On an axiomatization of the grey Banzhaf value

1.   Introduction

1.1.   Bilinear $\omega(t)$-type Calderón-Zygmund operators

In 1985, Yabuta ^[1] proposed the definitions of $\varpi(t)$-type Calderón-Zygmund operators, he introduced certain $\varpi(t)$-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 $\varpi(t)$, 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 $\varpi(t)$-type Calderón-Zygmund operators. we recall the so-called $\varpi(t)$-type Calderón-Zygmund operators.

Let $\varpi(t)$: $[0, \infty)\rightarrow [0, \infty)$ be a nondecreasing function with $0 < \varpi(1) < \infty$. For $\alpha > 0$, we say that $\varpi\in \rm{Dini}(a)$ if

It is evident that for $0 < \alpha_{1} < \alpha_{2}$, there is $\rm{Dini}(\alpha_{1}) < \rm{Dini}(\alpha_{2})$. If $\varpi\in \rm{Dini}(1)$, then

here and in what follows, for any quantities A and B, if there exists a constant $C > 0$ such that $A \leq CB$, we write $A\lesssim B$. If $A\lesssim B$ and $B\lesssim A$, we write $A\approx B$.

A measurable function $K(\cdot, \cdot, \cdot)$ on $\mathbb{R}^{n}\times \mathbb{R}^{n}\times \mathbb{R}^{n}\backslash \{(x, y_{1}, y_{2}):x = y_{1} = y_{2}\}$ is said to be a bilinear $\varpi(t)$-type Calderón-Zygmund kernel if it satisfies: for all $(x, y_{1}, y_{2})\in \mathbb{R}^{n}$ with $x\neq y_{i}, \; i = 1, 2$, if there exists a constant $A > 0$ such that

and for $(x, y_{1}, y_{2})\in (\mathbb{R}^{n})^{3}$ with $x\neq y_{1}, y_{2}$, and

whenever $2|x-z| < \max\{|x-y_{1}|, |x-y_{2}|\}$.

$\textbf{Definition 1.1.}$ (^[2]) Let $\varpi\in \rm{Dini}(1)$. One can say that $T_{\varpi}$ is a bilinear $\varpi(t)$-type operator with the kernel $K$ satisfying (1.2) and (1.3), for all $f_{1}$, $f_{2}\in C_{c}^{\infty}(\mathbb{R}^{n})$,

1.2.   Products of weighted Herz-Morrey spaces with variable exponents

In the following, for each $k\in \mathbb{Z}$, we define $B_{k} = \{x\in \mathbb{R}^{n}:|x|\leq 2^{k}\}$, $D_{k} = B_{k} \backslash B_{k-1}$, $\chi_{k} = \chi_{D_{k}}$, $m\geq 1$, $\widetilde{\chi}_{0} = \chi_{B_{0}}$.

Given a function $p(x)\in \mathcal{P}(\mathbb{R}^{n})$, the space $L^{p(x)}(\mathbb{R}^{n})$ is now defined by

Denote $\mathcal{P}(\mathbb{R}^{n})$ to be the set of the all measurable functions $p(x)$ with

and

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

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

Let $f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$. Then the standard Hardy-Littlewood maximal function of $f$ is defined by

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

then $\alpha(\cdot)$ is said local $\log$-Hölder continuous on $\mathbb{R}^{n}$.

(ⅱ) For all $x\in \mathbb{R}^{n}$, if

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

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

For $1 < p < \infty$, we say that $\omega$ is an $A_{p}$ weight if

$\textbf{Definition 1.3.}$ (^[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

$\textbf{Lemma 1.1.}$ (^[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 $f\in L^{p(\cdot)}(\omega)$,

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

where $\lambda^{\frac{1}{\infty}} = 1$. If $q^{+} < \infty$ or $q(\cdot)\leq p(\cdot)$, the above can be written as

The norm is

$\textbf{Definition 1.4.}$ (^[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

where

$\textbf{Lemma 1.2.}$ (^[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

Additionally, if $\alpha(\cdot)$ and $q(\cdot)$ are log-Hölder continuous at the origin, then

$\textbf{Definition 1.5.}$ (^[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

where

$\textbf{Lemma 1.3.}$ (^[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 $f\in L_{\rm{loc}}^{p(\cdot)}(\mathbb{R}^{n}\backslash \left \{ 0 \right \}, \omega)$,

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 $S\subset B$,

2.   Background knowledges and notations

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

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

$\textbf{Lemma 2.2.}$ (^[9]) Let $0 < p, \; \delta < \infty$ and $\omega\in A_{\infty}$. There exists a positive constant $C$ such that

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

If $p\in \mathcal{P}(\mathbb{R}^{n})$, $\omega$ is a weight with $\omega = \omega_{1}\times \omega_{2}$, there exists

$\textbf{Lemma 2.4.}$ (^[11]) Let $0 < p\leq \infty$, $\delta > 0$. Then for non-negative sequence $\{a_{j}\}_{j = -\infty}^{\infty}$, there exists

when $p = \infty$, above inequality stands for

$\textbf{Lemma 2.5.}$ (^[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

Then for all $0 < p < \infty$ and $\omega_{0}\in A_{\infty}$,

Furthermore, for every $p, q\in(0, \infty)$, $\omega_{0}\in A_{\infty}$, and sequences $\{(f_{j}, g_{j})\}\in \mathcal{F}$,

