Boundedness of sublinear operators on weighted grand Herz-Morrey spaces

1.   Introduction

Motivated by applications to fluid dynamics, variational integrals and partial differential equations with non-standard growth conditions, many research papers focus on the function spaces with variable exponent in harmonic analysis. As a special case of the Musielak-Orlicz spaces, variable exponent Lebesgue spaces $L^{q(\cdot)}$ (together with Sobolev spaces built upon them) were first studied in ^[1] and some of the properties of the Lebesgue spaces were readily generalized to $L^{q(\cdot)}$. Subsequently, besides the variable Lebesgue and Sobolev spaces, there are diverse spaces with variable exponent were introduced and studied. For example, see ^[2] for variable exponent Bessel potential spaces, the variable exponent Besov and Triebel-Lizorkin spaces were defined on ^[3,4,5], and the variable exponent Morrey spaces and Campanato spaces were also introduced on ^[6,7]. The list is not exhausted.

Herz spaces with variable exponent was initially defined by Izuki ^[8], who used the Haar functions to obtain wavelet characterization of this spaces. Later, Izuki ^[9] established the boundedness of sublinear operators and their commutators on Herz spaces with variable exponent. As a generalization, Herz-Morrey spaces with variable exponent also were introduced by Izuki in ^[10]. Indeed, Izuki established the boundedness of vector-valued sublinear operators satisfying a size condition on Herz-Morrey spaces with variable exponent $M\dot{K}_{p, q(\cdot)}^{\alpha, \lambda}$. Furthermore, Wang and Xu ^[11] generalized Izuki's result for the wighted Herz-Morrey spaces with variable exponent $M\dot{K}_{p(\cdot), q(\cdot)}^{\alpha(\cdot), \lambda}(\omega)$.

Recently, Sultan et al. ^[12] introduced weighted grand Herz space $\dot{K}_{q(\cdot)}^{\alpha, p), \theta}(\omega)$ and they obtained the boundedness of fractional integrals. Inspired by the results of ^[11,12], the aim of this paper is to introduce the weighted grand Herz-Morrey spaces and prove the boundedness of sublinear operators and their multilinear commutators on these spaces.

The paper is organized as follows. In Section 2, we will recall some related definitions and auxiliary lemmas, including some basic notions regarding Lebesgue space with variable exponent and grand Lebesgue sequence spaces which are the main ingredients to define weighted grand Herz-Morrey spaces. In Section 3, we introduce the concept of weighted grand Herz-Morrey spaces and investigate the relationship between weighted grand Herz-Morrey spaces and weighted Herz-Morrey spaces. In Section 4, we prove the boundedness of sublinear operators with certain weak size conditions on weighted grand Herz-Morrey spaces. As an application, we obtain the boundedness estimation for some classical sublinear operators on weighted grand Herz-Morrey spaces. Finally, basing on the theories of variable exponent and the generalization of BMO norm, we prove the boundedness for multilinear commutators of sublinear operators on weighted grand Herz-Morrey spaces in Section 5.

Throughout this paper, we denote $\chi_{k} = \chi_{R_{k}}, \; R_{k} = B_{k}\backslash B_{k-1}$ and $B_{k} = \{{x\in \mathbb{R}^{n}:|x|\leq2^{k}}\}$ for all $k\in \mathbb{Z}$. Constants (often different constant in the same series of inequalities) will mainly be denoted by $c$ or $C.\; f\lesssim g$ means that $f\leq Cg$ and $f\thickapprox g$ means that $f\lesssim g\lesssim f.$

2.   Preliminaries

In this section, we will recall some necessary definitions and auxiliary results.

