Bilinear $\theta$-type Calderón-Zygmund operators and its commutators on generalized variable exponent Morrey spaces

1.   Introduction

$\theta$-type Calderón-Zygmund operators, which used to study certain classes of pseudo-differential operators, was introduced by Peng ^[1] in 1985. Firstly, Yang and Tao obtained the boundedness of $\theta$-type Calderón-Zygmund operators on Variable Exponents Herz space ^[2] and Morrey-Herz-type Hardy spaces with variable exponents ^[3].

Guliyev further proved that the Calderón-Zygmund operators with kernels of Dini's type are bounded on generalized weighted variable exponent Morrey spaces (see ^[4]). Besides, Maldonado and Naibo developed a theory of the bilinear Calderón-Zygmund operators of type $\omega(t)$ in 2009 and generalized the results of Yabuta ^[5]. For comprehensive bilinear $\theta$-type Calderón-Zygmund operators references, interested readers may refer to Zheng ^[6,7] and Lu ^[8].

Variable exponent function spaces play a vital role in the fluid dynamics, elasticity dynamics, and differential equations with nonstandard growth, and thus have received a plenty of attention from researchers. For more details, one may refer to ^[9,10,11,12]. More specially, variable exponent Lebesgue spaces were studied in ^{[13,14,15,16,17,18,19]}, Morrey spaces with variable exponent were studied in ^[3,20,21,22], generalized Morrey spaces with variable exponent were studied in ^{[23,24,25,26,27]} and local "complementary" generalized variable exponent Morrey space were studied in ^[28,29].

Inspired by the work above, this paper devotes to studying the boundedness of bilinear $\theta$-type Calderón-Zygmund operator and its commutators on generalized variable exponent Morrey spaces.

Suppose that $\theta$ is a non-negative and non-decreasing function on $\mathbb{R}^{+} = (0, \infty)$ satisfying

A continuous function $K(\cdot, \cdot, \cdot)$ on $\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\setminus\{(x, y_{1}, y_{2}):x = y_{1} = y_{2}\}$ is said to be a bilinear $\theta$-type Calderón-Zygmund kernel if it satisfies: for all $(x, y_{1}, y_{2})\in\mathbb{R}^{n}$ with $x\neq y_{i}$, $i = 1, 2$,

and for all $x, z, y_{1}, y_{2}\in\mathbb{R}^{n}$ with $2\mid x-z\mid < \max\{\mid x-y_{1}\mid, \mid x-y_{2}\mid\}$, then exists a positive constant $C$ such that

Now we state the definition of bilinear $\theta$-type Calderón-Zygmund operator as follows.

Let $T_{\theta}$ be a linear operator from $\mathcal{S}(\mathbb{R}^{n})\times\mathcal{S}(\mathbb{R}^{n})$ into its dual $\mathcal{S}^{'}(\mathbb{R}^{n})$, where $\mathcal{S}$ denotes the Schwartz class. One can say that $T_{\theta}$ is a bilinear $\theta$-type Calderón-Zygmund operator with kernel $K$ satisfying $(1.2)$ and $(1.3)$, for all $f_{1}, f_{2}\in L^{\infty}_{c}(\mathbb{R}^{n})$ (the space of compactly supported bounded functions on $\mathbb{R}^{n}$) and $x\notin supp f_{1}\cap supp f_{2}$,

where $\theta$ satisfies $(1.1)$.

It is easy to see that the classical bilinear Calderón-Zygmund operator $T$ with standard kernel is a special case of $T_{\theta}$ as $\theta(t) = t^{\delta}$ with $0 < \delta\leq1$. Let $b_{1}$ and $b_{2}$ be locally integrable functions, the commutator generated by $b_{1}, \; b_{2}$ and $T_{\theta}$ is defined by

Also, $[b_{1}, T_{\theta}]$ and $[b_{2}, T_{\theta}]$ are defined by

and

respectively.

Due to the singularity of commutators generated by bilinear $\theta$-type Calderón-Zygmund operators with BMO function is stronger than that of bilinear $\theta$-type Calderón-Zygmund operators. Thus, we need to strength the condition of $\theta$ in $(1.1)$. Let $\theta$ be a non-negative and non-decreasing function on $(0, \infty)$ such that

Furthermore, the commutators of bilinear $\theta$-type Calderón-Zygmund operator are defined by

where $\theta$ satisfies $(1.5)$.

