The main purpose of this paper is to establish the weighted boundedness result of vector valued bilinear ϖ(t)-type Calderón-Zygmund operators in variable exponents Herz-Morrey spaces, where ϖ being nondecreasing and ϖ∈Dini(1).
Citation: Yanqi Yang, Qi Wu. Vector valued bilinear Calderón-Zygmund operators with kernels of Dini's type in variable exponents Herz-Morrey spaces[J]. AIMS Mathematics, 2023, 8(11): 25688-25713. doi: 10.3934/math.20231310
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The main purpose of this paper is to establish the weighted boundedness result of vector valued bilinear ϖ(t)-type Calderón-Zygmund operators in variable exponents Herz-Morrey spaces, where ϖ being nondecreasing and ϖ∈Dini(1).
In 1985, Yabuta [1] proposed the definitions of ϖ(t)-type Calderón-Zygmund operators, he introduced certain ϖ(t)-type Calderón-Zygmund operators to facilitate his study of certain classes of pseudodifferential operators. After that, Maldonado and Naibo [2] established the weighted norm inequalities for the bilinear Calderón-Zygmund operators of type ϖ(t), and applied them to the study of para-products and bilinear pseudo-differential operators with mild regularity. In 2009, Lu and Zhang [3] established the a number of results concerning boundedness of multi-linear ϖ(t)-type Calderón-Zygmund operators. we recall the so-called ϖ(t)-type Calderón-Zygmund operators.
Let ϖ(t): [0,∞)→[0,∞) be a nondecreasing function with 0<ϖ(1)<∞. For α>0, we say that ϖ∈Dini(a) if
|ϖ|Dini(α)=∫10ϖα(t)tdt<∞. | (1.1) |
It is evident that for 0<α1<α2, there is Dini(α1)<Dini(α2). If ϖ∈Dini(1), then
∞∑0ϖ(2−j)≈∫10ϖ(t)tdt<∞, |
here and in what follows, for any quantities A and B, if there exists a constant C>0 such that A≤CB, we write A≲B. If A≲B and B≲A, we write A≈B.
A measurable function K(⋅,⋅,⋅) on Rn×Rn×Rn∖{(x,y1,y2):x=y1=y2} is said to be a bilinear ϖ(t)-type Calderón-Zygmund kernel if it satisfies: for all (x,y1,y2)∈Rn with x≠yi,i=1,2, if there exists a constant A>0 such that
|K(x,y1,y2)|≤Aϖ(2∑i=1|x−yi|)−2n, | (1.2) |
and for (x,y1,y2)∈(Rn)3 with x≠y1,y2, and
|K(x,y1,y2)−K(z,,y1,y2)|≤Aω(|x−z|∑2i=1|x−yi|)[2∑i=1|x−yi|]−2n. | (1.3) |
whenever 2|x−z|<max{|x−y1|,|x−y2|}.
Definition 1.1. ([2]) Let ϖ∈Dini(1). One can say that Tϖ is a bilinear ϖ(t)-type operator with the kernel K satisfying (1.2) and (1.3), for all f1, f2∈C∞c(Rn),
Tϖ(f1,f2)(x)=∫Rn∫RnK(x,y1,y2)f1(y1)f2(y2)dy1dy2,x∉suppf1∩suppf2. | (1.4) |
In the following, for each k∈Z, we define Bk={x∈Rn:|x|≤2k}, Dk=Bk∖Bk−1, χk=χDk, m≥1, ˜χ0=χB0.
Given a function p(x)∈P(Rn), the space Lp(x)(Rn) is now defined by
‖ |
Denote \mathcal{P}(\mathbb{R}^{n}) to be the set of the all measurable functions p(x) with
p_{-} = :\mathrm{ess}\inf\limits_{x\in \mathbb{R}^{n}}p(x) > 1 |
and
p_{+} = :\mathrm{ess}\sup\limits_{x\in \mathbb{R}^{n}}p(x) < \infty, |
and \mathcal{B}(\mathbb{R}^{n}) to be the set of all functions p(\cdot)\in \mathcal{P}(\mathbb{R}^{n}) satisfying the condition that the Hardy-littlewood maximal operator M is bounded on L^{p(\cdot)}(\mathbb{R}^{n}) , \mathcal{P}^{0}(\mathbb{R}^{n}) the set of all measurable functions p(x) with p_{-} > 0 and p_{+} < \infty .
The space L_{\rm{loc}}^{p(\cdot)}(\mathbb{R}^{n}) is defined by
L_{\rm{loc}}^{p(\cdot)}(\mathbb{R}^{n}) = \{f:f_{\chi_{K}}\in L_{\rm{loc}}^{p(\cdot)}(\mathbb{R}^{n})\ \text{for all compact subsets K} \in \mathbb{R}^{n}\}, |
where and what follows, \chi_{S} denotes the characteristic function of a measurable set S \subset \mathbb{R}^{n} .
Let p(\cdot)\in \mathcal{P}(\mathbb{R}^{n}) and \omega be a nonnegative measurable function on \mathbb{R}^{n} . Then the weighted variable exponent Lebesgue space L^{p(\cdot)}(\omega) is the set of all complex-valued measurable functions f such that f\omega\in L^{p(\cdot)} . The space L^{p(\cdot)}(\omega) is a Banach space equipped with the norm
\|f\|_{L^{p(\cdot)}(\omega)} = \|f\omega\|_{L^{p(\cdot)}}. |
Let f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n}) . Then the standard Hardy-Littlewood maximal function of f is defined by
Mf(x) = \sup\limits_{x\in B}\frac{1}{|B|}\int_{B}f(y)\mathrm{d}y,\; \; \; \forall\; x\in \mathbb{R}^{n}, |
where the supremum is taken over all balls containing x in \mathbb{R}^{n} .
\textbf{Definition 1.2.} ([4]) Let \alpha(\cdot) be a real-valued function on \mathbb{R}^{n} .
(ⅰ) For any x, y\in \mathbb{R}^{n} , |x-y| < 1/2 , if
|\alpha(x)-\alpha(y)|\lesssim \frac{1}{\log(e+1/|x-y|)}, |
then \alpha(\cdot) is said local \log -Hölder continuous on \mathbb{R}^{n} .
(ⅱ) For all x\in \mathbb{R}^{n} , if
|\alpha(x)-\alpha(0)|\lesssim\frac{1}{\log(e+1/|x|)}, |
then \alpha(\cdot) is said \log -Hölder continuous functions at origin, denote by \mathcal{P}_{0}^{\rm{log}}(\mathbb{R}^{n}) the set of all \log -Hölder continuous at origin.
(ⅲ) If there exists \alpha_{\infty}\in \mathbb{R} , for x\in \mathbb{R}^{n} , if
|\alpha(x)-\alpha_{\infty}|\lesssim \frac{1}{\log(e+|x|)}, |
then \alpha(\cdot) is said \log -Hölder continuous at infinity, denote by \mathcal{P}_{\infty}^{\rm{log}}(\mathbb{R}^{n}) the set of all \log -Hölder continuous functions at infinity.
(ⅳ) The function \alpha(\cdot) is global \log -Hölder continuous if \alpha(\cdot) are both locally \log -Hölder continuous and \log -Hölder continuous at infinity. Denote by \mathcal{P}^{\rm{log}}(\mathbb{R}^{n}) the set of all global \log -Hölder continuous functions.
