omega(x-\cdot)\right).\chi_{B_{j}}\|_{L^{p^{\prime}(\cdot)}}\|h_{j}\|_{L^{p(\cdot)}}.
\end{eqnarray}
(20)
Substituting (13) into (20), together with Lemmas 1, 2, 3, 5 and \(\left\|\frac{2^{j\alpha }
|h\chi_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)q_{1}(\cdot)}}\leq1\), we get
\begin{align}\label{4.16}
&\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }
|\sum^{\infty}_{j=k+2}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C \sum^{\infty}_{k=-\infty}\left\|\frac{2^{k\alpha }
|\sum^{\infty}_{j=k+2}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}
\right\|_{L^{p_{1}(\cdot)}}^{(q^{3}_{2})_{k}}\notag\\
&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{\infty}_{j=k+2}2^{-jn}2^{(k-j)(v+n/s)}\right.\left.\times\left\|\frac{h_{j}}{\eta_{10}}
\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}
\|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{(q^{3}_{2})_{k}}\notag\\
&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{\infty}_{j=k+2}2^{-jn}2^{(k-j)(v+n/s)}\left\|\frac{h\chi_{j}}{\eta_{10}}
\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}|B_{j}|
\frac{\|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}(\mathbb{R}^{n})}}{
\|\chi_{B_{j}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}}\right)^{(q^{3}_{2})_{k}}\notag\\
&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{\infty}_{j=k+2}2^{(k-j)(v+n/s)}\left\|\frac{h\chi_{j}}{\eta_{10}}
\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\frac{\|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}(\mathbb{R}^{n})}}{
\|\chi_{B_{j}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}}\right)^{(q^{3}_{2})_{k}}\notag\\
\quad\quad&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{\infty}_{j=k+2}2^{(k-j)(v+n/s)}2^{-j\alpha}\left\|\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}
\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}
\frac{\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{j_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}}
\right)^{(q^{3}_{2})_{k}}\notag\\
&\leq C \sum^{\infty}_{k=-\infty}\left\{
\sum^{\infty}_{j=k+2}2^{(k-j)(\alpha+v+n/s+n\delta_{12})}\left\|\left(
\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}\right)^{q_{1}(\cdot)}
\right\|^{\frac{1}{(q_{1})+}}_{L^{p_{1}(\cdot)q_{1}(\cdot)}(\mathbb{R}^{n})}
\right\}^{(q^{3}_{2})_{k}},
\end{align}
(21)
where
$${(q^{3}_{2})_{k}}= \left\{\begin{array}{ll}
(q_{2})_{-}, \left\|\left(\frac{2^{k\alpha }
|\sum^{\infty}_{j=k+2}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\
(q_{2})_{+}, \left\|\left(\frac{2^{k\alpha }
|\sum^{\infty}_{j=k+2}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1.
\end{array}\right.$$
From above and by an argument similar to that of Step 2,we conclude
\begin{equation}\label{4.17}\eta_{13}\leq C\eta_{10}\leq C\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}.\end{equation}
(22)
The proof is completed.
Theorem 2. Suppose \(b\in \mathrm{BMO}(\mathbb{R}^{n}),m\in\mathbb{N},0< \rho< n,0(p_{1}’)_{+}\) and \(q_{1}(\cdot),q_{2}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) with \((q_{2})_{-}\geq(q_{1})_{+}\). If \(-n\delta_{1}-v-n/s< \alpha< n\delta_{2}-v-n/s\) with \(\delta_{1},\delta_{2}\) as defined in Lemma 2. Then the operator \([b^{m},\mu_{\Omega}^{\rho}]\) is bounded from
\( \dot{K}_{p_{1}(\cdot)}^{\alpha,q_{1}(\cdot)}(\mathbb{R}^{n})\) to
\(\dot{K}_{p_{1}(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n})\) and \(\left({K}_{p_{1}(\cdot)}^{\alpha,q_{1}(\cdot)}(\mathbb{R}^{n})\right)\) to \(
\left({K}_{p_{1}(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n})\right)\).