For a measurable subset $E\subseteq\mathbb{R}^{n}$, we define $\mathcal{P}^{0}(E)$ to be the set of measurable functions $p(\cdot):E\rightarrow (0, \infty)$ such that

Define $\mathcal{P}(E)$ to be the set of measurable functions $p(\cdot):E\rightarrow [1, \infty)$ such that

Define $\mathcal{P}^{1}(E)$ to be the set of measurable functions $p(\cdot):E\rightarrow [1, \infty)$ such that

By $p^{'}(x) = \frac{p(x)}{p(x)-1}$, we denote the conjugate exponent of $p(x)$.

Let $f\in L^{1}_{loc}(\mathbb{R}^{n})$, the Hardy-Littlewood maximal operator $M$ is defined by

where the supremum is taken over all balls $B$ containing $x$. Let $\mathcal{B}(E)$ be the set of $p(\cdot)\in\mathcal{P}(E)$ such that $M$ is bounded on $L^{p(\cdot)}(E)$.

A subset of $\mathcal{B}(\mathbb{R}^{n})$ is the class of globally log-Hölder continuous functions $p(\cdot)\in LH(\mathbb{R}^{n})$ and $p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})$. Recall that $p(\cdot)\in LH(\mathbb{R}^{n})$, if $p(\cdot)$ satisfies

and

where $p_{\infty} = \lim\limits_{x\rightarrow \infty}p(x) > 1$.

Definition 1.1. ^[13] Given an open set $E\subset\mathbb{R}^{n}$ and $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$ denotes the set of measurable functions $f$ on $E$ such that

This set becomes a Banach function space when equipped with the Luxemburg-Nakano norm

For all compact subsets $E\subset\Omega$, the space $L^{p(\cdot)}_{loc}(\Omega)$ is defined by

Definition 1.2. ^[25] Let $p(\cdot)\in \mathcal{P}^{1}(\mathbb{R}^{n})$, $\varphi$ be a positive measurable function on $\mathbb{R}^{n}\times(0, \infty)$. The generalized variable exponent Morrey space $\mathcal{M}^{p(\cdot), \varphi}(\mathbb{R}^{n})$ is defined by

where

We recall the definition of space of $BMO(\mathbb{R}^{n})$.

Definition 1.3. ^[30] Suppose that $b\in L^{1}_{loc}(\mathbb{R}^{n})$, and let

where

Define

Definition 1.4. ^[17] The $BMO_{p(\cdot)}(\mathbb{R}^{n})$ space is the set of all locally integrable functions $b$ with finite norm

The rest of this paper is organized as follows. Section 2 recalls some basic lemmas that will be used in the sequel. Section 3 demonstrates the boundedness of bilinear $\theta$-type Calderón-Zygmund operators on generalized variable exponent Morrey spaces. Finally, the corresponding results of its commutators are made in Section 4.

2.   Preliminaries

The following notions will be encountered often throughout the text. $C$ is denoted by a positive constant which is independent of the main parameters, but it may vary from line to line. $\mathbb{R}^{n}$ is the n-dimensional Euclidean space, $\chi_{E}(x)$ is the characteristic function of a set $E\subseteq\mathbb{R}^{n}$. $A\approx B$ means that $A\geq CB$ and $A\leq CB$. $B(x_{0}, r) = \{x\in\mathbb{R}^{n}:\mid x-x_{0}\mid < r\}$ denotes the open ball with center $x_{0}\in\mathbb{R}^{n}$ and radius $r > 0$. Let $\mathbb{B} = \{B(x_{0}, r):x_{0}\in\mathbb{R}^{n}, r > 0\}$.

Lemma 2.1. ^[31] Let $p(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$, Then there exists a positive constant $C$ such that

where

and $p_{\infty} = \lim\limits_{x\rightarrow \infty}p(x)$.