Let \omega be a weighted function on \mathbb{R}^{n} , that is, \omega is real-valued, non-negative and locally integrable. \omega is said to be a Muckenhoupt A_{1} weight if
M\omega(x)\lesssim \omega(x)\; \; \; \; a.e., x\in \mathbb{R}^{n}. |
For 1 < p < \infty , we say that \omega is an A_{p} weight if
\sup\limits_{B}\left(\frac{1}{|B|}\int_{B}\omega(x)\mathrm{d}x \right) \left(\frac{1}{|B|}\int_{B}\omega(x)^{1-p^{\prime}}\mathrm{d}x\right)^{p-1} < \infty. |
\textbf{Definition 1.3.} ([5]) Let p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) . For some constant C , a weight \omega is said to be an A_{p(\cdot)} weight, if for all balls B in \mathbb{R}^{n} such that
\frac{1}{|B|}\|\omega\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|\omega^{-1}\chi_{B}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})}\leq C. |
\textbf{Lemma 1.1.} ([5]) If p(\cdot)\in \mathcal{P}^{\rm{log}}(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n}) and \omega\in A_{p(\cdot)} , then for each f\in L^{p(\cdot)}(\omega) ,
\|(Mf)\omega\|_{L^{p(\cdot)}}\lesssim \|f\omega\|_{L^{p(\cdot)}}, |
Before give the definitions of the weighted Herz space and Herz-Morrey space with variable exponents, we also need the notation of the variable mixed sequence space \ell^{q}(L^{p(\cdot)}) , which was firstly defined in [6]. Let \omega be a nonnegative measurable function. Given a sequence of functions \{f_{j}\}_{j\in \mathbb{Z}} , we define the modular
\rho_{\ell^{q}(L^{p(\cdot)}(\omega))}((f_{j})_{j}) = \sum\limits_{j\in \mathbb{Z}}\inf\Big\{\lambda_{j}:\int_{\mathbb{R}^{n}}\Big(\frac{|f_{j}(x)\omega(x)|}{\lambda_{j}^{\frac{1}{q(x)}}}\Big)^{p(x)}\mathrm{d}x\leq 1\Big\}, |
where \lambda^{\frac{1}{\infty}} = 1 . If q^{+} < \infty or q(\cdot)\leq p(\cdot) , the above can be written as
\rho_{\ell^{q}(L^{p(\cdot)}(\omega))}((f_{j})_{j}) = \sum\limits_{j\in \mathbb{Z}}\left\|f_{j}\omega|^{q(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q(\cdot)}}}. |
The norm is
\|(f_{j})_{j}\|_{\rho_{\ell^{q}(L^{p(\cdot)}(\omega))}} = \inf\bigg\{\mu > 0:\rho_{\ell^{q}(L^{p(\cdot)}(\omega))}((\frac{f_{j}}{\mu})_{j})\leq 1\bigg\}. |
\textbf{Definition 1.4.} ([7]) Let p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) , q \in \mathcal{P}^{0}(\mathbb{R}^{n}) . Let \alpha(\cdot) be a bounded real-valued measurable function on \mathbb{R}^{n} . The homogeneous weighted Herz space \dot{K}^{\alpha(\cdot), q(\cdot)}_{p(\cdot)}(\omega) are defined by
\dot{K}^{\alpha(\cdot),q(\cdot)}_{p(\cdot)}(\omega) = \{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb{R}^{n}\backslash\{0\},\omega):\|f\|_{\dot{K}^{\alpha(\cdot),q(\cdot)}_{p(\cdot)}(\omega)} < \infty\}, |
where
\|f\|_{\dot{K}^{\alpha(\cdot),q(\cdot)}_{p(\cdot)}(\omega)} = \|(2^{j\alpha(\cdot)}f\chi_{j})_{j}\|_{\rho_{\ell^{q}(L^{p(\cdot)}(\omega))}}. |
\textbf{Lemma 1.2.} ([7]) Let \alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n}) , p(\cdot), q(\cdot)\in \mathcal{P}^{0}(\mathbb{R}^{n}) and \omega be a weight. If \alpha(\cdot) and q(\cdot) are \log -Hölder continuous at the origin, then T
\|f\|_{\dot{K}^{\alpha(\cdot),q(\cdot)}_{p(\cdot)}(\omega)} = \|f\|_{\dot{K}^{\alpha_{\infty},q_{\infty}}_{p(\cdot)}(\omega)}. |
Additionally, if \alpha(\cdot) and q(\cdot) are log-Hölder continuous at the origin, then
\|f\|_{\dot{K}^{\alpha(\cdot),q(\cdot)}_{p(\cdot)}(\omega)} \approx \Big(\sum\limits_{k\leq 0}\|2^{k\alpha(0)}f\chi_{k}\|_{L^{p(\cdot)}}^{q(0)}\Big)^{\frac{1}{q(0)}}+ \Big(\sum\limits_{k > 0}\|2^{k\alpha_{\infty}}f\chi_{k}\|_{L^{p(\cdot)}}^{q(0)}\Big)^{\frac{1}{q_{\infty}}}. |
\textbf{Definition 1.5.} ([8]) Let p(\cdot), q(\cdot)\in \mathcal{P}^{0}(\mathbb{R}^{n}) , \lambda\in[0, 1) . Let \alpha(\cdot) be a bounded real-valued measurable function on \mathbb{R}^{n} . The homogeneous weighted Herz-Morrey space M\dot{K}^{\alpha(\cdot), q(\cdot)}_{p(\cdot), \lambda}(\omega) are defined by
M\dot{K}^{\alpha(\cdot),q(\cdot)}_{p(\cdot),\lambda}(\omega) = \{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb{R}^{n}\backslash{\left \{0 \right \}},\omega):\|f\|_{M\dot{K}^{\alpha(\cdot),q(\cdot)}_{p(\cdot),\lambda}(\omega)} < \infty\}, |
where
\|f\|_{M\dot{K}^{\alpha(\cdot),q(\cdot)}_{p(\cdot),\lambda}(\omega)} = \sup\limits_{L\in \mathbb{Z}}2^{-L\lambda}\|(2^{k\alpha(\cdot)k}f\chi_{k})_{k\leq L}\|_{\rho_{\ell^{q}(L^{p(\cdot)}(\omega))}}. |
\textbf{Lemma 1.3.} ([8]) Let p(\cdot), q(\cdot)\in \mathcal{P}^{0}(\mathbb{R}^{n}) , \omega be a weight, \lambda\in [0, \infty) and \alpha\in L^{\infty}(\mathbb{R}^{n}) . If \alpha(\cdot) , q(\cdot)\in \mathcal{P}_{0}^{\rm{log}}(\mathbb{R}^{n})\cap \mathcal{P}_{\infty}^{\rm{log}}(\mathbb{R}^{n}) , then for any f\in L_{\rm{loc}}^{p(\cdot)}(\mathbb{R}^{n}\backslash \left \{ 0 \right \}, \omega) ,
\begin{align*} \|f\|_{M\dot{K}^{\alpha(\cdot),q(\cdot)}_{p(\cdot),\lambda}(\omega)} & \approx \max\bigg\{\sup\limits_{L\leq 0,L\in \mathbb{Z}}2^{-L\lambda}\|(2^{k\alpha(0)}f\chi_{k})_{k\leq L}\|_{l^{q_{0}} (L^{p(\cdot)}(\omega))}, \Big.\\ &\quad\Big.\sup\limits_{L > 0,L\in \mathbb{Z}}\Big[2^{-L\lambda}\|(2^{k\alpha(0)}f\chi_{k})_{k\leq L}\|_{\rho_{\ell^{q_{0}}(L^{p(\cdot)}(\omega))}}+2^{-L\lambda}\|(2^{k\alpha_{\infty}}f\chi_{k})_{k = 0}^{L}\|_{\rho_{\ell^{q_{0}}(L^{p(\cdot)}(\omega))}}\Big]\bigg\}, \end{align*} |
where and hereafter, q_{0} = q(0).
\textbf{Lemma 1.4.} ([8]) If p(\cdot)\in \mathcal{P}^{\rm{log}}(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n}) and \omega \in A_{p(\cdot)} , then there exist constants \delta_{1}, \delta_{2}\in (0, 1) , such that for all balls B in \mathbb{R}^{n} and all measurable subsets S\subset B ,
\frac{\|\chi_{S}\|_{L^{p(\cdot)}(\omega)}}{\|\chi_{B}\|_{L^{p(\cdot)}(\omega)}}\lesssim\left(\frac{|S|}{|B|}\right)^{\delta_{1}}, \; \; \frac{\|\chi_{S}\|_{L^{p^{'}(\cdot)}(\omega^{-1})}}{\|\chi_{B}\|_{L^{p^{'}(\cdot)}(\omega^{-1})}}\lesssim\left(\frac{|S|}{|B|}\right)^{\delta_{2}}. |
Before proving the main results, we need the following lemmas.
For \delta > 0 , we denote [M(|f|^{\delta})]^{\frac{1}{\delta}} by M_{\delta} . Let f\in L_{\rm{loc}}^{1}(\mathbb{R}^n) . Then the sharp maximal function is defined by
M^{\#}f(x) = \sup\limits_{Q}\frac{1}{Q}\int_{Q}|f(y)-f_{Q}|\rm{d}y, |
where the supremum is taken over all the cubes Q containing the point x , and where as usual f_{Q} denotes the average of f on Q . we denote [M^{\#}(|f|^{\delta})]^{\frac{1}{\delta}} by M^{\#}_{\delta} .
\textbf{Lemma 2.1.} ([3]) Let T_{\omega} be a bilinear \omega(t) -type Calderón-Zygmund operator with \varpi\in \rm{Dini}(1) and let 0 < \delta < \frac{1}{2} . Then, for any vector function \vec{f} = (f_{1}, f_{2}) , where each component is smooth and with compact support, the following inequality holds
M^{\#}_{\delta}(T_{\omega}(f_{1},f_{2}))(x)\lesssim M(f_{1})(x)M(f_{2})(x). |
\textbf{Lemma 2.2.} ([9]) Let 0 < p, \; \delta < \infty and \omega\in A_{\infty} . There exists a positive constant C such that
\int_{\mathbb{R}^{n}}[M_{\delta}f(x)]^{p}\omega(x)\rm{d}x\leq \int_{\mathbb{R}^{n}}[M^{\#}_{\delta}f(x)]^{p}\omega(x)\rm{d}x |
for every function f such that the left hand side is finite.