Proof.
Let \(h(x)\in \dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n}),b\in \mathrm{BMO}(\mathbb{R}^{n})\). We may write
\(h(x)=\sum_{j=-\infty}^{\infty}h(x)\chi_{j}=\sum_{j=-\infty}^{\infty}h_{j}(x).\)
By definition of \(\dot{K}^{\alpha,q(\cdot)}_{p(\cdot)}(\mathbb{R}^{n})\), we have
$$\|[b^{m},\mu_{\Omega}^{\rho}](h)\|_{\dot{K}^{\alpha,q_{2}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}
=\inf\left\{\eta>0 :
\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }|[b^{m},\mu_{\Omega}^{\rho}](h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1\right\}.$$
Since
\begin{eqnarray*}\left\|\left(\frac{2^{k\alpha}|[b^{m},\mu_{\Omega}^{\rho}](h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq\left\|\left(\frac{2^{k\alpha}|\sum^{k-2}_{j=-\infty}
[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\sum^{3}_{i=1}\eta_{2i}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\end{eqnarray*}
\begin{eqnarray*}&\leq&\left\|\left(\frac{2^{k\alpha}|\sum^{\infty}_{j=-\infty}
[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{21}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}+\left\|\left(\frac{2^{k\alpha}|\sum^{k+2}_{j=k-2}
[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{22}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\\&&+\left\|\left(\frac{2^{k\alpha}|\sum^{\infty}_{j=k+2}
[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{23}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}.
\end{eqnarray*}
Let
\begin{eqnarray*}
&&\eta_{21}=\left\|\left\{2^{k\alpha
}|\sum^{k-2}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|\right\}^{\infty}_{k=-\infty}
\right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})},\\
&&\eta_{22}=\left\|\left\{2^{k\alpha }|\sum^{k+2}_{j=k-2}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|\right\}^{\infty}_{k=-\infty}
\right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})},\\
&&\eta_{23}=\left\|\left\{2^{k\alpha }|\sum^{\infty}_{j=k+2}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|\right\}^{\infty}_{k=-\infty}
\right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})},
\end{eqnarray*}
where we put
$$\eta=\eta_{21}+\eta_{22}+\eta_{23}=\sum^{3}_{i=1}\eta_{2i}.$$
Hence,
$$\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }|[b^{m},\mu_{\Omega}^{\rho}](h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C.$$
So, it follows that
$$\|[b^{m},\mu_{\Omega}^{\rho}](h)\|_{\dot{K}^{\alpha,q_{2}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}\leq C\eta= C \sum^{3}_{i=1}\eta_{1i}.$$
Hence, \(\eta_{21},\eta_{22}\text{and}\eta_{23}\leq C \|b\|_{\ast}\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})} \). Denoting that \(\eta_{10}= C\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}.\)\\
{\textbf Step 1.} We estimate \(\eta_{22}\). The proof of Theorem 2 is the same to that of Theorem 1 and we use the similar notation as in the proof \(\eta_{12}\) of Theorem 1. By Lemma 5 and \(\left(L^{p(\cdot)}(\mathbb{R}^{n}),L^{p(\cdot)}(\mathbb{R}^{n})\right)\)-boundedness of the operators \([b^{m},\mu_{\Omega}^{\rho}]\)
, we directly arrive at
$$\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha}|\sum^{k+2}_{j=k-2}
[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}\|b\|_{\ast}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C,$$
which, implies that
\begin{equation}\label{4.18}\eta_{21}\leq C\eta_{10}\|b\|_{\ast}\leq C\|b\|_{\ast}\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}.\end{equation}
(23)
Step 2.