Lemma 2.2. ^[32] Let $k$ be a positive integer. Then one has that, for all $b\in BMO(\mathbb{R}^{n})$ and all $i, j\in \mathbb{Z}$ with $j > i$,

Lemma 2.3. ^[33] Let $p(\cdot)\in \mathcal{B}(\mathbb{R}^{n})$, there exists a positive constant $C$ such that

Lemma 2.4. ^[27] Let $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$, for all $f\in L^{p(\cdot)}(\mathbb{R}^{n})$ and $g\in L^{p^{'}(\cdot)}(\mathbb{R}^{n})$, then

where $r_{p} = 1+\frac{1}{p_{-}}-\frac{1}{p_{+}}$. This inequality is named the generalized Hölder inequality with respect to the variable Lebesgue spaces.

Lemma 2.5. ^[34] Let $p(\cdot), \; p_{1}(\cdot), \; p_{2}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$, so that $\frac{1}{p(\cdot)} = \frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}$. Then the inequality

holds for any $f_{i}\in L^{p_{i}(\cdot)}(\mathbb{R}^{n})$ and $i = 1, 2.$

We will use the following two Lemmas on the boundedness of weighted Hardy operator

where $\omega$ is a weight.

Lemma 2.6. ^[35] Let $v_{1}, v_{2}$ and $\omega$ be weights on $(0, \infty)$ and $v_{1}(s)$ be bounded outside a neighborhood at the origin. The inequality

holds for some $C > 0$ for all non-negative and non-decreasing functions g on $(0, \infty)$ if and only if

Lemma 2.7. ^[36] Let $v_{1}, v_{2}$ and $\omega$ be weights on $(0, \infty)$ and $v_{1}(s)$ be bounded outside a neighborhood at the origin. The inequality

holds for some $C > 0$ for all non-negative and non-decreasing functions g on $(0, \infty)$ if and only if

Lemma 2.8. ^[17] Let $p(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$. Then $\parallel\cdot\parallel_{BMO_{p(\cdot)}}\approx\parallel\cdot\parallel_{BMO}$.

Lemma 2.9. ^[18] Let $T$ be a bilinear Calderón-Zygmund operators. If $p(\cdot), p_{1}(\cdot), p_{2}(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{p(\cdot)} = \frac{1}{p_{1}(\cdot)} +\frac{1}{p_{2}(\cdot)}$, then for all $f_{i}\in L^{p_{i}(\cdot)}(\mathbb{R}^{n})$, $i = 1, 2$, we have

Lemma 2.10. ^[21] Let $T$ be a bilinear Calderón-Zygmund operators, $b_{1}, \; b_{2}\in BMO(\mathbb{R}^{n})$. If $p(\cdot), p_{1}(\cdot), p_{2}(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{p(\cdot)} = \frac{1}{p_{1}(\cdot)} +\frac{1}{p_{2}(\cdot)}$, then for all $f_{i}\in L^{p_{i}(\cdot)}(\mathbb{R}^{n})$, $i = 1, 2$, we have

3.   Bilinear $\theta$-type Calderón-Zygmund operators on $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$

The main results of this section are stated as follows.

Lemma 3.1. Let $T_{\theta}$ be a bilinear $\theta$-type Calderón-Zygmund operator defined by $(1.4)$ with $\theta$ satisfies $(1.1)$. Suppose $p_{1}(\cdot), \; p_{2}(\cdot), \; q(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{q(\cdot)} = \frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}$. Then for all $f_{i}\in L^{p_{i}(\cdot)}(\mathbb{R}^{n})$, $i = 1, 2$, we have

with the constant $C > 0$ independent of $f_{1}$ and $f_{2}$.

The above result can be proved by using a similar proof method with that of Lemma $2.9$, which is omitted here for brevity.

We are now ready to extend the definition of $T_{\theta}(f_{1}, f_{2})$ when $f_{i}\in \mathcal{M}^{p_{i}(\cdot), \varphi_{i}}(\mathbb{R}^{n})$ $(i = 1, 2)$ and $T_{\theta}$ be a bilinear $\theta$-type Calderón-Zygmund operator defined by $(1.4)$ with $\theta$ satisfies $(1.1)$.

