Let \(0<\rho<n\) and \(\mu_{\Omega}^{\rho}\) be the Parametrized Marcinkiewicz integrals operator. In this work, the bondedness of \(\mu_{\Omega}^{\rho}\) is discussed on Herz spaces \(\dot{K}_{p(\cdot)}^{\alpha,q(\cdot)}(\mathbb{R}^{n})\), where the two main indices are variable exponent. The boundedness of the commutators generated by BOM function, Lipschitz function and parametrized Marcinkiewicz integrals operator is also discussed.
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
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;
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 $$L^{p(\cdot)}(\Upsilon)= \left\{{ h \mbox{is measurable} : \int_{\Omega}\left(\frac{|h(x)|}{\eta}\right)^{p(x)} dx < \infty} \mbox{for some constant } \eta > 0\right\}$$
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 byDefinition 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 $$ \dot{K}_{p(\cdot)}^{\alpha, q(\cdot)}(\mathbb{R}^{n})= \{h\in {L_{loc}^{p(\cdot)}}(\mathbb{R}^{n}\backslash\{0\}) : \|h\|_{\dot{K}_{p(\cdot)}^{\alpha, q(\cdot)}(\mathbb{R}^{n})}< \infty \},$$ where \begin{eqnarray*} \|h\|_{\dot{K}_{p_{(\cdot)}}^{\alpha,q(\cdot)}(\mathbb{R}^{n})}&=&\left\| \{ 2^{k \alpha}|h\chi_{k}|\}_{k=0}^{\infty}\right\|_{l^{q(\cdot)}(L^{p(\cdot)})}=\inf\left\{ \eta> 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 \(v\in \mathbb{N},a_{v}\geq 0,1\leq p_{v} < \infty\), then $$\sum\limits_{v=0}^{\infty} a_{v}\leq \left(\sum\limits_{v=0}^{\infty} a_{v} \right)^{p_{\ast}},$$ where $$ p_{\ast}= \left\{\begin{array}{ll} \min\limits_{v\in \mathbb{N} }p_{v},\quad\quad\quad\sum\limits_{v=0}^{\infty} a_{v}\leq 1,\\ \max\limits_{v\in \mathbb{N} }p_{v},\quad\quad\quad\sum\limits_{v=0}^{\infty} a_{v}>1. \end{array}\right.$$
Remark 2.[26]
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 \( C_{k}= B_{k}\backslash B_{k-1} , \chi_{k}= \chi_{C_{k}} , \) \; \( k \in{\mathbb{Z}}.\)Proposition 1. [27] Let a function \(p(\cdot): \mathbb{R}^{n} \rightarrow [ 1 , \infty).\) If \(p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) satisfies:
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:
Lemma 2. [18, [19]] Let \(p(\cdot) \in \mathcal{B}(\mathbb{R}^{n})\). If there exist 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 subsets \(R\subset B\), we have:
\[ \frac{\|\chi_{R}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\leq C \frac{|R|}{|B|}, \quad \frac{\|\chi_{R}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p_{u}^{\prime}(\cdot)}(\mathbb{R}^{n})}}\leq C\left(\frac{|R|}{|B|}\right)^{\delta_{2}}, \quad \frac{\|\chi_{R}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\leq C\left(\frac{|R|}{|B|}\right)^{\delta_{1}}. \]Lemma 3. [28] Let \(p(\cdot) \in \mathcal{B}(\mathbb{R}^{n})\), then there exists a constant \(C > 0\) such that for any ball \(B\) 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:
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 \(\tilde{q}(\cdot)\) be 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 \(p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}),\Omega\in L^{s}(\mathbb{S}^{n-1})\) and \(0<\rho0\) independent of \(h\), then \(\mu_{\Omega}^{\rho}\) is bounded from \(L^{p(\cdot)}\) to itself.