Next we estimate \(\eta_{21}\). Since
\begin{eqnarray*}
&|[b^{m},\mu_{\Omega}^{\rho}](h_{j})(x)|&:= \left(\int^{\infty}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}[b(x)-b(y)]^{m}h_{j}(y)\text{d}y
\right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\
&&\leq \left(\int^{|x|}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}[b(x)-b(y)]^{m}h_{j}(y)\text{d}y
\right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\
&&\quad+\left(\int_{|x|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}[b(x)-b(y)]^{m}h_{j}(y)\text{d}y
\right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\
&&:=\mathfrak{\eta’}_{22}+\mathfrak{\eta”}_{22}.
\end{eqnarray*}
Observe that \(|x-y|\approx|x|\) for each \(x\in A_{k},y\in A_{j}\) and \(j \leq k-2.\) From (9) and applying the Minkowski’s and the generalized H{\”o}lder’s inequality, we get
\begin{eqnarray}\label{4.19}
\mathfrak{\eta’}_{22}&\leq& C\int_{\mathbb{R}^{n}}\frac{\left|\Omega(x-y)\right|}{|x-y|^{n-\rho}}
[b(x)-b(y)]^{m}|h_{j}(y)|\left(\int^{|x|}_{|x-y|}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\text{d}y\notag\\
&\leq& C\int_{\mathbb{R}^{n}}\frac{\left|\Omega(x-y)\right|}{|x-y|^{n-\rho}}
[b(x)-b(y)]^{m}|h_{j}(y)|\left|\frac{1}{|x-y|^{2\rho}}-\frac{1}{|x|^{2\rho}}\right|^{1/2}\text{d}y\notag\\
&\leq& C\int_{\mathbb{R}^{n}}\frac{\left|\Omega(x-y)\right|}{|x-y|^{n-\rho}}
[b(x)-b(y)]^{m}|h_{j}(y)|\frac{|y|^{1/2}}{|x-y|^{\rho+1/2}}\text{d}y\notag\\
&\leq& C\frac{2^{j/2}}{|x|^{n+1/2}}\left\{[b(x)-b_{B_{j}}]^{m}\int_{A_{j}}\left|\Omega(x-y)\right||h_{j}(y)|\text{d}y\right.\left.+\int_{A_{j}}\left|\Omega(x-y)\right|[b_{B_{j}}-b(y)]^{m}|h_{j}(y)|\text{d}y\right\}\notag\\
&\leq& C2^{j/2}2^{-k(n+1/2)}\left\{[b(x)-b_{B_{j}}]^{m}\|\left(\Omega(x-\cdot)\right)
.\chi_{B_{j}}\|_{L^{p_{1}^{\prime}(\cdot)}}\|h_{j}\|_{L^{p_{1}(\cdot)}}\right.\notag\\
&&\quad\left.+\|\Omega(x-\cdot)(b_{B_{j}}-b(\cdot))^{m}(h_{j}).\chi_{B_{j}}\|_{L^{p_{1}^{\prime}(\cdot)}}
\|h_{j}\|_{L^{p_{1}(\cdot)}}\right\}.
\end{eqnarray}
(24)
Similarly, we consider \(\mathfrak{\eta”}_{22}\)
\begin{eqnarray}\label{4.20}
\mathfrak{\eta”}_{22}&\leq& C\int_{\mathbb{R}^{n}}\frac{\left|\Omega(x-y)\right|}{|x-y|^{n-\rho}}
[b(x)-b_{B_{j}}]^{m}|h_{j}(y)|\left(\int^{\infty}_{|x|}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\text{d}y\notag\\
&\leq& C\int_{\mathbb{R}^{n}}\frac{\left|\Omega(x-y)\right|}{|x-y|^{n-\rho}}
[b(x)-b_{B_{j}}]^{m}|h_{j}(y)|\left(\frac{1}{|x|^{2\rho}}\right)^{1/2}\text{d}y\notag\\
&\leq& C2^{-nk}\left\{[b(x)-b_{B_{j}}]^{m}\int_{A_{j}}\left|\Omega(x-y)\right||h_{j}(y)|\text{d}y\right.\left.+\int_{A_{j}}\left|\Omega(x-y)\right|[b_{B_{j}}-b(y)]^{m}|h_{j}(y)|\text{d}y\right\}\notag\\
&\leq& C 2^{-nk}\left\{[b(x)-b_{B_{j}}]^{m} \|\left(\Omega(x-\cdot)\right)\cdot\chi_{{j}}(\cdot)\|_{L^{p_{1}^{\prime}(\cdot)}}
\|h_{j}\|_{L^{p_{1}(\cdot)}}\right.\notag\\
&&\quad\left.+\|\Omega(x-\cdot)(b_{B_{j}}-b(\cdot))^{m}(h_{j})\cdot\chi_{{j}}(\cdot)\|_{L^{p_{1}^{\prime}(\cdot)}}
\|h_{j}\|_{L^{p_{1}(\cdot)}}\right\}\;.