Definition 3.2. Let $T_{\theta}$ be a bilinear $\theta$-type Calderón-Zygmund operator defined by $(1.4)$ with $\theta$ satisfies $(1.1)$. If $q(\cdot), p_{1}(\cdot), p_{2}(\cdot) \in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{q(\cdot)} = \frac{1}{p_{1}(\cdot)} +\frac{1}{p_{2}(\cdot)}$, $\varphi, \varphi_{1}, \varphi_{2}$ satisfy the condition

and denote $\varphi(x_{0}, r) = \varphi_{1}(x_{0}, r)\varphi_{2}(x_{0}, r)$, where $C$ does not depend on $r$. Suppose that $T_{\theta}$ is a bounded linear operator on $L^{q(\cdot)}(\mathbb{R}^{n})$. For any $f_{i}\in \mathcal{M}^{p_{i}(\cdot), \varphi_{i}}(\mathbb{R}^{n}), \; i = 1, 2$, and $x\in B = B(x_{0}, r)\in\mathbb{B}$, we define

We need to show that $\mathcal{T}_{\theta}(f_{1}, f_{2})$ is well defined. That is, the above definition is independent of the selection of $B(x_{0}, r)$. Its proof is similar to the Theorem 3.1 in ^[37].

Theorem 3.3. Let $T_{\theta}$ be a bilinear $\theta$-type Calderón-Zygmund operator defined by $(1.4)$ with $\theta$ satisfies $(1.1)$. If $q(\cdot), p_{1}(\cdot), p_{2}(\cdot) \in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{q(\cdot)} = \frac{1}{p_{1}(\cdot)} +\frac{1}{p_{2}(\cdot)}$, $\varphi, \varphi_{1}, \varphi_{2}$ satisfy the condition $(3.2)$ and $\varphi = \varphi_{1}\varphi_{2}$. If $T_{\theta}$ is a bounded linear operator on $L^{q(\cdot)}(\mathbb{R}^{n})$, then $\mathcal{T}_{\theta}$ is a well defined linear operator on $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$.

Since $\mathcal{T}_{\theta}$ is well defined on $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$, we are allowed to study the boundedness of $\mathcal{T}_{\theta}$ on $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$.

Theorem 3.4. Let $T_{\theta}$ be a bilinear $\theta$-type Calderón-Zygmund operator defined by $(1.4)$ with $\theta$ satisfies $(1.1)$. Suppose $p_{1}(\cdot), \; p_{2}(\cdot), \; q(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{q(\cdot)} = \frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}$. Then for any ball $B = B(x_{0}, r)$ and $f_{i}\in L^{p_{i}(\cdot)}_{loc}(\mathbb{R}^{n})$, $i = 1, 2$, the following inequality

holds, where the constant $C > 0$ independent of $f_{1}$ and $f_{2}$.

Now, we present the boundedness of bilinear $\theta$-type Calderón-Zygmund operators on the generalized variable exponent Morrey spaces based on Lemma 3.1 and Theorem 3.4.

Theorem 3.5. Let $T_{\theta}$ be a bilinear $\theta$-type Calderón-Zygmund operator defined by $(1.4)$ with $\theta$ satisfies $(1.1)$. Suppose $p_{1}(\cdot), \; p_{2}(\cdot), \; q(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{q(\cdot)} = \frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}$, $\varphi, \varphi_{1}, \varphi_{2}$ satisfy the condition $(3.2)$ and $\varphi = \varphi_{1}\varphi_{2}$. Then $\mathcal{T}_{\theta}$ is bounded from the place $\mathcal{M}^{p_{1}(\cdot), \varphi_{1}}(\mathbb{R}^{n})\times\mathcal{M}^{p_{2}(\cdot), \varphi_{2}}(\mathbb{R}^{n})$ to the place $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$.