2.1.   Lebesgue space with variable exponent

To begin with, we recall some basic definitions and results on the variable exponent Lebesgue spaces. We can refer to the monographs ^[13,14] for more informations. Let $q(\cdot):\mathbb{R}^{n}\rightarrow [1, \infty)$ be a measurable function. $L^{q(\cdot)}(\mathbb{R}^{n})$ denotes the spaces of all measurable functions $f$ on $\mathbb{R}^{n}$ such that

This is Banach space with respect to the norm

Given an open set $\Omega\subseteq\mathbb{R}^{n}$, the space $L^{q(\cdot)}_{\rm loc}(\Omega)$ is defined by

For conciseness, we denote by $\mathcal{P}(\mathbb{R}^{n})$ the set of all measurable functions $q(x)$ on ${\mathbb{R}}^n$ with range in $[1, \infty)$ such that

where $M$ is the Hardy-Littlewood maximal operator defined by

A real-valued measurable function $g(\cdot): \mathbb{R}^{n}\rightarrow (0, \infty)$ is called globally log-Hölder continuous if there exists a constant $C_{\rm log} > 0$ such that

if, for some $g_{\infty}\in (0, \infty)$ and $C_{\rm log} > 0$, there hold

then we say $g(\cdot)$ is log-Hölder continuous at the origin (or has a log decay at the origin) and at infinity (or has a log decay at infinity), respectively.

The set $\mathcal{P}_{0}(\mathbb{R}^{n})$ consists of all measurable functions $q(\cdot)$ satisfying $q_{-} > 0$ and $q_{+} < \infty$. By $\mathcal{P}^{\rm log}_{0}(\mathbb{R}^{n})$ and $\mathcal{P}^{\rm log}_{\infty}(\mathbb{R}^{n})$ can be denoted the class of exponents $q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$, which satisfies conditions $(2.3)$ and $(2.4)$, respectively. $\mathcal{P}^{\rm log}(\mathbb{R}^{n})$ is the set of functions $q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$ satisfying conditions $(2.3)$ and $(2.4)$, with $q_{\infty}: = \lim_{|x|\rightarrow \infty}q(x)$. In particular, we note that if $q(\cdot)\in \mathcal{P}^{\rm log}(\mathbb{R}^{n})$ with $1 < q_{-}\leq q_{+} < \infty$, then the Hardy-Littlewood maximal operator $M$ is bounded on $L^{q(\cdot)}(\mathbb{R}^{n})$, namely, $q(\cdot)\in \mathcal{B}(\mathbb{R}^{n})$, see ^[14,15,16].

Lemma 2.1. ^[1] (Generalized Hölder's inequality) Let $q(\cdot)\in \mathcal{P}(\mathbb{R}^{n}), f\in L^{q(\cdot)}(\mathbb{R}^{n})$ and $g\in L^{q'(\cdot)}(\mathbb{R}^{n}),$ the generalized Hölder's inequality holds in the form

where $r_{p} = 1+1/q_{-}-1/q_{+}$.

2.2.   Weighted Lebesgue spaces with variable exponent

Let $q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$ and $\omega$ be a nonnegative measurable function on $\mathbb{R}^{n}$. Then the weighted variable exponent Lebesgue space $L^{q(\cdot)}(\omega)$ is the set of all complex-valued measurable functions $f$ such that $f\omega\in L^{q(\cdot)}$. The space $L^{q(\cdot)}(\omega)$ is a Banach space equipped with the norm

Lemma 2.2. ^[11] Let $q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$. A positive measurable function $\omega \in A_{q(\cdot)}$, if there exists a positive constant $C$ for all balls $B$ in $\mathbb{R}^{n}$ such that

The variable Muckenhoupt $A_{q(\cdot)}$ was introduced by Cruz-Uribe et al. in ^[17]. It is easy to see that if $q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$ and $\omega\in A_{q(\cdot)}$, then $\omega^{-1}\in A_{q'(\cdot)}$.