\end{eqnarray}
(25)
Therefore,
\begin{eqnarray}\label{4.21}
|[b^{m},\mu_{\Omega}^{\rho}](h_{j})(x)|&\leq& C2^{-nk}\left\{[b(x)-b_{B_{j}}]^{m}
\|\left(\Omega(x-\cdot)\right)\cdot\chi_{{j}}(\cdot)\|_{L^{p_{1}^{\prime}(\cdot)}}
\|h_{j}\|_{L^{p(\cdot)}}\right.\notag\\
&&\left.+\|\Omega(x-\cdot)(b_{B_{j}}-b(\cdot))^{m}\cdot\chi_{{j}}(\cdot)\|_{L^{p_{1}^{\prime}(\cdot)}}
\|h_{j}\|_{L^{p_{1}(\cdot)}}\right\}.\end{eqnarray}
(26)
By (13) and Lemmas 6 and 7, we get
\begin{eqnarray}\label{4.22}
&&\notag\|\Omega(x-\cdot)(b_{B_{j}}-b(\cdot))^{m}
\cdot\chi_{{j}}(\cdot)\|_{L^{p_{1}^{\prime}(\cdot)}}\leq\|\Omega(x-\cdot)\cdot\chi_{{j}}(\cdot)\|_{L^{s}}\|(b_{B_{j}}-b(\cdot))^{m}
\cdot\chi_{{j}}(\cdot)\|_{L^{\widetilde{p}_{1}^{\prime}(\cdot)}}\\
&&\leq 2^{-jv}2^{k(v+n/s)}\|b\|^{m}_{\ast}\|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}
\|\chi_{B_{j}}\|_{L^{\tilde{p}_{1}'(\cdot)}}\leq 2^{(k-j)(v+n/s)}\|b\|^{m}_{\ast}\|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}
\|\chi_{B_{j}}\|_{L^{{p}_{1}'(\cdot)}}.
\end{eqnarray}
(27)
From this, we deduced
\begin{eqnarray}\label{4.23}
\|[b^{m},\mu_{\Omega}^{\rho}](h_{j})(x)\cdot\chi_{B_{k}}
\|_{L^{p_{1}(\cdot)}}&\leq& C \|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}2^{-nk}2^{(k-j)(v+n/s)}\|h_{j}\|_{L^{p_{1}(\cdot)}}
\|(b(\cdot)-b_{B_{j}})^{m}\cdot\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}
\|\chi_{B_{j}}\|_{L^{p_{1}^{\prime}(\cdot)}}\notag\\
&&\quad+ C \|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}2^{-nk}2^{(k-j)(v+n/s)} \|b\|^{m}_{\ast}
\|h_{j}\|_{L^{p_{1}(\cdot)}}\|\chi_{B_{j}}\|_{L^{{p}_{1}'(\cdot)}}
\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}.