Collory 3.6. Let $T$ be a classical bilinear Calderón-Zygmund operators. If $q(\cdot), p_{1}(\cdot), p_{2}(\cdot) \in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{q(\cdot)} = \frac{1}{p_{1}(\cdot)} +\frac{1}{p_{2}(\cdot)}$, $\varphi, \varphi_{1}, \varphi_{2}$ satisfy the condition $(3.2)$ and $\varphi = \varphi_{1}\varphi_{2}$. Then $\mathcal{T}$ is bounded from the place $\mathcal{M}^{p_{1}(\cdot), \varphi_{1}}(\mathbb{R}^{n})\times\mathcal{M}^{p_{2}(\cdot), \varphi_{2}}(\mathbb{R}^{n})$ to the place $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$.

Proof of Theorem 3.3. Let $f_{i}\in\mathcal{M}^{p_{i}(\cdot), \varphi_{i}}(\mathbb{R}^{n}), \; i = 1, 2$. As $T_{\theta}$ is bounded on $L^{q(\cdot)}(\mathbb{R}^{n})$, $E_{1}$ is well defined.

Noting that $\mid x-y_{1}\mid +\mid x-y_{2}\mid\approx\mid x_{0}-y_{2}\mid$ for $x\in B(x_{0}, r)$, $y_{1}\in 2B$ and $y_{2}\in(2B)^{c}$. Applying Lemma 2.1, Lemma 2.4 and Lemma 2.3, $E_{2}$ can be estimated as

Similar to the estimates for $E_{2}$, it is easy to get

For $E_{4}$. Noting that $\mid x-y_{1}\mid +\mid x-y_{2}\mid\approx\mid x_{0}-y_{1}\mid\approx\mid x_{0}-y_{2}\mid$ for $x\in B(x_{0}, r)$ and $y_{1}, y_{2}\in(2B)^{c}$. By applying Lemma 2.1, Lemma 2.4 and Lemma 2.3, it follows that

Therefore, on the right hand side $(3.3)$ is well defined.

Finally, it remains to show that the definition is independent of $B(x_{0}, r)\in\mathbb{B}$. That is, for any $x\in B(x_{0}, r)\cap B(\omega, R)$ with $B(x_{0}, r), B(\omega, R)\in\mathbb{B}$ and $B(x_{0}, r)\cap B(\omega, R)\neq\varnothing$, we have

Suppose $B(S, M)\in\mathbb{B}$ be selected so that $B(x_{0}, 2r)\cap B(\omega, 2R)\subset B(S, M)$. According to the estimate of $E_{1}, \; E_{2}, \; E_{3}$ and $E_{4}$, for any $x\in B(x_{0}, r)\cap B(\omega, R)$, we can get

Because of $\chi_{B(x_{0}, 2r)}f_{i}, \; \chi_{B(S, M)\setminus B(x_{0}, 2r)}f_{i}\in L^{p_{i}(\cdot)}(\mathbb{R}^{n})$, where $i = 1, 2$, the linearity of $T_{\theta}$ on $L^{q(\cdot)}(\mathbb{R}^{n})$ implies that

Similarly, we also get

Therefore, $\mathcal{T}_{\theta}(f_{1}, f_{2})$ is well defined when $f_{i}\in\mathcal{M}^{p_{i}(\cdot), \varphi_{i}}(\mathbb{R}^{n})$, $i = 1, 2$. Obviously, because of $(3.3)$, $\mathcal{T}_{\theta}$ is a linear operator on $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$.

When $f\in L^{p_{i}(\cdot)}(\mathbb{R}^{n})\cap\mathcal{M}^{p_{i}(\cdot), \varphi_{i}}(\mathbb{R}^{n})$ $(i = 1, 2)$, $E_{2}, E_{3}$ and $E_{4}$ guarantee that

Consequently, $\chi_{\mathbb{R}^{n}\setminus B(x_{0}, 2r)}f_{i}\in L^{p_{i}(\cdot)}(\mathbb{R}^{n})$ $(i = 1, 2)$ and the linearity of $T_{\theta}$ on $L^{q(\cdot)}(\mathbb{R}^{n})$ implies that

That is, $\mathcal{T}_{\theta}$ reduces to $T_{\theta}$ on $L^{q(\cdot)}\cap\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$. Therefore, $\mathcal{T}_{\theta}$ is an extension of $T_{\theta}$.

So, we can get the precise definition of bilinear $\theta$-type Calderón-Zygmund operators on generalized variable exponent Morrey spaces.