Lemma 9. Let \(b\in\mathrm{BMO}(\mathbb{R}^{n})\) and \(m\in\mathbb{N}\). Further let \(p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}),\Omega\in L^{s}(\mathbb{S}^{n-1})\), 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 \(p(\cdot)\in \mathcal{P}(\Omega)\) and \(h:\Omega\times \Omega\rightarrow \mathbb{R}\) is a measurable function (with respect to the product measure) such that \(y\in \Omega, h(\cdot,y)\in L^{p(\cdot)}(\Omega)\), then we have: \[ \left\|\int_{\Omega}h(\cdot,y)dy\right\|_{L^{p(\cdot)}(\Omega)}\leq C \int_{\Omega}\left\|h(\cdot,y)\right\|_{L^{p(\cdot)}(\Omega)}dy. \]
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: \[ \|\mu^{\rho}_{\Omega}(h)\|_{\dot{K}^{\alpha,q_{2}(\cdot)}_{p_{1}(\cdot)}(\mathbb{R}^{n})}=\inf\left\{\eta>0 : \sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha }|\mu^{\rho}_{\Omega}(h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}} \leq1\right\}. \] 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: \[ \eta_{11}=\left\|\left\{2^{k\alpha }|\sum^{k-2}_{j=-\infty}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}| \right\}^{\infty}_{k=-\infty}\right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})}, \] \[ \eta_{12}=\left\|\left\{2^{k\alpha }|\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|\right\}^{\infty}_{k=-\infty} \right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})}, \] \[ \eta_{13}=\left\|\left\{2^{k\alpha }|\sum^{\infty}_{j=k+2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|\right\}^{\infty}_{k=-\infty} \right\|_{\ell^{q_{2}(\cdot)}(L^{p_{1}(\cdot)})}, \] 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:
where:
\[ {(q^{1}_{2})_{k}}= \begin{cases} (q_{2})_{-}, & \left\|\left(\frac{2^{k\alpha} |\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\ (q_{2})_{+}, & \left\|\left(\frac{2^{k\alpha} |\sum^{k+2}_{j=k-2}\mu^{\rho}_{\Omega}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1. \end{cases} \]So, by using 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:
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:
Step 2. Now, let us deal with \(\eta_{11}\). Since:
\[ \begin{aligned} &|\mu_{\Omega}^{\rho}(h_{j})(x)| := \left(\int^{\infty}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\mathrm{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2} \\ &\leq \left(\int^{|x|}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2} \\ &\quad+\left(\int_{|x|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2} \\ &:=\eta_{11}’+\eta_{11}”. \end{aligned} \]Now we estimate \(\eta_{11}’\) and \(\eta_{11}”\). For \(\eta_{11}’\), note that \(x\in A_{k}, y\in A_{j}\), and \(j\leq k-2.\) Since \(|x-y|\sim|x|\), by virtue of the Mean Value Theorem, we have:
Substituting the inequality (9) into \(\eta_{11}’\), and by virtue of Minkowski’s inequality, we deduce that:
Similarly, we obtain:
Combining inequality (11) with Lemma 1, we get:
Now, consider \(\tilde{p}_{1}'(\cdot)>1\) and \(1/p_{1}'(x)=1/\tilde{p}’_{1}(x)+1/s\). Since \(s>(p_{1}’)_{+}\), by virtue of Lemma 1 and Lemma 8, we get:
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:
where:
\[ {(q^{2}_{2})_{k}}= \begin{cases} (q_{2})_{-}, & \left\|\left(\frac{2^{k\alpha} |\sum^{k-2}_{j=-\infty}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\ (q_{2})_{+}, & \left\|\left(\frac{2^{k\alpha} |\sum^{k-2}_{j=-\infty}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1. \end{cases} \]Which, together with \((q_{1})_{+}< 1\) and \((p_{1})_{+}\leq(p_{2})_{-}\leq(q^{2}_{2})_{k}\), gives:
\[ \sum^{\infty}_{k=-\infty}\left\|\left(\frac{2^{k\alpha} |\sum^{k-2}_{j=-\infty}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq C \left\{\sum^{\infty}_{j=-\infty} \left\|\left(\frac{|2^{j\alpha}h\chi_{j}|}{\eta_{10}}\right)^{q_{1}(\cdot)} \right\|_{L^{p_{1}(\cdot)q_{1}(\cdot)}}\sum^{\infty}_{k=j+2}2^{(k-j)(\alpha+v+n/s-n\delta_{2})} \right\}^{q_{*}} \]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 and applying the generalized Hölder's inequality, we get:
where \(q_{*}= \min\limits_{k\in\mathbb{N}}\frac{(q^{2}_{2})_{k}}{(q_{1})_{+}}.\) Hence, we have:
Step 3. Finally, we estimate \(\eta_{13}\). For each \(x\in A_{j}\) and \(j\geq k+2\), we have:
\[ \begin{aligned} |\mu_{\Omega}^{\rho}(h_{j})(x)|&:= \left(\int^{\infty}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2} \\ &\leq \left(\int^{|y|}_{0}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2} \\ &\quad+\left(\int_{|y|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}h_{j}(y)\text{d}y \right|^{2}\frac{\text{d}t}{t^{2\rho+1}}\right)^{1/2} \\ &:=\eta_{13}’+\eta_{13}”. \end{aligned} \]The estimates of \(\eta_{13}’\) and \(\eta_{13}”\) can be obtained similarly as those of \(\eta_{11}’\) and \(\eta_{11}”\) in Step 2, and we get:
Thus, we have:
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:
where:
\[ {(q^{3}_{2})_{k}}= \begin{cases} (q_{2})_{-}, & \left\|\left(\frac{2^{k\alpha} |\sum^{\infty}_{j=k+2}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}\leq1, \\ (q_{2})_{+}, & \left\|\left(\frac{2^{k\alpha} |\sum^{\infty}_{j=k+2}\mu_{\Omega}^{\rho}(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{{\frac{p_{1}(\cdot)}{q_{2}(\cdot)}}}}>1. \end{cases} \]From the above and by an argument similar to that of Step 2, we conclude:
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
The author(s) do not have any competing interests in the manuscript.