\end{eqnarray}
(28)
Applying Lemmas 1, 3, 4 and 5, we have
\begin{align*}
&\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha}|\sum^{k-2}_{j=-\infty}
[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}\|b\|^{m}_{\ast}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C \sum^{\infty}_{k=-\infty}\left\|\frac{2^{k\alpha}|\sum^{k-2}_{j=-\infty}
[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}\|b\|^{m}_{\ast}}
\right\|^{(q^{2}_{2})_{k}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\
&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{k-2}_{j=-\infty}2^{-kn}2^{(k-j)(v+n/s)}\right.\left.\left\|\frac{| h_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}
\frac{1}{\|b\|^{m}_{\ast}}
\|(b(\cdot)-b_{B_{j}})^{m}\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}\|\chi_{B_{j}}\|_{L^{p_{1}^{\prime}(\cdot)}}
\right)^{(q^{2}_{2})_{k}}
\end{align*}
\begin{align*}
\\
&\quad+ C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{k-2}_{j=-\infty}(k-j)^{m}2^{-kn}2^{(k-j)(v+n/s)}\right.\left.\left\|\frac{| h_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}
\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}
\|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}}\right)^{(q^{2}_{2})_{k}}\\
&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{k-2}_{j=-\infty}(k-j)^{m}2^{-kn}2^{(k-j)(v+n/s)}\right.\left.\left\|\frac{| h_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}
\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}
\|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}}\right)^{(q^{2}_{2})_{k}}\\
&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{k-2}_{j=-\infty}(k-j)^{m}2^{(k-j)(v+n/s)}2^{-j\alpha}
\left\|\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}
\frac{\|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}}}{
\|\chi_{B_{k}}\|_{L^{p^{\prime}_{1}(\cdot)}}}\right)^{(q^{2}_{2})_{k}}.
\end{align*}
Now, by Lemma 2, we have
\begin{align}\label{4.24}
&\notag\sum^{\infty}_{k=-\infty}\left\|\left(
\frac{2^{k\alpha}|\sum^{\infty}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}
\|b\|_{\ast}}\right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\\
&\leq C \sum^{\infty}_{k=-\infty}\left\{
\sum^{k-2}_{j=-\infty}(k-j)^{m}2^{(k-j)(\alpha+v+n/s-n\delta_{2})}\left\|\left(\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}
\right)^{q_{1}(\cdot)}\right\|^{\frac{1}{(q_{1})+}}_{L^{p_{1}(\cdot)q_{1}(\cdot)}(\mathbb{R}^{n})}
\right\}^{(q^{2}_{2})_{k}},
\end{align}
(29)
where
$${(q^{2}_{2})_{k}}= \left\{\begin{array}{ll}
(q_{2})_{-},\quad\quad\quad\left\|\left(\frac{2^{k\alpha }
|\sum^{k-2}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}}
\right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\
(q_{2})_{+},\quad\quad\quad\left\|\left(\frac{2^{k\alpha }
|\sum^{k-2}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}}
\right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1.