Proof of Theorem 3.4. For arbitrary ball $B = B(x_{0}, r)$, we represent $f$ as $f_{i} = f^{1}_{i}+f^{2}_{i}$ for $i = 1, 2$, where $f^{1}_{i} = f_{i}\chi_{2B}$ and $f^{2}_{i} = f_{i}\chi_{\mathbb{R}^{n}\backslash2B}$.

Then, it can be rewritten as

According to Lemma 3.1, we conclude that

where

According to Lemma 2.1 and the estimate of $E_{2}$, $I_{2}$ can be estimated as

Similar to the estimates for $I_{2}$, it is easy to get

For $I_{4}$. On the basis of Lemma 2.1 and $E_{4}$, it follows that

On account of the above estimates for $I_{1}, I_{2}, I_{3}$ and $I_{4}$, (3.4) is obtained.

Proof of Theorem 3.5. Let $f_{i}\in\mathcal{M}^{p_{i}(\cdot), \varphi_{i}}(\mathbb{R}^{n}), \; i = 1, 2$. According to Theorem 3.4, Lemma 2.6 and (3.2), we get

thus, the proof of the Theorem 3.5 is completed.

4.   Commutators of bilinear $\theta$-type Calderón-Zygmund operators on $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$

Now we formulate the main results of this section.

Lemma 4.1. Let $T_{\theta}$ be a bilinear $\theta$-type Calderón-Zygmund operator defined by $(1.4)$ with $\theta$ satisfies $(1.5)$. Suppose $p_{1}(\cdot), \; p_{2}(\cdot), \; p(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{p(\cdot)} = \frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}$, $b_{1}, \; b_{2}\in BMO(\mathbb{R}^{n})$. Then for all $f_{i}\in L^{p_{i}(\cdot)}(\mathbb{R}^{n})$, $i = 1, 2$, we have

with the constant $C > 0$ independent of $f_{1}$ and $f_{2}$.

As this result can be proved in a way similar to Lemma $2.10$, we do not present the proof here for the sake of brevity.

We now ready to study the boundedness of the commutator $[b_{1}, b_{2}, T_{\theta}]$ on $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$ by $(3.3)$.

Definition 4.2. Let $T_{\theta}$ be a bilinear $\theta$-type Calderón-Zygmund operator defined by $(1.4)$ with $\theta$ satisfies $(1.5)$. Suppose $p_{1}(\cdot), \; p_{2}(\cdot), \; p(\cdot), \; q(\cdot)$, $s_{1}(\cdot), \; s_{2}(\cdot), \; s(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)}+\frac{1}{s(\cdot)}, \; \frac{1}{p(\cdot)} = \frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}, \; \frac{1}{s(\cdot)} = \frac{1}{s_{1}(\cdot)}+\frac{1}{s_{2}(\cdot)}$, $b_{1}, \; b_{2}\in BMO(\mathbb{R}^{n})$, $\varphi, \varphi_{1}, \varphi_{2}$ satisfy the condition

and denote $\varphi(x_{0}, r) = \varphi_{1}(x_{0}, r)\varphi_{2}(x_{0}, r)$, where $C$ does not on $r$. Suppose that $T_{\theta}$ and $[b_{1}, b_{2}, T_{\theta}]$ is a bounded linear operator on $L^{p(\cdot)}(\mathbb{R}^{n})$. For any $f_{i}\in \mathcal{M}^{p_{i}(\cdot), \varphi_{i}}(\mathbb{R}^{n})$, $i = 1, 2$, and $x\in B = B(x_{0}, r)\in\mathbb{B}$, we define