Lemma 2.3. ^[18] Let $q(\cdot)\in \mathcal{P}^{\rm log}(\mathbb{R}^{n}), \omega\in A_{q(\cdot)}$. Then there exist constants $\delta_{1}, \delta_{2}\in (0, 1)$ and $C > 0$, such that for all balls in $\mathbb{R}^{n}$ and all measurable subsets $S\subset B$,

A locally integrable function $b$ is called a BMO function, if it satisfies

where $B$ is a ball-centered at $x$ and radius of $r$, $b_{B} = \frac{1}{|B|}\int_{B}b(t){\rm d}t$.

Given a positive integer $m$ and $1\leq j\leq m,$ we denote by $C_{j}^{m}$ the family of all finite subsets $\sigma = \{\sigma(1), \cdot\cdot\cdot, \sigma(j)\}$ of $\{1, \cdot\cdot\cdot, m\}$ with $j$ different elements. For any $\sigma\in C_{j}^{m},$ the complementary sequence $\sigma^{c} = \{1, \cdot\cdot\cdot, m\}\backslash \sigma.$ For $\boldsymbol{b} = (b_{1}, \cdot\cdot\cdot, b_{m})$ and $\sigma = \{\sigma(1), \cdot\cdot\cdot, \sigma(j)\}\in C_{j}^{m}$ with $1\leq j\leq m$, we denote

In particular,

Lemma 2.4. Let $\omega\in A_{q(\cdot)}, q(\cdot)\in \mathcal{B}(\mathbb{R}^{n}), \; b_{i}\in {\rm BMO}(\mathbb{R}^{n}), \; i = 1, 2, ..., m, \; k > j(k, \; j\in \mathbb{N}).$ Then we have

and

Lemma 2.4 is a generalization of the well-known properties for BMO spaces (see ^[19]) and it's also a generalized version of Izuki's and Wang's results in ^[9,20].

2.3.   Weighted Herz-Morrey spaces with variable exponent

In this subsection, we recall definition of the weighted Herz-Morrey spaces with variable exponent.

Definition 2.1. ^[11] Let $0\leq\lambda < \infty, \; 0 < p < \infty, \; q(\cdot)\in \mathcal{P}_{0}(\mathbb{R}^{n})$, $\alpha(\cdot):\mathbb{R}^{n}\rightarrow \mathbb{R}, and \; \alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})$.

(i) The homogeneous weighted Herz-Morrey space $M\dot{K}_{p, q(\cdot)}^{\alpha(\cdot), \lambda}(\omega)$ with variable exponent is defined by

where

(ii) The non-homogeneous weighted Herz-Morrey space $MK_{p, q(\cdot)}^{\alpha(\cdot), \lambda}(\omega)$ with variable exponent is defined by

where

2.4.   Grand Lebesgue sequence space

In this subsection, we introduce grand Lebesgue sequence space. $\mathbb{X}$ stands for one of the sets $\mathbb{Z}^{n}, \; \mathbb{Z}, \; \mathbb{N}$ and $\mathbb{N}_{0}$.

Definition 2.2. ^[21] Let $1\leq p < \infty$ and $\theta > 0$. The grand Lebesgue sequence space $l^{p), \theta}$ is defined by the norm

where $\boldsymbol{x} = {\{x_{k}\}}_{k\in \mathbb{X}}$.

Note that the following nesting properties hold:

for $0 < \varepsilon < \frac{1}{p}, \; \delta > 0$ and $0 < \theta_{1}\leq\theta_{2}$.

3.   Weighted grand Herz-Morrey spaces with variable exponent

In this section, we give the definition of weighted grand Herz-Morrey spaces in a natuaral way from Definition 2.2.