\textbf{Lemma 2.3.} ([10]) Let p(\cdot), \; p_{1}(\cdot), \; p_{2}(\cdot)\in \mathcal{P}^{0}(\mathbb{R}^{n}) such that \frac{1}{p(x)} = \frac{1}{p_{1}(x)}+\frac{1}{p_{2}(x)} . Then for every f\in L^{p_{1}(\cdot)}(\mathbb{R}^{n}) and g\in L^{p_{2}(\cdot)}(\mathbb{R}^{n}) , there exists
\|fg\|_{L^{p(\cdot)}}\lesssim \|f\|_{L^{p_{1}(\cdot)}}\|g\|_{L^{p_{2}(\cdot)}} |
If p\in \mathcal{P}(\mathbb{R}^{n}) , \omega is a weight with \omega = \omega_{1}\times \omega_{2} , there exists
\|fg\|_{L^{p(\cdot)}(\omega)}\lesssim\|f\|_{L^{p_{1}(\cdot)}(\omega_{1})}\|g\|_{L^{p_{2}(\cdot)}(\omega_{2})}. |
\textbf{Lemma 2.4.} ([11]) Let 0 < p\leq \infty , \delta > 0 . Then for non-negative sequence \{a_{j}\}_{j = -\infty}^{\infty} , there exists
\bigg(\sum\limits_{j = -\infty}^{\infty}\bigg(\sum\limits_{k = -\infty}^{\infty}2^{-|k-j|^{\delta}}a_{k}\bigg)^{p}\bigg)^{\frac{1}{p}}\lesssim (\sum\limits_{j = -\infty}^{\infty}a_{j}^{p})^{\frac{1}{p}}, |
when p = \infty , above inequality stands for
\sum\limits_{k = -\infty}^{\infty}(2^{-|k-j|^{\delta}}a_{k})\lesssim \sup\limits_{j\in \mathbb{Z}}a_{j}. |
\textbf{Lemma 2.5.} ([12]) Assume that for some p_{0}\in(0, \infty) and every \omega_{0}\in A_{\infty} , let \mathcal{F} be a family of pairs of non-negative functions such that
\begin{align} \int_{\mathbb{R}^{n}}f(x)^{p_{0}}\omega_{0}(x)\mathrm{d}x\lesssim \int_{\mathbb{R}^{n}}g_{0}(x)^{p_{0}}\omega_{0}(x)\mathrm{d}x,\; \; (f,g)\in \mathcal{F}. \end{align} | (2.1) |
Then for all 0 < p < \infty and \omega_{0}\in A_{\infty} ,
\int_{\mathbb{R}^{n}}f(x)^{p}\omega_{0}(x)\mathrm{d}x\lesssim \int_{\mathbb{R}^{n}}g_{0}(x)^{p}\omega_{0}(x)\mathrm{d}x,\; \; (f,g)\in \mathcal{F}. |
Furthermore, for every p, q\in(0, \infty) , \omega_{0}\in A_{\infty} , and sequences \{(f_{j}, g_{j})\}\in \mathcal{F} ,
\begin{align} \bigg\|\bigg(\sum\limits_{j = 1}^{\infty}(f_{j})\bigg)^{q}\bigg\|_{L^{p}(\omega_{0})}\lesssim \bigg\|\bigg(\sum\limits_{j = 1}^{\infty}(g_{j})\bigg)^{q}\bigg\|_{L^{p}(\omega_{0})}. \end{align} | (2.2) |
\textbf{Lemma 2.6.} ([8]) Assume that for some p_{0} and let F be a family of pairs of non-negative functions such that (2.1) holds. Let p(\cdot)\in \mathcal{P}_{0}(\mathbb{R}^{n}) . If there exists s\leq p_{-} such that \omega^{s}\in A_{\frac{p(\cdot)}{s}} and M is bounded on L^{(\frac{p(\cdot)}{s})^{'}}(\omega^{-s}) . Then for every q\in(1, \infty) and sequence \{(f_{j}, g_{j})\}_{j\in \mathbb{N}}\subset \mathcal{F}
\bigg\|\bigg(\sum\limits_{j = 1}^{\infty}(f_{j})\bigg)^{q}\bigg\|_{L^{p(\cdot)}(\omega)}\lesssim \bigg\|\bigg(\sum\limits_{j = 1}^{\infty}(g_{j})\bigg)^{q}\bigg\|_{L^{p(\cdot)}(\omega)}. |
\textbf{Lemma 2.7.} ([13]) Let p(\cdot)\in \mathcal{P}(\mathbb{R}^{n}) , and \omega be a weight. If the maximal operator M is bounded both on L^{p(\cdot)}(\omega) and L^{p^{'}(\cdot)}(\omega^{-1}) , q\in (1.\infty) , then
\bigg\|\bigg(\sum\limits_{j = 1}^{\infty}(Mf_{j})^{q}\bigg)^{\frac{1}{q}}\bigg\|_{L^{p(\cdot)}(\omega)}\lesssim \bigg\|\bigg(\sum\limits_{j = 1}^{\infty}|f_{j}|^{q}\bigg)^{\frac{1}{q}}\bigg\|_{L^{p(\cdot)}(\omega)}. |
\textbf{Lemma 2.8.} Let T_{\varpi} be a bilinear Calderón-Zygmund operator with \varpi\in \rm{Dini}(1) and p(\cdot)\in \mathcal{P}_{0} such that there exists s\leq p_{-} such that \omega^{s}\in A_{\frac{p(\cdot)}{s}} and M is bounded on L^{(\frac{p(\cdot)}{s})^{'}}(\omega^{-s}) . Suppose that \omega = \omega_{1}\times \omega_{2} and \omega_{i}\in A_{p_{i}(\cdot)}, \; i = 1, 2 . If p_{i}\in \mathcal{P}^{\mathrm{log}}(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})\; (i = 1, 2) satisfying
\frac{1}{p(x)} = \frac{1}{p_{1}(x)}+\frac{1}{p_{2}(x)} |
for x\in \mathbb{R}^{n} . Then for compactly supported bounded functions f_{1}^{j}, f_{2}^{j}\in L^{p_{0}}(\mathbb{R}^{n}) , j\in \mathbb{N} such that
\bigg\|\bigg(\sum\limits_{j = 1}^{\infty}|T_{\varpi}(f_{1}^{j},f_{2}^{j})|^{q}\bigg)^{\frac{1}{q}}\bigg\|_{L^{p(\cdot)}(\omega)}\lesssim \prod\limits_{i = 1}^{2}\bigg\|\bigg(\sum\limits_{j = 1}^{\infty}|f_{i}^{j}|^{q_{i}}\bigg)^{\frac{1}{q_{i}}}\bigg\|_{L^{p_{i}(\cdot)}(\omega_{i})}, |
where q_{i}\in (1, \infty) for i = 1, 2 and
\frac{1}{q} = \frac{1}{q_{1}}+\frac{1}{q_{2}}. |
Proof of Lemma 2.8. Since f_{1}^{j}, f_{2}^{j} are bounded functions with compact support, T_{\varpi}(f_{1}^{j}, f_{2}^{j})\in L^{p}(\mathbb{R}^{n}) for every 0 < p < \infty . With Lemmas 2.1 and 2.2, Lu and Zhang [3] showed that for all \omega\in A_{\infty} ,
\int _{\mathbb{R}^{n}}|T_{\varpi}(f_{1},f_{2})(x)|^{p}\omega(x)\mathrm{d}x\lesssim \int _{\mathbb{R}^{n}}(Mf_{1}(x)Mf_{2}(x))^{p}\omega(x)\mathrm{d}x. |
Therefore, by Lemmas 2.5 and 2.6, we have
\bigg\|\bigg(\sum\limits_{j = 1}^{\infty}|T_{\varpi}(f_{1}^{j},f_{2}^{j})|^{q}\bigg)^{\frac{1}{q}}\bigg\|_{L^{p(\cdot)}(\omega)}\lesssim \bigg\|\bigg(\sum\limits_{j = 1}^{\infty}|Mf^{j}_{1}(x)Mf^{j}_{2}(x)|^{q}\bigg)^{\frac{1}{q}}\bigg\|_{L^{p(\cdot)}(\omega)}. |
Since
\frac{1}{q} = \frac{1}{q_{1}}+\frac{1}{q_{2}},\ \ \ \ \frac{1}{p} = \frac{1}{p_{1}}+\frac{1}{p_{2}} |
and \omega = \omega_{1}\omega_{2} , together with Hölders inequality, Lemmas 2.3 and 2.7, we have
\begin{align*} \Big\|\Big(\sum\limits_{j = 1}^{\infty}|Mf^{j}_{1}(x)Mf^{j}_{2}(x)|^{q}\Big)^{\frac{1}{q}}\Big\|_{L^{p(\cdot)}(\omega)}&\lesssim \prod\limits_{i = 1}^{2}\Big\|\Big(\sum\limits_{j = 1}^{\infty}|Mf^{j}_{i}|^{q_{i}}\Big)^{\frac{1}{q_{i}}}\Big\|_{L^{p_{i}(\cdot)}(\omega_{i})}\\ &\lesssim \prod\limits_{i = 1}^{2}\Big\|\Big(\sum\limits_{j = 1}^{\infty}|f^{j}_{i}|^{q_{i}}\Big)^{\frac{1}{q_{i}}}\Big\|_{L^{p_{i}(\cdot)}(\omega_{i})}. \end{align*} |
We complete the proof of Lemma 2.8.