Theorem 4.3. Let $T_{\theta}$ be a bilinear $\theta$-type Calderón-Zygmund operator defined by $(1.4)$ with $\theta$ satisfies $(1.5)$. Suppose $p_{1}(\cdot), \; p_{2}(\cdot), \; p(\cdot), \; q(\cdot), \; s_{1}(\cdot), \; s_{2}(\cdot), \; s(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)}+\frac{1}{s(\cdot)}, \; \frac{1}{p(\cdot)} = \frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}, \; \frac{1}{s(\cdot)} = \frac{1}{s_{1}(\cdot)}+\frac{1}{s_{2}(\cdot)}$, $b_{1}, \; b_{2}\in BMO(\mathbb{R}^{n})$, $\varphi, \varphi_{1}, \varphi_{2}$ satisfy the condition $(4.2)$ and $\varphi = \varphi_{1}\varphi_{2}$. If $T_{\theta}$ and $[b_{1}, b_{2}, T_{\theta}]$ is a bounded linear operator on $L^{p(\cdot)}(\mathbb{R}^{n})$, then $[b_{1}, b_{2}, \mathcal{T}_{\theta}]$ is a well defined linear operator on $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$.

The result can be proved by using a similar proof method with that of Theorem 3.3, which is omitted here for brevity.

Additionally, for any $f\in L^{p_{i}(\cdot)}(\mathbb{R}^{n})\cap\mathcal{M}^{p_{i}(\cdot), \varphi_{i}}(\mathbb{R}^{n})$, where $i = 1, 2$, we have $[b_{1}, b_{2}, \mathcal{T}_{\theta}] = [b_{1}, b_{2}, T_{\theta}]$.

Since $[b_{1}, b_{2}, \mathcal{T}_{\theta}]$ is well defined on $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$, we are allowed to study the boundedness of commutators of bilinear $\theta$-type Calderón-Zygmund operators on $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$.

Theorem 4.4. Let $T_{\theta}$ be a bilinear $\theta$-type Calderón-Zygmund operator defined by $(1.4)$ with $\theta$ satisfies $(1.5)$. Suppose $p_{1}(\cdot), \; p_{2}(\cdot), \; p(\cdot), \; q(\cdot), \; s_{1}(\cdot), \; s_{2}(\cdot), \; s(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)}+\frac{1}{s(\cdot)}, \; \frac{1}{p(\cdot)} = \frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}, \; \frac{1}{s(\cdot)} = \frac{1}{s_{1}(\cdot)}+\frac{1}{s_{2}(\cdot)}$, $b_{1}, \; b_{2}\in BMO(\mathbb{R}^{n})$. Then for any ball $B = B(x_{0}, r)$ and $f_{i}\in L^{p_{i}(\cdot)}_{loc}(\mathbb{R}^{n})$, $i = 1, 2$, the following inequality

holds, where the constant $C > 0$ independent of $f_{1}$ and $f_{2}$.

Combining Lemma 4.1 with Theorem 4.4, the following Theorem shows the boundedness of commutators of bilinear $\theta$-type Calderón-Zygmund operators on the generalized variable exponent Morrey spaces.

Theorem 4.5. Let $T_{\theta}$ be a bilinear $\theta$-type Calderón-Zygmund operator defined by $(1.4)$ with $\theta$ satisfies $(1.5)$. Suppose $p_{1}(\cdot), \; p_{2}(\cdot), \; p(\cdot), \; q(\cdot), \; s_{1}(\cdot), \; s_{2}(\cdot), \; s(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)}+\frac{1}{s(\cdot)}, \; \frac{1}{p(\cdot)} = \frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}, \; \frac{1}{s(\cdot)} = \frac{1}{s_{1}(\cdot)}+\frac{1}{s_{2}(\cdot)}$, $b_{1}, \; b_{2}\in BMO(\mathbb{R}^{n})$, $\varphi, \varphi_{1}, \varphi_{2}$ satisfy the condition $(4.2)$ and $\varphi = \varphi_{1}\varphi_{2}$. Then $[b_{1}, b_{2}, \mathcal{T}_{\theta}]$ is bounded from the place $\mathcal{M}^{p_{1}(\cdot), \varphi_{1}}(\mathbb{R}^{n})\times\mathcal{M}^{p_{2}(\cdot), \varphi_{2}}(\mathbb{R}^{n})$ to the place $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$.