\end{array}\right.$$
So, together with \((q_{1})_{+}< 1\), \((p_{1})_{+}\leq(p_{2})_{-}\leq(q^{2}_{2})_{k}\), along with Remark 1, gives
\begin{align}\label{4.25}
\quad\quad\quad\quad&\notag\sum^{\infty}_{k=-\infty}\left\|\left(
\frac{2^{k\alpha}|\sum^{k-2}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}
\|b\|_{\ast}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\\
&\notag\leq C \left\{\sum^{\infty}_{j=-\infty}
\left\|\left(\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}\right)^{q_{1}(\cdot)}
\right\|_{L^{p_{1}(\cdot)q_{1}(\cdot)}}\sum^{\infty}_{k=j+2}(k-j)^{m}2^{(k-j)(\alpha+v+n/s-n\delta_{2})}
\right\}^{q_{*}}\\
&\leq C,
\end{align}
(30)
where \(q_{*}= \min\limits_{k\in N}\frac{(q^{2}_{2})_{k}}{(q_{1})_{+}}.\)
If \((q_{1})_{+} \leq 1\), then by Hölder’s inequality and Remark 1, we have
\begin{align}\label{4.36}
&\notag\sum^{\infty}_{k=-\infty}\left\|\left(
\frac{2^{k\alpha}|\sum^{k-2}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}
\|b\|_{\ast}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\\
&\notag\leq C \sum^{\infty}_{k=-\infty}\left\{\sum^{k-2}_{j=-\infty}
2^{(k-j)(\alpha+v+n/s-n\delta_{2})(q_{1})_{+}/2}
\left\|\left(\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^{p_{1}(\cdot)q_{1}(\cdot)}}
\right\}^{\frac{(q^{2}_{2})_{k}}{(q_{1})+}}\\
&\notag\times \left(\sum^{k-2}_{j=-\infty}(k-j)^{m}2^{(k-j)(\alpha+v+n/s-n\delta_{2})((q_{1})_{+})^{\prime}/2}
\right)^{\frac{(q^{2}_{2})_{k}}{((q_{1})+)^{\prime}}}\\
&\leq C \left\{\sum^{\infty}_{j=-\infty}
\left\|\left(\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^{p_{1}(\cdot)q_{1}(\cdot)}}
\sum^{\infty}_{k=j+2}2^{(k-j)(\alpha+v+n/s-n\delta_{2})(q_{1})_{+}/2}
\right\}^{q_{*}}\leq C,\end{align}
(31)
where \(q_{*}= \min\limits_{k\in N}\frac{(q^{2}_{2})_{k}}{(q_{1})_{+}}.\)
This implies that
\begin{equation}\label{4.27}\eta_{21}\leq C\eta_{10}\|b\|_{\ast}\leq C\|b\|_{\ast}\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}.\end{equation}
(32)
Finally we estimate \(\eta_{23}\). For any \(x\in A_{j},j\geq k+2\), by the same argument as in \(\eta_{21}\), we obtain
\begin{eqnarray*}
|[b^{m},\mu_{\Omega}^{\rho}](h_{j})(x)|
&&:= \left(\int^{\infty}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}[b(x)-b(y)]^{m}h_{j}(y)\text{d}y
\right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\
\quad&&\leq \left(\int^{|y|}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}[b(x)-b(y)]^{m}h_{j}(y)\text{d}y
\right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\
&&\quad+\left(\int_{|y|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}[b(x)-b(y)]^{m}h_{j}(y)\text{d}y