\textbf{Theorem 3.1.} Let T_{\varpi} be a bilinear \varpi -type Calderón-Zygmund operator with \varpi\in \rm{Dini}(1) , p_{1} and p_{2} \in \mathcal{P}^{\rm{log}}(\mathbb{R}^{n})\cap \mathcal{P}^{\rm{log}}(\mathbb{R}^{n}) santisfying
\frac{1}{p(x)} = \frac{1}{p_{1}(x)}+\frac{1}{p_{2}(x)} |
and p(\cdot)\in \mathcal{P}_{0} such that there exists s\leq p_{-} such that \omega^{s}\in A_{\frac{p(\cdot)}{s}} and M is bounded on L^{(\frac{p(\cdot)}{s})^{'}}(\omega^{-s}) , where \omega = \omega_{1}\omega_{2} and \omega_{i}\in A_{p_{i}(\cdot)} , i = 1, 2 . Suppose that
\alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\cap \mathcal{P}_{0}^{\rm{log}}(\mathbb{R}^{n})\cap \mathcal{P}_{\infty}^{\rm{log}}(\mathbb{R}^{n}),\quad \alpha(0) = \alpha_{1}(0)+\alpha_{2}(0), |
\alpha_{\infty} = \alpha_{1\infty}+\alpha_{2\infty},\quad q(\cdot)\in \mathcal{P}_{0}^{\rm{log}}(\mathbb{R}^{n})\cap \mathcal{P}_{\infty}^{\rm{log}}(\mathbb{R}^{n}), |
\frac{1}{q(0)} = \frac{1}{q_{1}(0)}+\frac{1}{q_{2}(0)},\quad \frac{1}{q_{\infty}} = \frac{1}{q_{1 \infty}}+\frac{1}{q_{2 \infty}}, |
\lambda = \lambda_{1}+\lambda_{2},\quad 0\leq \lambda_{i} < \infty,\quad \delta_{i1}, \delta_{i2}\in (0,1) |
are the constants in Lemma 1.4 for exponents p_{i}(\cdot) and weights \omega_{i}(\; i = 1, 2) . Let r_{i}\in (1, \infty) and
\frac{1}{r} = \frac{1}{r_{1}}+\frac{1}{r_{2}}. |
If \lambda_{i}-n\delta_{i1} < \alpha_{i\infty} , \alpha_{i}(0)\leq n \delta_{i2} , then
\Big\|\Big(\sum\limits_{j = 1}^{\infty}|T_{\varpi}(f_{1}^{j},f_{2}^{j})|^{r}\Big)^{\frac{1}{r}}\Big\|_{M\dot{K}^{\alpha(\cdot),q(\cdot)}_{p(\cdot),\lambda}(\omega)}\lesssim \prod\limits_{i = 1}^{2}\Big\|\Big(\sum\limits_{j = 1}^{\infty}|(f_{i}^{j})|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}^{\alpha_{i}(\cdot),q_{i}(\cdot)}_{p_{i}(\cdot),\lambda_{i}}(\omega_{i})} |
for all f_{i}^{j}\in M\dot{K}^{\alpha_{i}(\cdot), q_{i}(\cdot)}_{p_{i}(\cdot), \lambda_{i}}(\omega_{i}) , j\in \mathbb{N} , i = 1, 2 .
Proof of Theorem 3.1. We only consider bounded compact supported functions for the set of all bounded compactly supported functions is dense in weighted variable Lebesgue spaces (see [13]). Let f_{1}^{v} and f_{2}^{v} be bounded functions with compact support for v\in{\mathbb{N}} and write
f_{i}^{v} = \sum\limits_{l = -\infty}^{\infty}f_{il}^{v}\chi_{l} = \sum\limits_{l = -\infty}^{\infty}f_{il}^{v},\quad i = 1,2,v\in{\mathbb{N}}. |
By Lemma 1.3, we have
\begin{align*} \Big\|\Big(\sum\limits_{v = 1}^{\infty}|T_{\varpi}(f_1^v,f_2^v)|^r)^{\frac{1}{r}}\Big\|_{M\dot{K}_{p(\cdot),\lambda}^{\alpha(\cdot),q(\cdot)}(w)} &\approx\max\Big\{\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big\|(2^{k\alpha(0)}(\sum\limits_{v = 1}^{\infty}|T_{\varpi}(f_1^v,f_2^v)|^r)^{\frac{1}{r}}\chi_k\Big)_{k \leq L}\Big\|_{\ell^{q_0}(L^{p(\cdot)}(w))}\\ &\ \ \ \ \sup\limits_{L > 0,L\in\mathbb{Z}}\Big[2^{-L\lambda}\Big\|\Big(2^{k\alpha(0)}(\sum\limits_{v = 1}^{\infty}|T_{\varpi}(f_1^v,f_2^v)|^r)^{\frac{1}{r}}\chi_k\Big)_{k < 0}\Big\|_{\ell^{q_0}(L^{p(\cdot)}(w))}\\ &\ \ \ \ \ \Big.\Big.+2^{-L\lambda}\Big\|\Big(2^{k\alpha_{\infty}}\Big(\sum\limits_{v = 1}^{\infty}\Big|T_{\varpi}(f_1^v,f_2^v)\Big|^r\Big)^{\frac{1}{r}}\chi_k\Big)_{k = 0}^L\Big\|_{\ell^{q_{\infty}}(L^{p(\cdot)}(w))}\Big]\Big\}\\ & = \max\{E,F\}, \end{align*} |
where
\begin{eqnarray*} E& = & \sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big\|\Big(2^{k\alpha(0)}\Big(\sum\limits_{v = 1}^{\infty}\Big|T_{\varpi}(f_{1}^{v},f_{2}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big)_{k\leq L}\Big\|_{\ell^{q_{0}}(L^{p(\cdot)}(w))},\\ F& = &\sup\limits_{L > 0,L\in\mathbb{Z}}\{G+H\},\\ G& = &2^{-L\lambda}\Big\|\Big(2^{k\alpha(0)}\Big(\sum\limits_{v = 1}^{\infty}\Big|T_{\varpi}(f_{1}^{v},f_{2}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big)_{k < 0}\Big\|_{\ell^{q_{0}}{(L^{p(\cdot)}(w))}},\\ H& = &2^{-L\lambda}\Big\|\Big(2^{k\alpha_{\infty}}\Big(\sum\limits_{v = 1}^{\infty}\Big|T_{\varpi}(f_{1}^{v},f_{2}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big)_{k = 0}^{L}\Big\|_{\ell^{q_{\infty}}{(L^{p(\cdot)}(w))}}. \end{eqnarray*} |
Since to estimate G is essentially similar to estimate E , it is suffice to obtain that E and H are bounded in Herz-Morrey space with variable exponents. It is easy to see that
E\lesssim\sum\limits_{i = i}^9E_i,\quad H\lesssim\sum\limits_{i = i}^9H_i, |
where
\begin{eqnarray*} &E_1 = &\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^L2^{k\alpha(0)q(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = -\infty}^{k-2}\sum\limits_{j = -\infty}^{k-2}T_{\varpi}(f_{1l}^v,f_{2j}^v)\Big|^r\Big)^{\frac{1}{r}}\chi_k\Big\|_{L^{p(\cdot)}(w)}^{q(0)}\Big)^{\frac{1}{q(0)}},\\ &E_2 = &\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^L2^{k\alpha(0)q(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = -\infty}^{k-2}\sum\limits_{j = k-1}^{k+1}T_{\varpi}(f_{1l}^v,f_{2j}^v)\Big|^r\Big)^{\frac{1}{r}}\chi_k\Big\|_{L^{p(\cdot)}(w)}^{q(0)}\Big)^{\frac{1}{q(0)}},\\ &E_3 = &\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^L2^{k\alpha(0)q(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = -\infty}^{k-2}\sum\limits_{j = k+2}^{\infty}T_{\varpi}(f_{1l}^v,f_{2j}^v)\Big|^r\Big)^{\frac{1}{r}}\chi_k\Big\|_{L^{p(\cdot)}(w)}^{q(0)}\Big)^{\frac{1}{q(0)}},\\ &E_{4} = &\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k-1}^{k+1}\sum\limits_{j = -\infty}^{k-2}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q(0)}\Big)^{\frac{1}{q(0)}},\\ &E_{5} = &\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k-1}^{k+1}\sum\limits_{j = k-1}^{k+1}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q(0)}\Big)^{\frac{1}{q(0)}},\\ &E_{6} = &\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k-1}^{k+1}\sum\limits_{j = k+2}^{\infty}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q(0)}\Big)^{\frac{1}{q(0)}},\\ &E_{7} = &\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\mid\sum\limits_{l = k+2}^{\infty}\sum\limits_{j = -\infty}^{k-2}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q(0)}\Big)^{\frac{1}{q(0)}},\\ &E_{8} = &\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k+2}^{\infty}\sum\limits_{j = k-1}^{k+1}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q(0)}\Big)^{\frac{1}{q(0)}},\\ &E_{9} = &\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k+2}^{\infty}\sum\limits_{j = k+2}^{\infty}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q(0)}\Big)^{\frac{1}{q(0)}},\\ \end{eqnarray*} |
\begin{eqnarray*} &H_{1} = &2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = -\infty}^{k-2}\sum\limits_{j = -\infty}^{k-2}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}},\\ &H_{2} = &2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = -\infty}^{k-2}\sum\limits_{j = k-1}^{k+1}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q_{\infty}}\Big)^{\frac{1}{q\infty}},\\ &H_{3} = &2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = -\infty}^{k-2}\sum\limits_{j = k+2}^{\infty}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}},\\ &H_{4} = &2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k-1}^{k+1}\sum\limits_{j = -\infty}^{k-2}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}},\\ &H_{5} = &2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k-1}^{k+1}\sum\limits_{j = k-1}^{k+1}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}},\\ &H_{6} = &2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k-1}^{k+1}\sum\limits_{j = k+2}^{\infty}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q_{\infty}}\Big)^{\frac{1}{q\infty}},\\ &H_{7}: = &2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k+2}^{\infty}\sum\limits_{j = -\infty}^{k-2}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}},\\ &H_{8}: = &2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k+2}^{\infty}\sum\limits_{j = k-1}^{k+1}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}},\\ &H_{9}: = &2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k+2}^{\infty}\sum\limits_{j = k+2}^{\infty}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}. \end{eqnarray*} |
We will use the following estimates. If l\leq k-1 , by Hölder's inequality, Lemma 1.4 and Definition 1.3, we have
\begin{eqnarray} \Big\|2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{il}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\mathrm{d}y_{i}\chi_{k}\Big\|_{L^{p_{i}(\cdot)}(w_{i})} &\leq& C2^{-kn}\|\chi_{B_{k}}\|_{L^{p_{i}(\cdot)}(w_{i})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}w_{i}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}}\|\chi_{l}w_{i}^{-1}\|_{L^{p_{i}^{\prime}(\cdot)}}\\ &\leq& C2^{-kn}|B_{k}|\|\chi_{B_{k}}\|_{L^{p_{i}^{\prime}(\cdot)}(w_{i}^{-1})}^{-1}\|\chi_{B_{l}}\|_{L^{p_{i}^{\prime}(\cdot)}(w_{i}^{-1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}\\ &\leq& C2^{(l-k)n\delta_{2i}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}. \end{eqnarray} | (3.1) |
If l = k , then
\begin{eqnarray} \Big\|2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{il}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\mathrm{d}y_{i}\chi_{k}\Big\|_{L^{p_{i}(\cdot)}(w_{i}} &\leq&C2^{-kn}\|\chi_{B_{k}}\|_{L^{p_{i}(\cdot)}(w_{i})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}w_{i}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}}\|\chi_{l}w_{i}^{-1}\|_{L^{p_{i}^{\prime}(\cdot)}}\\ &\leq&C2^{-kn}\|\chi_{B_{k}}\|_{L^{p_{i}(\cdot)}(w_{i})}\|\chi_{B_{l}}\|_{L^{p_{i}^{\prime}(\cdot)}(w_{i}^{-1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}\\ &\leq&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}. \end{eqnarray} | (3.2) |
If l\ge k+1 , then