Collory 4.6. Let $T$ be a classical bilinear Calderón-Zygmund operators. Suppose $p_{1}(\cdot), \; p_{2}(\cdot), \; p(\cdot), \\\; q(\cdot)$, $s_{1}(\cdot), \; s_{2}(\cdot), \; s(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}(\mathbb{R}^{n})$ such that $\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)}+\frac{1}{s(\cdot)}, \; \frac{1}{p(\cdot)} = \frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}, \; \frac{1}{s(\cdot)} = \frac{1}{s_{1}(\cdot)}+\frac{1}{s_{2}(\cdot)}$, $b_{1}, \; b_{2}\in BMO(\mathbb{R}^{n})$, $\varphi, \varphi_{1}, \varphi_{2}$ satisfy the condition $(4.2)$ and $\varphi = \varphi_{1}\varphi_{2}$. Then $[b_{1}, b_{2}, \mathcal{T}]$ is bounded from the place $\mathcal{M}^{p_{1}(\cdot), \varphi_{1}}(\mathbb{R}^{n})\times\mathcal{M}^{p_{2}(\cdot), \varphi_{2}}(\mathbb{R}^{n})$ to the place $\mathcal{M}^{q(\cdot), \varphi}(\mathbb{R}^{n})$.

Proof of Theorem 4.4. We decompose the function $f_{i}$ in the form $f_{i} = f^{1}_{i}+f^{2}_{i}$ in the proof of Theorem 3.4, where $i = 1, 2$. For all $f_{i}\in L^{p_{i}(\cdot)}_{loc}(\mathbb{R}^{n})$, then

Let $\frac{1}{p(\cdot)} = \frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}, \; \frac{1}{q(\cdot)} = \frac{1}{p(\cdot)}+\frac{1}{s(\cdot)}$. From Lemma 2.5, Theorem 4.1 and Lemma 2.1, it follows that

Now we deal with $H_{2}$. $H_{2}$ can be decomposed into four terms.

Noting that $\mid x-y_{1}\mid +\mid x-y_{2}\mid\approx\mid x_{0}-y_{2}\mid$ for $x\in B(x_{0}, r), \; y_{1}\in2B$ and $y_{2}\in(2B)^{c}$. Let $\frac{1}{p(\cdot)} = \frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}, \; \frac{1}{s(\cdot)} = \frac{1}{s_{1}(\cdot)}+\frac{1}{s_{2}(\cdot)}, \; \frac{1}{q(\cdot)} = \frac{1}{p(\cdot)}+\frac{1}{s(\cdot)}$. By Lemma 2.5, Lemma 2.1, lemma 2.8 and Lemma 2.4, one has

From Lemma 2.5, Lemma 2.8, Lemma 2.1, Lemma 2.4 and Lemma 2.3, it follows that

By applying Lemma 2.5, Lemma 2.1, Lemma 2.8, Lemma 2.4 and Lemma 2.3, we can deduce that

Further, according to Lemma 2.1, Lemma 2.4, Lemma 2.8 and Lemma 2.3, we obtain that

Which, together with the estimates for $H_{21}, H_{22}, H_{23}$ and $H_{24}$, we get

Similarly, it is not difficult to obtain

It remains to estimate $H_{4}$. Noting that $\mid x-y_{1}\mid +\mid x-y_{2}\mid\approx\mid x-y_{1}\mid\approx\mid x-y_{2}\mid$ for $x\in B(x_{0}, r)$ and $y_{1}, y_{2}\in(2B)^{c}$. By using Lemma 2.1, Lemma 2.4, Lemma 2.8 and Lemma 2.3, we can deduce that

which, combining the estimates of $H_{1}$, $H_{2}$ and $H_{3}$, implies (4.4).

Proof of Theorem 4.5. Let $f_{i}\in\mathcal{M}^{p_{i}(\cdot), \varphi_{i}}(\mathbb{R}^{n}), \; i = 1, 2$. By Theorem 4.4, Lemma 2.7 and (4.2), we obtain

which is our desire result.

5.   Conclusions

In this paper, I mainly obtain the boundedness of bilinear $\theta$-type Calderón-Zygmund operator $T_{\theta}$ and its commutator $[b_{1}, b_{2}, T_{\theta}]$ on the generalized variable exponent Morrey spaces.

Acknowledgments

The author would like to express sincere thanks to reviewers for their helpful comments and suggestions.

Conflict of interest

The author declares that he has no conflicts of interest.