Definition 3.1. Let $0\leq\lambda < \infty, \; k\in \mathbb{Z}, \; 1\leq p < \infty, \; q(\cdot)\in \mathcal{P}_{0}(\mathbb{R}^{n}), \; \alpha(\cdot):\mathbb{R}^{n}\rightarrow \mathbb{R}, \; \alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})$ and $\theta > 0$. We define the homogeneous weighted grand Herz-Morrey space by

where

The non-homogeneous weighted grand Herz-Morrey space by

where

Remark 3.1. From Definition 3.1, it is not hard to see that if $\alpha(\cdot) = c$ and $\lambda = 0$, $M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega) = \dot{K}_{q(\cdot)}^{\alpha, p), \theta}(\omega)$ is weighted grand Herz space with variable exponent in ^[12]. If $\alpha(\cdot), q(\cdot)$ are constants and $\lambda = 0$, then $M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega) = \dot{K}_{q}^{\alpha, p), \theta}(\omega)$ in ^[22]. When $\theta = 0$, the weighed grand Herz-Morrey space $M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega)$ is weighed Herz-Morrey space in ^[11]. In the meantime, there is a analog for the non-homogeneous case.

In the following theorem, we prove that homogeneous weighed Herz-Morrey space is contained in homogeneous weighed grand Herz-Morrey space.

Theorem 3.1. For $q(\cdot)\in \mathcal{P}_{0}(\mathbb{R}^{n})$, $\lambda\in [0, \infty)$, $k\in \mathbb{Z}$, $\alpha(\cdot):\mathbb{R}^{n}\rightarrow \mathbb{R}, \; \alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})$ and $1\leq p < \infty$. We have ${M\dot{K}_{p, q(\cdot)}^{\alpha(\cdot), \lambda}(\omega)}\subset{M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega)}$, $\theta > 0$.

Proof. Let $f\in {M\dot{K}_{p, q(\cdot)}^{\alpha(\cdot), \lambda}(\omega)}$, from (2.6), we can obtained that

4.   Boundedness of sublinear operators

In this section, we show that sublinear operators are bounded on homogeneous weighted grand Herz-Morrey spaces. To this end, we need the following lemma.

Lemma 4.1. Let $q(\cdot)\in \mathcal{P}_{0}(\mathbb{R}^{n})$, $\omega$ be a weight, $\lambda\in [0, \infty)$, $\alpha(\cdot):\mathbb{R}^{n}\rightarrow \mathbb{R}, \; \alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})$, $k\in \mathbb{Z}$ and $1\leq p < \infty$. If $\alpha(\cdot)$ is log-Hölder continuous both at the origin and at infinity, then

The proof of this lemma is essentially similar to the proof of Proposition 3.8 in ^[23] of $\dot{K}_{p, q(\cdot)}^{\alpha(\cdot), \lambda}(\mathbb{R}^{n})$ space. Indeed, when $\alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})$ and $\alpha(\cdot)$ is log-Hölder continuous both at the origin and at infinity, there exist positive constants $C_{1}$, $C_{2}$ such that if $k\leq0$ and $x\in R_{k}$, then $C_{1}2^{k\alpha(0)}\leq 2^{k\alpha(x)}\leq C_{2}2^{k\alpha(0)}$; if $k > 0$ and $x\in R_{k}$, then $C_{1}2^{k\alpha_{\infty}}\leq 2^{k\alpha(x)}\leq C_{2}2^{k\alpha_{\infty}}$. Thus, we obtain Lemma 4.1, which is also true for non-homogeneous weighted grand Herz-Morrey space.

Theorem 4.2. Let $1 < p < \infty, \; q(\cdot)\in \mathcal{P}^{\rm log}(\mathbb{R}^{n}), \omega\in A_{q(\cdot)}$, $\alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\cap \mathcal{P}^{\rm log}_{0}(\mathbb{R}^{n})\cap \mathcal{P}^{\rm log}_{\infty}(\mathbb{R}^{n})\cap \mathcal{P}_{0}(\mathbb{R}^{n})$ such that $-n\delta_{1} < \alpha(0), \alpha_{\infty} < n\delta_{2}$, where $0 < \delta_{1}, \delta_{2} < 1$ be the constants in Lemma 2.2. Suppose that sublinear operator $T$ satisfies the size conditions

when ${\rm supp}f\subseteq R_{k}$ and $|x|\geq 2^{k+1}$ with $k\in \mathbb{Z}$ and

when ${\rm supp}f\subseteq R_{k}$ and $|x|\leq 2^{k-2}$ with $k\in \mathbb{Z}$. If $T$ is bounded on $L^{q(\cdot)}(\omega)$, then $T$ is bounded on ${M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega)}$.