\right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2}\\
&&:=\mathfrak{\eta’}_{23}+\mathfrak{\eta”}_{23}.
\end{eqnarray*}
Noticing that \(j \geq k+2\). To estimate \(\eta_{23}’\) and \(\eta_{23}”\) we will use same method as that of \(\eta_{21}’\) and \(\eta_{21}”\) in Step 2. Since
\begin{eqnarray}\label{4.28}
\mathfrak{\eta’}_{23}
&\leq& C 2^{(k-j)/2}2^{-jn}\left\{[b(x)-b_{B_{j}}]^{m} \|\left(\Omega(x-\cdot)\right)\cdot\chi_{{j}}(\cdot)\|_{L^{p^{\prime}(\cdot)}}
\|h_{j}\|_{L^{p(\cdot)}}\right.\notag\\
&&\left.+\|\Omega(x-\cdot)(b_{B_{j}}-b(\cdot))^{m}(h_{j})\cdot\chi_{{j}}(\cdot)\|_{L^{p^{\prime}(\cdot)}}
\|h_{j}\|_{L^{p(\cdot)}}\right\}\end{eqnarray}
(33)
and
\begin{eqnarray}\label{4.29}
\mathfrak{\eta”}_{23}
&\leq& C 2^{-jn}\left\{[b(x)-b_{B_{j}}]^{m} \|\left(\Omega(x-\cdot)\right)\cdot\chi_{{j}}(\cdot)\|_{L^{p^{\prime}(\cdot)}}
\|h_{j}\|_{L^{p(\cdot)}}\right.\notag\\
&&\left.+\|\Omega(x-\cdot)(b_{B_{j}}-b(\cdot))^{m}(h_{j})\cdot\chi_{{j}}(\cdot)\|_{L^{p^{\prime}(\cdot)}}
\|h_{j}\|_{L^{p(\cdot)}}\right\}.\end{eqnarray}
(34)
Thus,
\begin{eqnarray}\label{4.30}
|[b^{m},\mu_{\Omega}^{\rho}](h_{j})(x)|&\leq& C2^{-jn}\left\{[b(x)-b_{B_{j}}]^{m}\|\left(\Omega(x-\cdot)\right)\cdot\chi_{{j}}(\cdot)\|_{L^{p^{\prime}(\cdot)}}
\|h_{j}\|_{L^{p(\cdot)}}\right.\notag\\
&&\left.+\|\Omega(x-\cdot)(b_{B_{j}}-b(\cdot))^{m}\cdot\chi_{{j}}(\cdot)\|_{L^{p^{\prime}(\cdot)}}
\|h_{j}\|_{L^{p(\cdot)}}\right\}.
\end{eqnarray}
(35)
From (13), by using Lemma 7 and Lemma 2, we get
\begin{eqnarray}\label{4.31}
\|\Omega(x-\cdot)(b_{B_{j}}-b(\cdot))^{m}\cdot\chi_{{j}}(\cdot)\|_{L^{p^{\prime}(\cdot)}}
&\leq&\|\Omega(x-\cdot)\|_{L^{s}}\|(b_{B_{j}}-b(\cdot))^{m}
\cdot\chi_{{j}}(\cdot)\|_{L^{\widetilde{p}^{\prime}(\cdot)}}\notag\\
&\leq& 2^{-jv}2^{k(v+n/s)}\|b\|^{m}_{\ast}\|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}
\|\chi_{B_{j}}\|_{L^{\tilde{p}'(\cdot)}}.
\end{eqnarray}
(36)
Hence, we plug the inequality (36) into (35) and obtain
\begin{eqnarray}\label{4.32}
\|[b^{m},\mu_{\Omega}^{\rho}](h_{j})(x)\chi_{B_{k}}
\|_{L^{p_{1}(\cdot)}}&\leq& C \|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}2^{-jn}2^{(k-j)(v+n/s)}\|h_{j}\|_{L^{p_{1}(\cdot)}}
\|(b(\cdot)-b_{B_{j}})^{m}\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}
\|\chi_{B_{j}}\|_{L^{p_{1}^{\prime}(\cdot)}}\notag\\
&&+ C \|\Omega\|_{L^{s}(\mathbb{S}^{n-1})}2^{-jn}2^{(k-j)(v+n/s)} \|b\|^{m}_{\ast}\|h_{j}\|_{L^{p_{1}(\cdot)}}\|\chi_{B_{j}}\|_{L^{{p}_{1}'(\cdot)}}
\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}.