\begin{eqnarray} \Big\|2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{il}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\mathrm{d}y_{i}\chi_{k}\Big\|_{L^{p_{i}(\cdot)}(w_{i})} &\leq&C2^{-kn}\|\chi_{B_{k}}\|_{L^{p_{i}(\cdot)}(w_{i})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}w_{i}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}}\|\chi_{l}w_{i}^{-1}\|_{L^{p_{i}^{\prime}(\cdot)}}\\ &\leq&C2^{-kn}\|\chi_{B_{k}}\|_{L^{p_{i}(\cdot)}(w_{i})}\|\chi_{B_{l}}\|_{L^{p_{i}(\cdot)}(w_{i})}\|\chi_{B_{l}}\|_{L^{p_{i}(\cdot)}(w_{i})}^{-1}\\ &&\times\|\chi_{B_{l}}\|_{L^{p_{i}^{\prime}(\cdot)}(w_{i}^{-1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}\\ &\leq&C2^{(l-k)n\Big(1-\delta_{1i}\Big)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|f_{i}^{v}|^{r_{i}})^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}. \end{eqnarray} | (3.3) |
Reverse the order of f_{1} and f_{2} , it is obviously that the estimates of E_{2} , E_{3} and E_{6} are similar to those of E_{4} , E_{7} and E_{8} , respectively. Thus We just need to estimate E_{1} – E_{3}, E_{5}, E_{6} and E_{9} .
For E_{1} , since l , j\leq k-2 , then for i = 1, 2,
|x-y_{i}|\geq|x|-|y_{i}| > 2^{k-1}-2^{\min\{l,j\}}\geq2^{k-2},\quad x\in D_{k},\ y_{1}\in D_{l},\ y_{2}\in D_{j}. |
Therefore, for x\in D_{k} , we have
|K(x,y_{1},y_{2})|\leq C(|x-y_{1}|+|x-y_{2}|)^{-2n}\leq C2^{-2kn}. |
Thus, for any x\in D_{k}\; \; \rm{and} \; \; l, j\leq k-2 , we have
\begin{eqnarray*} |T(f_{1l}^{v},f_{2j}^{v})(x)|&\lesssim&\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\mathrm{d}y_{1}\mathrm{d}y_{2}\\ &\lesssim&2^{-2kn}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|\mathrm{d}y_{1}\mathrm{d}y_{2}. \end{eqnarray*} |
Hence, together with the Hölder's and Minkowski's inequality, we have
\begin{eqnarray} &&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = -\infty}^{k-2}\sum\limits_{j = -\infty}^{k-2}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\sum\limits_{j = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2})^{r})^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\Big)^{r_{1}}\Big)^{\frac{1}{r_{1}}}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{j = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2})^{r_{2}})^{\frac{1}{r_{2}}}\chi_{k}\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &\lesssim&\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\sum\limits_{j = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}. \end{eqnarray} | (3.4) |
Since
\frac{1}{q_(0)} = \frac{1}{q_{1}(0)}+\frac{1}{q_{2}(0)}\; \; \; \text{and}\; \; \; \lambda = \lambda_{1}+\lambda_{2}, |
by Hölder's inequality, we have
\begin{eqnarray*} E_{1}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}})^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q(0)}\Big.\\ &&\times\Big\|\sum\limits_{j = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q(0)}\Big)^{\frac{1}{q(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{1}} \times(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{1}(0)q_{1}(0)}\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}})^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1}(0)})^{\frac{1}{q_{1}(0)}}\\ &&\times\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}} \times(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{2}(0)q_{2}(0)}\|\sum\limits_{j = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}})^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2}(0)})^{\frac{1}{q_{2}(0)}}\\ & = &E_{1,1}\times E_{1,2}. \end{eqnarray*} |
For convenience's sake, we write
\begin{eqnarray*} E_{1,i} = \sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{i}}\times\Big\{\sum\limits_{k = -\infty}^{L}2^{k\alpha_{i}(0)q_{i}(0)}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{il}(y_{i})|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\mathrm{d}y_{i}\chi_{k}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i}(0)}\Big\}^{\frac{1}{q_{i}(0)}}. \end{eqnarray*} |
For n\delta_{i2}-\alpha_{i}(0) > 0 , by (3.1) and Lemma 2.4 we have
\begin{eqnarray*} E_{1,i}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{i}}\Big\{\sum\limits_{k = -\infty}^{L}2^{k\alpha_{i}(0)q_{i}(0)}\Big.Big.\times\Big(\sum\limits_{l = -\infty}^{k-2}2^{(l-k)n\delta_{i2}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}\Big)^{q_{i}(0)}\Big\}^{\frac{1}{q_{i}(0)}}\\ & = &\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{i}} \times\Big\{\sum\limits_{k = -\infty}^{L}\Big(\sum\limits_{l = -\infty}^{k-2}2^{l\alpha_{i}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)\Big(n\delta_{i2}-\alpha_{i}(0)\Big)}\Big)^{q_{i}(0)}\Big\}^{\frac{1}{q_{i}(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{i}}\Big(\sum\limits_{l = -\infty}^{L-2}2^{l\alpha_{i}(0)q_{i}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i}(0)}\Big)^{\frac{1}{q_{i}(0)}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}(w_{i})}, \end{eqnarray*} |
where we write 2^{-|k-l|(n\delta_{i2}-\alpha_{i}(0))} = 2^{-|k-l|\varepsilon_{i}} for \varepsilon_{i} = n\delta_{i2}-\alpha_{i}(0) > 0 , then we have
E_{1}\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. |
To estimate E_{2} , since l\leq k-2 , k-1\leq j\leq k+1, then we have
|x-y_{2}|\geq|x-y_{1}|\geq|x|-|y_{1}|\geq2^{k-2},\quad x\in D_{k},\ y_{1}\in D_{l},\ y_{2}\in D_{j}. |
Therefore, for x\in D_{k} , we have
|K(x,y_{1},y_{2})|\leq C(|x-y_{1}|+|x-y_{2}|)^{-2n}\leq C2^{-2kn}. |
Thus, for any x\in D_{k}, l\leq k-2, k-1\leq j\leq k+1 , we have
\begin{eqnarray*} |T(f_{1l}^{v},f_{2j}^{v})(x)|&\lesssim&\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\mathrm{d}y_{1}\mathrm{d}y_{2}\\ &\lesssim&2^{-2kn}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|\mathrm{d}y_{1}\mathrm{d}y_{2}. \end{eqnarray*} |
Combining the Hölder's with and Minkowski's inequality, hence we obtain
\begin{eqnarray} &&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = -\infty}^{k-2}\sum\limits_{j = k-1}^{k+1}T_{\varpi}(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\sum\limits_{j = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2}\Big)^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\Big)^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{j = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2}\Big)^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &\lesssim&\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\sum\limits_{j = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}. \end{eqnarray} | (3.5) |
Since
\frac{1}{q(0)} = \frac{1}{q_{1}(0)}+\frac{1}{q_{2}(0)}\; \; \; \text{and}\; \; \; \lambda = \lambda_{1}+\lambda_{2}, |
by Hölder's inequality, we have
\begin{eqnarray*} E_{2}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q(0)}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q(0)}\Big)^{\frac{1}{q(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{1}} \times\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{1}(0)q_{1}(0)}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1}(0)}\Big)^{\frac{1}{q_{1}(0)}}\\ &&\times\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}} \Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{2}(0)q_{2}(0)}\Big\|\sum\limits_{j = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2}\Big)\Big|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ & = &E_{2,1}\times E_{2,2}. \end{eqnarray*} |
It is obvious that
E_{2,1} = E_{1,1}\lesssim\|(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}})^{\frac{1}{r_{1}}}\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}. |
Now we turn to estimate E_{2, 2}. By (3.1)–(3.3), we have
\begin{eqnarray*} E_{2,2}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{2}(0)q_{2}(0)}\Big\|\sum\limits_{j = k-1}^{k+1}2^{(j-k)n}\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L+1}2^{k\alpha_{2}(0)q_{2}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}, \end{eqnarray*} |
where we use 2^{-n\delta_{22}} < 1 and 2^{(j-k)n(1-\delta_{12})} < 2^{(j-k)n} < 2^{2n}, j\in\{k-1, k, k+1\} for (3.1) and (3.3) respectively. Thus, we obtain
E_{2}\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. |
To estimate E_{3} , since l\leq k-2 and j\geq k+2 , we have
|x-y_{1}|\geq|x|-|y_{1}|\geq2^{k-2},\quad|x-y_{2}|\geq|y_{2}|-|x| > 2^{j-2},\quad x\in D_{k},\; y_{1}\in D_{l},\; y_{2}\in D_{j}. |
Therefore, for any x\in D_{k}, l\leq k-2, j\geq k+2, we get
\begin{eqnarray*} |T_{\varpi}(f_{1l},f_{2j})(x)|&\lesssim&\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\mathrm{d}y_{1}\mathrm{d}y_{2}\\ &\lesssim&2^{-kn}2^{-jn}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|\mathrm{d}y_{1}\mathrm{d}y_{2}. \end{eqnarray*} |
Thus, by Hölder's inequality and Minkowski's inequality, we have
\begin{eqnarray} &&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = -\infty}^{k-2}\sum\limits_{j = k+2}^{\infty}T(f_{1l}^{v},f_{2j}^{v})|^{r})^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2}\Big)^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\Big)^{r_{1}}\Big)^{\frac{1}{r_{1}}}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2}\Big)^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &\lesssim&\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}. \end{eqnarray} | (3.6) |
Since
\frac{1}{q(0)} = \frac{1}{q_{1}(0)}+\frac{1}{q_{2}(0)}\; \; \; \text{and}\; \; \; \lambda = \lambda_{1}+\lambda_{2}, |
by Hölder's inequality, we have
\begin{eqnarray*} E_{3}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q(0)}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}})^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q(0)}\Big)^{\frac{1}{q(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{1}} \times\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{1}(0)q_{1}(0)}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1}(0)}\Big)^{\frac{1}{q_{1}(0)}}\\ &&\times\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}} \times\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{2}(0)q_{2}(0)}\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ & = &E_{3,1}\times E_{3,2}. \end{eqnarray*} |