Proof. Let $f\in{M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega)}$. We decompose

From Lemma 4.1, we have

where

Since the estimate of $\rm F$ is essentially similar to the estimate of $\rm E$, it suffices to show that $\rm E$ and $\rm G$ are bounded on homogeneous weighted grand Herz-Morrey space. It is easy to see that

where

Firstly, we consider $\rm E_{1}$. For a.e. $x\in R_{k}$ with $k\in \mathbb{Z}$ and $l\leq k-2$, from size condition of $T$ and generalized Hölder's inequality, it follows that

By Lemma 2.2 and Lemma 2.3, we get

From $(4.4)$, it follows that

where $v: = n\delta_{2}-\alpha(0) > 0$. And then, by Hölder's inequality, Fubini's theorem for series and $2^{-p(1+\varepsilon)} < 2^{-p}$, we obtain that

Next we show $\rm E_{2}$, since the operator $T$ is bounded on $L^{q(\cdot)}(\omega)$, we get

Now we turn to estimate $\rm E_{3}$. For each $k\in \mathbb{Z}, \; l\geq k+2$ and a.e. $x\in R_{k}$, size condition of $T$ and generalized Hölder's inequality imply that

Splitting $\rm E_{3}$ by means of Minkowski's inequality, we deduce

By Lemma 2.2 and Lemma 2.3, we have

For $\rm E_{31}$, using $(4.6)$ we get

where $d: = n\delta_{1}+\alpha(0) > 0$. Then applying Hölder's inequality, Fubini's theorem for series and $2^{-p(1+\varepsilon)} < 2^{-p}$, we obtain that

On the other hand, applying Hölder's inequality, $(4.6)$ and note that $h: = n\delta_{1}+\alpha_{\infty} > 0$, for $\rm E_{32}$ we get

This combine the estimate of $\rm E_{31}$ implies that ${\rm E_{3}}\lesssim\|f\|_{M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega)}.$ Furthermore, we can get ${\rm E}\lesssim \; \|f\|_{M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega)}.$

Next, we consider $\rm G_{1}$. By the notion of $\rm G_{1}$, splitting $\rm G_{1}$ as follows

To estimate $\rm G_{11}$, by $(4.4)$ and the fact that $e: = n\delta_{2}-\alpha_{\infty} > 0$, we get

According to Hölder inequality and noting that $v: = n\delta_{2}-\alpha(0) > 0$, we further have

The estimate for $\rm G_{12}$ follows from a similar method to $\rm E_{1}$ and using the fact that $e: = n\delta_{2}-\alpha_{\infty} > 0, k > 0$. So we cancel the proof of $\rm G_{12}$.

For $\rm G_{2}$, in the view of the boundedness of $T$ on $L^{q(\cdot)}(\omega)$, we obtain that

We cancel the proof of $\rm G_{3}$. Since the estimate for $\rm G_{3}$ can be obtained by similar way to $\rm E_{31}$ and note that $h: = n\delta_{1}+\alpha_{\infty} > 0, \; k > 0.$

Therefore, combining the estimates for $\rm E$ and $\rm G$ to deduce that

This finishes the proof of Theorem 4.2.

Corollary 4.1. Let $p, \alpha(\cdot)$ and $q(\cdot)$ as in Theorem 4.2. If a sublinear operator $T$ satisfies the condition

for any integrable function $f$ with compact support and $T$ is bounded on $L^{q(\cdot)}(\omega)$, then $T$ is bounded on ${M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega)}$.