\end{eqnarray}
(37)
By Lemma 5 and the above inequality, we have
\begin{align*}
&\sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha}|\sum^{\infty}_{j=k+2}
[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}\|b\|^{m}_{\ast}}\right)^{q_{2}(\cdot)}
\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C \sum^{\infty}_{k=-\infty}\left\|\frac{2^{k\alpha}|\sum^{\infty}_{j=k+2}
[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}\|b\|^{m}_{\ast}}
\right\|^{(q^{2}_{2})_{k}}_{L^{p_{1}(\cdot)}}\notag\\
&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{\infty}_{j=k+2}2^{-jn}2^{(k-j)(v+n/s)}\right.\left.\left\|\frac{| h_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}
\frac{1}{\|b\|^{m}_{\ast}}
\|(b(\cdot)-b_{B_{j}})^{m}\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}
\|\chi_{B_{j}}\|_{L^{p_{1}^{\prime}(\cdot)}}\right)^{(q^{2}_{2})_{k}}\end{align*}
\begin{align}\label{4.33}
&\quad+ C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{\infty}_{j=k+2}(j-k)^{m}2^{-jn}2^{(k-j)(v+n/s)}\left\|\frac{| h_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}
\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}
\|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}}\right)^{(q^{2}_{2})_{k}}\notag\\
&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{\infty}_{j=k+2}(j-k)^{m}2^{-jn}2^{(k-j)(v+n/s)}\left\|\frac{| h_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}
\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}\|\chi_{B_{j}}\|_{L^{p^{\prime}_{1}(\cdot)}}\right)^{(q^{2}_{2})_{k}}\notag\\
&\leq C \sum^{\infty}_{k=-\infty}\left(2^{k\alpha }
\sum^{\infty}_{j=k+2}(j-k)^{m}2^{(k-j)(v+n/s)}2^{-j\alpha}
\left\|\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}\right\|_{L^{p_{1}(\cdot)}}
\frac{\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}}{
\|\chi_{B_{j}}\|_{L^{p_{1}(\cdot)}}}\right)^{(q^{2}_{2})_{k}}\notag\\
&\leq C \sum^{\infty}_{k=-\infty}\left\{
\sum^{\infty}_{j=k+2}(j-k)^{m}2^{(k-j)(\alpha+v+n/s+n\delta_{12})}\left\|\left(\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}
\right)^{q_{1}(\cdot)}\right\|^{\frac{1}{(q_{1})+}}_{L^{p_{1}(\cdot)q_{1}(\cdot)}}
\right\}^{(q^{3}_{2})_{k}}
\end{align}
(38)
where
$${(q^{3}_{2})_{k}}= \left\{\begin{array}{ll}
(q_{2})_{-},\quad\quad\quad\left\|\left(\frac{2^{k\alpha }
|\sum^{k-2}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}}
\right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\
(q_{2})_{+},\quad\quad\quad\left\|\left(\frac{2^{k\alpha }
|\sum^{k-2}_{j=-\infty}[b^{m},\mu_{\Omega}^{\rho}](h_{j})\chi_{k}|}{\eta_{10}}
\right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1.
\end{array}\right.$$
Hence, by the similar argument to Theorem 1, we arrive at \(\eta_{23}\leq C\eta_{10}\|b\|_{\ast}\leq C\|b\|_{\ast}\|h\|_{\dot{K}^{\alpha,q_{1}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}\;.\) This completes the proof.
Theorem 3.
Let \(b\in\dot{\Lambda}_{\gamma}(\mathbb{R}^{n}),0< \gamma\leq1,m\in\mathbb{N},0< \rho< n,0< v\leq1.\) Suppose that \(q^{+}_{1}q^{+}_{2})\) with \(1\leq r’(p_{1}’)_{+}\) and \(q_{1}(\cdot),q_{2}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) with \((q_{2})_{-}\geq(q_{1})_{+}\). If \(-n\delta_{1}-v-n/s< \alpha< n\delta_{2}-v-n/s\) with \(\delta_{1},\delta_{2}\) as defined in Lemma 2, then the operator \([b^{m},\mu_{\Omega}^{\rho}]\) is bounded from
\( \dot{K}_{p_{1}(\cdot)}^{\alpha,q_{1}(\cdot)}(\mathbb{R}^{n})\)
to
\(\dot{K}_{p_{1}(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n}\) and from \(\left({K}_{p_{1}(\cdot)}^{\alpha,q_{1}(\cdot)}(\mathbb{R}^{n})\right)\) to
\(\left({K}_{p_{1}(\cdot)}^{\alpha, q_{2}(\cdot)}(\mathbb{R}^{n})\right)\).
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the
final manuscript.
Competing Interests
The author(s) do not have any competing interests in the manuscript.