It is obvious that
E_{3,1} = E_{1,1}\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}. |
Since n\delta_{21}+\alpha_{2}(0) > 0 , by (3.3), we obtain
\begin{eqnarray*} E_{3,2}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{2}(0)q_{2}(0)}\Big. \Big.\times\Big(\sum\limits_{j = k+2}^{\infty}2^{(k-j)n\delta_{21}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}} \times\Big(\sum\limits_{k = -\infty}^{L}\Big(\sum\limits_{j = k+2}^{L}2^{j\alpha_{2}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}2^{(k-j)(n\delta_{21}+\alpha_{2}(0))}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &&+\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}} \times\Big(\sum\limits_{k = -\infty}^{L}\Big(2^{k\alpha_{2}(0)}\sum\limits_{j = L+1}^{0}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}2^{(k-j)n\delta_{21}}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &&+\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}} \times\Big(\sum\limits_{k = -\infty}^{L}\Big(2^{k\alpha_{2}(0)}\sum\limits_{j = 1}^{\infty}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}2^{(k-j)n\delta_{21}}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ & = &I_{1}+I_{2}+I_{3}. \end{eqnarray*} |
First, we consider I_{1} . By Lemma 2.4, we have
\begin{eqnarray*} I_{1}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}} \times\Big(\sum\limits_{k = -\infty}^{L}\Big(\sum\limits_{j = k+2}^{L}2^{j\alpha_{2}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}2^{(k-j)(n\delta_{21}+\alpha_{2}(0))}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}}\Big(\sum\limits_{j = -\infty}^{L+2}2^{j\alpha_{2}(0)q_{2}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)(w_{2})}}^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}\Big(w_{2}\Big)}, \end{eqnarray*} |
where we write 2^{-|k-j|(n\delta_{21}+\alpha_{2}(0))} = 2^{-|k-j|\eta_{2}} for \eta_{2} = n\delta_{21}+\alpha_{2}(0) > 0 . Next, we consider I_{2} . Since n\delta_{21}+\alpha_{2}(0)-\lambda_{2} > 0 , we obtain
\begin{eqnarray*} I_{2}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}\Big(2^{k(n\delta_{21}+\alpha_{2}(0))}\sum\limits_{j = L+1}^{0}2^{j\alpha_{2}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &&\times2^{-j(n\delta_{21}+\alpha_{2}(0))}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}} \times2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}\Big(2^{k(n\delta_{21}+\alpha_{2}(0))}\sum\limits_{j = L+1}^{0}2^{-j(n\delta_{21}+\alpha_{2}(0)-\lambda_{2})}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}\sup\limits_{j\leq0}2^{-j\lambda_{2}}2^{j\alpha_{2}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{L\Big(-n\delta_{21}-\alpha_{2}(0)\Big)}\Big(\sum\limits_{k = -\infty}^{L}2^{k\Big(n\delta_{21}+\alpha_{2}(0)\Big)q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}.\\ \end{eqnarray*} |
Then, we consider I_{3} . Since \delta_{21}+\alpha_{2}(0)-\lambda_{2} > 0 , we obtain
\begin{eqnarray*} I_{3}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}\Big(2^{k(n\delta_{21}+\alpha_{2}(0))}\Big.\Big. \times\sum\limits_{j = 1}^{\infty}2^{j\alpha_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}2^{-j(n\delta_{21}+\alpha_{2\infty})}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}\sup\limits_{j\geq1}2^{-j\lambda_{2}}2^{j\alpha_{\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}\Big(2^{k(n\delta_{21}+\alpha_{2}(0))}\sum\limits_{j = 1}^{\infty}2^{-j(n\delta_{21}+\alpha_{2\infty}-\lambda_{2})}\Big)^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}2^{k(n\delta_{21}+\alpha_{2}(0))q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{L\Big(-\lambda_{2}+n\delta_{21}+\alpha_{2}(0)\Big)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. \end{eqnarray*} |
Thus, we have
E_{3}\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. |
To estimate E_{5} , using Hölder's inequality and Lemma 2.8, we have
\begin{eqnarray*} E_{5}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\sum\limits_{l = k-1}^{k+1}\sum\limits_{j = k-1}^{k+1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|T_{\varpi}(f_{1l},f_{2j})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q(0)}\Big)^{\frac{1}{q(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big(\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\Big.\Big. \times\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\Big)^{q(0)}\Big)^{\frac{1}{q(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{1}}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{1}(0)q_{1}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1}(0)}\Big)^{\frac{1}{q_{1}(0)}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{2}(0)q_{2}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. \end{eqnarray*} |
To estimate E_{6} , since k-1\leq l\leq k+1 and j\geq k+2 , we obtain
|x-y_{1}| > 2^{k-2},\quad|x-y_{2}| > 2^{j-2},\quad x\in D_{k},\; y_{1}\in D_{l},\; y_{2}\in D_{j}. |
Thus, for any x\in D_{k}, k-1\leq l\leq k+1 and j\geq k+2 , we obtain
\begin{eqnarray*} |T(f_{1l}^{v},f_{2j}^{v})(x)|&\lesssim&\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\mathrm{d}y_{1}\mathrm{d}y_{2}\\ &\lesssim&2^{-kn}2^{-jn}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|\mathrm{d}y_{1}\mathrm{d}y_{2}. \end{eqnarray*} |
Therefore, by Hölder's inequality and Minkowski's inequality, we obtain
\begin{eqnarray} &&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k-1}^{k+1}\sum\limits_{j = k+2}^{\infty}T(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2}\Big)^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\Big)^{r_{1}}\Big)^{\frac{1}{r_{1}}}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2}\Big)^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &\lesssim&\Big\|\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}. \end{eqnarray} | (3.7) |
Since
\frac{1}{q_(0)} = \frac{1}{q_{1}(0)}+\frac{1}{q_{2}(0)}\; \; \; \text{and}\; \; \; \lambda = \lambda_{1}+\lambda_{2}, |
by Hölder's inequality, we have
\begin{eqnarray*} E_{6}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q(0)}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q(0)}\Big)^{\frac{1}{q(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{1}} \times\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{1}(0)q_{1}(0)}\Big\|\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1}(0)}\Big)^{\frac{1}{q_{1}(0)}}\\ &&\times\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}} \times\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{2}(0)q_{2}(0)}\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}}(\cdot)(w_{2})}^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ & = &E_{6,1}\times E_{6,2}. \end{eqnarray*} |
By the interchange of f_{1} and f_{2} , we find the estimateof E_{6, 1} and E_{2, 2} are similar, and E_{6, 2} = E_{3, 2} . To estimate E_{9} , since l, j\geq k+2 , we get
|x-y_{i}| > 2^{k-2},\quad x\in D_{k},\; y_{1}\in D_{l},\; y_{2}\in D_{j}. |
Therefore, for any x\in D_{k}, \ l, j\geq k+2 , we have
\begin{eqnarray*} |T_{\varpi}(f_{1l}^{v},f_{2j}^{v})(x)|&\lesssim&\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\mathrm{d}y_{1}\mathrm{d}y_{2}\\ &\lesssim&2^{-ln}2^{-jn}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})||f_{2j}^{v}(y_{2})|\mathrm{d}y_{1}\mathrm{d}y_{2}. \end{eqnarray*} |
Thus, by Hölder's inequality and Minkowski's inequality, we have
\begin{eqnarray} &&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|\sum\limits_{l = k+2}^{\infty}\sum\limits_{j = k+2}^{\infty}T(f_{1l}^{v},f_{2j}^{v})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = k+2}^{\infty}2^{-ln}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2}\Big)^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{l = k+2}^{\infty}2^{-\ln}\int_{\mathbb{R}^{n}}|f_{1l}^{v}(y_{1})|\mathrm{d}y_{1}\Big)^{r_{1}}\Big)^{\frac{1}{r_{1}}}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big(\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}|f_{2j}^{v}(y_{2})|\mathrm{d}y_{2}\Big)^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\\ &\lesssim&\Big\|\sum\limits_{l = k+2}^{\infty}2^{-ln}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}\\ &&\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}. \end{eqnarray} | (3.8) |
Since
\frac{1}{q_(0)} = \frac{1}{q_{1}(0)}+\frac{1}{q_{2}(0)}\; \; \; \text{and}\; \; \; \lambda = \lambda_{1}+\lambda_{2}, |
by Hölder's inequality, we have
\begin{eqnarray*} E_{9}&\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda}\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha(0)q(0)}\Big\|\sum\limits_{l = k+2}^{\infty}2^{-ln}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}})^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q(0)}\Big.\\ &&\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q(0)}\Big)^{\frac{1}{q(0)}}\\ &\lesssim&\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{1}} \times\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{1}(0)q_{1}(0)}\Big\|\sum\limits_{l = k+2}^{\infty}2^{-ln}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1}(0)}\Big)^{\frac{1}{q_{1}(0)}}\\ &&\times\sup\limits_{L\leq0,L\in\mathbb{Z}}2^{-L\lambda_{2}} \times\Big(\sum\limits_{k = -\infty}^{L}2^{k\alpha_{2}(0)q_{2}(0)}\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2}(0)}\Big)^{\frac{1}{q_{2}(0)}}\\ & = &E_{9,1}\times E_{9,2}. \end{eqnarray*} |
Obviously, the estimates of E_{9, i} are similar to those of E_{3, 2}(i = 1, 2) .