$\textbf{Lemma 2.6.}$ (^[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 $\{(f_{j}, g_{j})\}_{j\in \mathbb{N}}\subset \mathcal{F}$

$\textbf{Lemma 2.7.}$ (^[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

$\textbf{Lemma 2.8.}$ 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

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

where $q_{i}\in (1, \infty)$ for $i = 1, 2$ and

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 $\omega\in A_{\infty}$,

Therefore, by Lemmas 2.5 and 2.6, we have

Since

and $\omega = \omega_{1}\omega_{2}$, together with Hölders inequality, Lemmas 2.3 and 2.7, we have

We complete the proof of Lemma 2.8.

3.   Boundedness of the vector valued bilinear $\varpi(t)$-type Calderón-Zygmund operators

$\textbf{Theorem 3.1.}$ 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

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

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

If $\lambda_{i}-n\delta_{i1} < \alpha_{i\infty}$, $\alpha_{i}(0)\leq n \delta_{i2}$, then

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

By Lemma 1.3, we have

where

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

where

We will use the following estimates. If $l\leq k-1$, by Hölder's inequality, Lemma 1.4 and Definition 1.3, we have

If $l = k$, then

If $l\ge k+1$, then

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 $i = 1, 2,$

Therefore, for $x\in D_{k}$, we have

Thus, for any $x\in D_{k}\; \; \rm{and} \; \; l, j\leq k-2$, we have

Hence, together with the Hölder's and Minkowski's inequality, we have

Since

by Hölder's inequality, we have

For convenience's sake, we write

For $n\delta_{i2}-\alpha_{i}(0) > 0$, by (3.1) and Lemma 2.4 we have

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

To estimate $E_{2}$, since $l\leq k-2$, $k-1\leq j\leq k+1,$ then we have

Therefore, for $x\in D_{k}$, we have

Thus, for any $x\in D_{k}, l\leq k-2, k-1\leq j\leq k+1$, we have

Combining the Hölder's with and Minkowski's inequality, hence we obtain

Since

by Hölder's inequality, we have

It is obvious that

Now we turn to estimate $E_{2, 2}.$ By (3.1)–(3.3), we have

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

To estimate $E_{3}$, since $l\leq k-2$ and $j\geq k+2$, we have

Therefore, for any $x\in D_{k}, l\leq k-2, j\geq k+2,$ we get

Thus, by Hölder's inequality and Minkowski's inequality, we have

Since

by Hölder's inequality, we have

It is obvious that

Since $n\delta_{21}+\alpha_{2}(0) > 0$, by (3.3), we obtain

First, we consider $I_{1}$. By Lemma 2.4, we have

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

Then, we consider $I_{3}$. Since $\delta_{21}+\alpha_{2}(0)-\lambda_{2} > 0$, we obtain

Thus, we have

To estimate $E_{5}$, using Hölder's inequality and Lemma 2.8, we have

To estimate $E_{6}$, since $k-1\leq l\leq k+1$ and $j\geq k+2$, we obtain

Thus, for any $x\in D_{k}, k-1\leq l\leq k+1$ and $j\geq k+2$, we obtain

Therefore, by Hölder's inequality and Minkowski's inequality, we obtain

Since

by Hölder's inequality, we have

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

Therefore, for any $x\in D_{k}, \ l, j\geq k+2$, we have

Thus, by Hölder's inequality and Minkowski's inequality, we have

Since

by Hölder's inequality, we have

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

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

To estimate $H_{1}$, since

and $\lambda = \lambda_{1}+\lambda_{2}$, by (3.4) and Hölder's inequality, we have

where

By (3.1), we obtain

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

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

We consider $I_{5}$. Since $n\delta_{i2}-\alpha_{i\infty} > 0$, by Lemma 2.4, we have

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

To estimate $H_{2}$, since

and $\lambda = \lambda_{1}+\lambda_{2}$, by (3.6) and Hölder's inequality, we have

It isobvious that

Now we estimate $H_{2, 2}$. Combining (3.1)–(3.3), we have

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

To estimate $H_{3}$, since

and $\lambda = \lambda_{1}+\lambda_{2}$, together (3.6) with the Hölder's inequality, we have

It is easy to see that

Since $n\delta_{21}+\alpha_{2\infty} > 0$, by (3.3), we obtain

For $I_{6}$, by Lemma 2.4, we obtain

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

Thus, we get

To estimate ${H}_{5}$, using Hölder's inequality and Lemma 2.8, we have

To estimate $H_{6}$, since

and $\lambda = \lambda_{1}+\lambda_{2}$, by (3.7) and Hölder's sinequality, we have

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

and $\lambda = \lambda_{1}+\lambda_{2}$, by (3.8) and Hölder's inequality, we have

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

This completes the proof.

4.   Conclusions

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.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

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).

Conflict of interest

The authors declare that there are no conflicts of interest.