All estimates for E_{i} i = 1, 2, \cdots, 9 considered, we have
E\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. |
Finally, we estimate H . By the interchange of f_{1} and f_{2} , we see that the estimates of H_{2}, H_{3} and H_{6} are similar to those of H_{4}, H_{7} and H_{8} , respectively. Thus we just need to estimate H_{1} – H_{3} , H_{5}, H_{6} and H_{9} .
For the subsequent proof process, we need following further preparation. If l < 0 , by Lemma 1.3, we have
\begin{eqnarray} \Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})} & = &2^{-l\alpha_{i}(0)}\Big(2^{l\alpha_{i}(0)q_{i}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i}(0)}\Big)^{\frac{1}{q_{i}(0)}}\\ &\lesssim&2^{-l\alpha_{i}(0)}\Big(\sum\limits_{t = -\infty}^{l}2^{t\alpha_{i}(0)q_{i}(0)}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{t}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i}(0)}\Big)^{\frac{1}{q_{i}(0)}}\\ &\lesssim&2^{l(\lambda-\alpha_{i}(0))}2^{-l\lambda}\Big(\sum\limits_{t = -\infty}^{l}\Big\|2^{t\alpha_{i}(0)}\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{t}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i}(0)}\Big)^{\frac{1}{q_{i}(0)}}\\ &\lesssim&2^{l(\lambda-\alpha_{i}(0))}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}(w_{i})}. \end{eqnarray} | (3.9) |
To estimate H_{1} , since
l,j\leq k-2,\ \ \ \ \frac{1}{q_{\infty}} = \frac{1}{q_{1\infty}}+\frac{1}{q_{2\infty}} |
and \lambda = \lambda_{1}+\lambda_{2} , by (3.4) and Hölder's inequality, we have
\begin{eqnarray*} H_{1}&\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{\infty}}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda_{1}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{1\infty}q_{1\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1\infty}}\Big)^{\frac{1}{q_{1\infty}}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\sum\limits_{j = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ & = &H_{1,1}\times H_{1,2}, \end{eqnarray*} |
where
H_{1,i} = 2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{il}(y_{i})|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\mathrm{d}y_{i}\chi_{k}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}. |
By (3.1), we obtain
\begin{eqnarray*} H_{1,i}&\lesssim&2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big(\sum\limits_{l = -\infty}^{k-2}2^{(l-k)n\delta_{i2}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ &\lesssim&2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big(\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)n\delta_{i2}}\Big.\Big.\\ &&+\sum\limits_{l = 0}^{k}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)n\delta_{i2}}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\&\lesssim&2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big(\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)n\delta_{i2}}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ &&+2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big(\sum\limits_{l = 0}^{k}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)n\delta_{i2}}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ & = &I_{4}+I_{5}. \end{eqnarray*} |
If q_{i\infty}\geq1 , since n\delta_{i2}-\alpha_{i\infty} > 0 and n\delta_{i2}-\alpha_{i}(0) > 0 , by the Minkowski's inequality and (3.9), we obtain
\begin{eqnarray*} I_{4}& = &2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big(\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)n\delta_{i2}}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ &&\lesssim2^{-L\lambda_{i}}\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}\Big\{\sum\limits_{k = 0}^{L}\Big(2^{k\alpha_{i\infty}}2^{(l-k)n\delta_{i2}}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ &&\lesssim2^{-L\lambda_{i}}\sum\limits_{l = -\infty}^{-1}2^{ln\delta_{i2}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}\Big\{\sum\limits_{k = 0}^{L}2^{-k\Big(n\delta_{i2}-\alpha_{i\infty}\Big)q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ &&\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}(w_{i})}2^{-L\lambda_{i}}\sum\limits_{l = -\infty}^{-1}2^{l\Big(n\delta_{i2}+\lambda_{i}-\alpha_{i}(0)\Big)}\\ &&\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}(w_{i})}. \end{eqnarray*} |
If q_{i\infty} < 1 , since n\delta_{i2}-\alpha_{i\infty} > 0 and n\delta_{i2}-\alpha_{i}(0) > 0 , by (3.9), we have
\begin{eqnarray*} I_{4}&\lesssim&2^{-L\lambda_{i}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i\infty}}2^{(l-k)n\delta_{i2}q_{i\infty}}\Big)^{\frac{1}{q_{i\infty}}}\\ & = &2^{-L\lambda_{i}}\Big(\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i\infty}}2^{ln\delta_{i2}q_{i\infty}}\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}2^{-kn\delta_{i2}q_{i\infty}}\Big)^{\frac{1}{q_{i\infty}}}\\ & = &2^{-L\lambda_{i}}\Big(\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i\infty}}2^{\ln\delta_{i2}q_{i\infty}}\sum\limits_{k = 0}^{L}2^{-k\Big(n\delta_{i2}-\alpha_{i\infty}\Big)q_{i\infty}}\Big)^{\frac{1}{q_{i\infty}}}\\ &\lesssim&2^{-L\lambda_{i}}\Big(\sum\limits_{l = -\infty}^{-1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i\infty}}2^{\ln\delta_{i2}q_{i\infty}}\Big)^{\frac{1}{q_{i\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}(w_{i})}2^{-L\lambda_{i}}\Big(\sum\limits_{l = -\infty}^{-1}2^{l\Big(n\delta_{i2}+\lambda_{i}-\alpha_{i}(0)\Big)q_{i\infty}}\Big)^{\frac{1}{q_{i\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}(w_{i})}. \end{eqnarray*} |
We consider I_{5} . Since n\delta_{i2}-\alpha_{i\infty} > 0 , by Lemma 2.4, we have
\begin{eqnarray*} I_{5}& = &2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}2^{k\alpha_{i\infty}q_{i\infty}}\Big(\sum\limits_{l = 0}^{k}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)n\delta_{i2}}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ & = &2^{-L\lambda_{i}}\Big\{\sum\limits_{k = 0}^{L}\Big(\sum\limits_{l = 0}^{k}2^{l\alpha_{i\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}2^{(l-k)(n\delta_{i2}-\alpha_{i\infty})}\Big)^{q_{i\infty}}\Big\}^{\frac{1}{q_{i\infty}}}\\ &\lesssim&2^{-L\lambda_{i}}\Big(\sum\limits_{l = 0}^{k}2^{l\alpha_{i\infty}q_{i\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\chi_{l}\Big\|_{L^{p_{i}(\cdot)}(w_{i})}^{q_{i\infty}}\Big)^{\frac{1}{q_{i\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{i}^{v}|^{r_{i}}\Big)^{\frac{1}{r_{i}}}\Big\|_{M\dot{K}_{p_{i}(\cdot),\lambda_{i}}^{\alpha_{i}(\cdot),q_{i}(\cdot)}\Big(w_{i}\Big)}, \end{eqnarray*} |
where we write 2^{-|k-l|(n\delta_{i2}-\alpha_{i\infty})}\lesssim2^{-|k-l|\eta_{i}} for \eta_{i} = n\delta_{i2}-\alpha_{i\infty} .
Thus, we get
H_{1}\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. |
To estimate H_{2} , since
l\leq k-2,\ \ \ \ k-1\leq j\leq k+1,\ \ \ \ \frac{1}{q_{\infty}} = \frac{1}{q_{1\infty}}+\frac{1}{q_{2\infty}} |
and \lambda = \lambda_{1}+\lambda_{2} , by (3.6) and Hölder's inequality, we have
\begin{eqnarray*} H_{2}&\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}\Big(y_{1}\Big)|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{\infty}}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda_{1}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{1\infty}q_{1\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1\infty}}\Big)^{\frac{1}{q_{1\infty}}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\sum\limits_{j = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r}\Big)^{\frac{1}{r}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ & = &H_{2,1}\times H_{2,2}. \end{eqnarray*} |
It isobvious that
H_{2,1} = H_{1,1}\lesssim\|(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}})^{\frac{1}{r_{1}}}\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}. |
Now we estimate H_{2, 2} . Combining (3.1)–(3.3), we have
\begin{eqnarray*} H_{2,2}&\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\sum\limits_{j = k-1}^{k+1}2^{(j-k)n}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{k = -1}^{L+1}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}, \end{eqnarray*} |
where we use 2^{-n\delta_{22}} < 1 and 2^{(j-k)n(1-\delta_{21})} < 2^{(j-k)n} for (3.6) and (3.8), respectively. Thus, we obtain
H_{2}\lesssim\|(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}})^{\frac{1}{r_{1}}}\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. |
To estimate H_{3} , since
l\leq k-2,\ \ \ \ j\geq k+2,\ \ \ \ \frac{1}{q_{\infty}} = \frac{1}{q_{1\infty}}+\frac{1}{q_{2\infty}} |
and \lambda = \lambda_{1}+\lambda_{2} , together (3.6) with the Hölder's inequality, we have
\begin{eqnarray*} H_{3}&\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{\infty}}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{\rho_{\infty}}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda_{1}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{1\infty}q_{1\infty}}\Big\|\sum\limits_{l = -\infty}^{k-2}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1\infty}}\Big)^{\frac{1}{q_{1\infty}}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ & = &H_{3,1}\times H_{3,2}. \end{eqnarray*} |
It is easy to see that
H_{3,1} = H_{1,1}\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}. |
Since n\delta_{21}+\alpha_{2\infty} > 0 , by (3.3), we obtain
\begin{eqnarray*} H_{3,2}&\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big(\sum\limits_{j = k+2}^{\infty}2^{(k-j)n\delta_{21}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\Big)^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}\Big(\sum\limits_{j = k+2}^{L+2}2^{j\alpha_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}2^{(k-j)\Big(n\delta_{21}+\alpha_{2\infty}\Big)}\Big)^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &&+2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}\Big(2^{k\alpha_{2\infty}}\sum\limits_{j = L+3}^{\infty}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}2^{(k-j)n\delta_{21}}\Big)^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ & = &I_{6}+I_{7}. \end{eqnarray*} |
For I_{6} , by Lemma 2.4, we obtain
\begin{eqnarray*} I_{6}&\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}\Big(\sum\limits_{j = k+2}^{L+2}2^{j\alpha_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}2^{(k-j)\Big(n\delta_{21}+\alpha_{2\infty}\Big)}\Big)^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{j = 0}^{L+2}2^{j\alpha_{2\infty}q_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)(w_{2})}}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha(\cdot),q_{2}(\cdot)}\Big(w_{2}\Big)}, \end{eqnarray*} |
where we write 2^{-|k-j|(n\delta_{21}+\alpha_{2\infty})} = 2^{-|k-j|\vartheta_{2}} for \vartheta_{2} = n\delta_{21}+\alpha_{2\infty} > 0.
For I_{7} , since n\delta_{21}+\alpha_{2\infty}-\lambda_{2} > 0, we have
\begin{eqnarray*} I_{7}&\lesssim&2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}\Big(2^{k(n\delta_{21}+\alpha_{2\infty})}\sum\limits_{j = L+3}^{\infty}2^{j\alpha_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})} \times2^{-j(n\delta_{21}+\alpha_{2\infty})}\Big)^{q_{2\infty}}\Big)^{\frac{1}{q_{2}\infty}}\\ &\lesssim&\sup\limits_{j\geq1}2^{-j\lambda_{2}}2^{j\alpha_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{j}\Big\|_{L^{p_{2}(\cdot)}(w_{2})} \times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}\Big(2^{k(n\delta_{21}+\alpha_{2\infty})}\sum\limits_{j = L+3}^{\infty}2^{-j(n\delta_{21}+\alpha_{2\infty}-\lambda_{2})}\Big)^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}2^{-L\lambda_{2}+\Big(n\delta_{21}+\alpha_{2\infty}\Big)L-L\Big(n\delta_{21}+\alpha_{2\infty}-\lambda_{2}\Big)}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. \end{eqnarray*} |
Thus, we get
H_{3}\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. |
To estimate {H}_{5} , using Hölder's inequality and Lemma 2.8, we have
\begin{eqnarray*} H_{5}&\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\sum\limits_{l = k-1}^{k+1}\sum\limits_{j = k-1}^{k+1}\Big\|\Big(\sum\limits_{v = 1}^{\infty}\Big|T(f_{1l},f_{2j})\Big|^{r}\Big)^{\frac{1}{r}}\chi_{k}\Big\|_{L^{p(\cdot)}(w)}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big(\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|\|_{L^{p_{1}(\cdot)}(w_{1})} \Big.\Big.\times\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}\Big)^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda_{1}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{1\infty}q_{1\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1\infty}}\Big)^{\frac{1}{q_{1\infty}}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ &\lesssim&\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),q_{1}(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. \end{eqnarray*} |
To estimate H_{6} , since
k-1\leq l\leq k+1,\ \ \ \ j\geq k+2,\ \ \ \ \frac{1}{q_{\infty}} = \frac{1}{q_{1\infty}}+\frac{1}{q_{2\infty}} |
and \lambda = \lambda_{1}+\lambda_{2} , by (3.7) and Hölder's sinequality, we have
\begin{eqnarray*} H_{6}&\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{\infty}}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda_{1}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{1\infty}q_{1\infty}}\Big\|\sum\limits_{l = k-1}^{k+1}2^{-kn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1\infty}}\Big)^{\frac{1}{q_{1\infty}}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ & = &H_{6,1}\times H_{6,2}. \end{eqnarray*} |
By the interchange of f_{1} and f_{2} , we see that that of H_{6, 1} is similar to the estimate of H_{2, 2} and H_{6, 2} = H_{3, 2}.
To estimate H_{9} , since
l,j\geq k+2,\ \ \ \ \frac{1}{q_{\infty}} = \frac{1}{q_{1\infty}}+\frac{1}{q_{2\infty}} |
and \lambda = \lambda_{1}+\lambda_{2} , by (3.8) and Hölder's inequality, we have
\begin{eqnarray*} H_{9}&\lesssim&2^{-L\lambda}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{\infty}q_{\infty}}\Big\|\sum\limits_{l = k+2}^{\infty}2^{-ln}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{\infty}}\Big.\\ &&\Big.\times\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{\infty}}\Big)^{\frac{1}{q_{\infty}}}\\ &\lesssim&2^{-L\lambda_{1}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{1\infty}q_{1\infty}}\Big\|\sum\limits_{l = k+2}^{\infty}2^{-ln}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{1l}^{v}(y_{1})|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\mathrm{d}y_{1}\chi_{k}\Big\|_{L^{p_{1}(\cdot)}(w_{1})}^{q_{1\infty}}\Big)^{\frac{1}{q_{1\infty}}}\\ &&\times2^{-L\lambda_{2}}\Big(\sum\limits_{k = 0}^{L}2^{k\alpha_{2\infty}q_{2\infty}}\Big\|\sum\limits_{j = k+2}^{\infty}2^{-jn}\int_{\mathbb{R}^{n}}\Big(\sum\limits_{v = 1}^{\infty}|f_{2j}^{v}(y_{2})|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\mathrm{d}y_{2}\chi_{k}\Big\|_{L^{p_{2}(\cdot)}(w_{2})}^{q_{2\infty}}\Big)^{\frac{1}{q_{2\infty}}}\\ & = &H_{9,1}\times H_{9,2}. \end{eqnarray*} |
Obviously, the estimates of H_{9, i} are similar to those of H_{3, 2} for i = 1, 2 , respectively.
Taking all estimates for H_{i} together, i = 1, 2, \cdots, 9 , we obtain
H\lesssim\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{1}^{v}|^{r_{1}}\Big)^{\frac{1}{r_{1}}}\Big\|_{M\dot{K}_{p_{1}(\cdot),\lambda_{1}}^{\alpha_{1}(\cdot),(\cdot)}(w_{1})}\Big\|\Big(\sum\limits_{v = 1}^{\infty}|f_{2}^{v}|^{r_{2}}\Big)^{\frac{1}{r_{2}}}\Big\|_{M\dot{K}_{p_{2}(\cdot),\lambda_{2}}^{\alpha_{2}(\cdot),q_{2}(\cdot)}(w_{2})}. |
This completes the proof.
On the basis of vector valued bilinear Calderón-Zygmund operators with kernels of Dini's type are bounded on variable Lebesgue spaces, with the help of properties of the \varpi(t) and space decomposition methods for variable exponents Herz-Morrey spaces. We establish the weighted boundedness result of vector valued bilinear \varpi(t) -type Calderón-Zygmund operators in variable exponents Herz-Morrey spaces, this is a new and meaningful result.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work is supported by the Doctoral Scientific Research Foundation of Northwest Normal University (202003101203), Young Teachers Scientific Research Ability Promotion Project of Northwest Normal University (NWNU-LKQN2021-03) and Open Foundation of Hubei Key Laboratory of Applied Mathematics (Hubei University) (HBAM202205).
The authors declare that there are no conflicts of interest.
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