Remark 4.1. We remark (4.7) is satisfied by many operators in harmonic analysis, such as Calder$\acute{o}$n-Zygmund operators, the Carleson maximal operator and Bochner-Riesz means and so on. In particular, the Hardy-Littlewood maximal function also satisfies the hypotheses of Theorem 4.2. Due to the proof for the non-homogenous case can be treated by the similar method, in this article, our results are valid for the homogenous weighted grand Herz-Morrey space.

5.   Boundedness for multilinear commutators of sublinear operators

In this section, we study the boundedness for multilinear commutators of sublinear operators on homogeneous weighted grand Herz-Morrey space.

Let $\boldsymbol{b} = (b_{1}, b_{2}, \cdot\cdot\cdot, b_{m}), \; {b_{j}}\in {\rm BMO}(\mathbb{R}^{n}), \; j\in \{1, 2, \cdot\cdot\cdot, m\}, \; m\in \mathbb{N}, \; x\notin \rm supp \; f$. The multilinear commutators of sublinear operators $T^{\boldsymbol{b}}$ is defined as

where $K(x, y)$ is the integral kernel of the operator $T$, see ^[24].

Theorem 5.1. Let $\boldsymbol{b} = (b_{1}, b_{2}, \cdot\cdot\cdot, b_{m}), \; {b_{j}}\in {\rm BMO}(\mathbb{R}^{n}), \; j\in \{1, 2, \cdot\cdot\cdot, m\}, \; m\in \mathbb{N}.$ For $1 < p < \infty, \; q(\cdot)\in \mathcal{P}^{\rm log}(\mathbb{R}^{n}), \; \omega\in A_{q(\cdot)}$, $\alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\cap \mathcal{P}^{\rm log}_{0}(\mathbb{R}^{n})\cap \mathcal{P}^{\rm log}_{\infty}(\mathbb{R}^{n})\cap \mathcal{P}_{0}(\mathbb{R}^{n})$, such that $-n\delta_{1} < \alpha(0), \alpha_{\infty} < n\delta_{2}$, where $0 < \delta_{1}, \delta_{2} < 1$ be the constants in Lemma 2.3. Suppose that sublinear operator $T$ satisfying the size conditions (4.1) and (4.2). If $T^{\boldsymbol{b}}$ is bounded on $L^{q(\cdot)}(\omega)$, then $T^{\boldsymbol{b}}$ is bounded on ${M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega)}$.

Proof. Let $f\in{M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega)}$, we decompose

From Lemma 4.1, we have

where

Since the estimate of $\rm N$ is essentially similar to the estimate of $\rm A$, it suffices to show that $\rm A$ and $\rm S$ are bounded on homogeneous grand weighted Herz-Morrey space. It is easy to see that

where

Firstly, we consider $\rm A_{1}$. For a.e. $x\in R_{k}$ with $k\in \mathbb{Z}$ and $l\leq k-2$, from size condition of $T$ and generalized Hölder's inequality, it follows that

By Lemmas 2.2–2.4, we get

From $(5.2)$, it follows that

where $v: = n\delta_{2}-\alpha(0) > 0$. And then, by Hölder's inequality, Fubini's theorem for series and $2^{-p(1+\varepsilon)} < 2^{-p}$, we obtain that

To show $\rm A_{2}$, if $T^{\boldsymbol{b}}$ is bounded on $L^{q(\cdot)}(\omega)$, we get

Now we turn to estimate $\rm A_{3}$. For each $k\in \mathbb{Z}, \; l\geq k+2$ and a.e. $x\in R_{k}$, size condition of $T$ and generalized Hölder's inequality imply that

Applying Lemmas 2.2–2.4, we get

Splitting $\rm A_{3}$ by means of Minkowski's inequality, we deduce

For $\rm A_{31}$, according to (5.4), we have

where $d: = n\delta_{1}+\alpha(0) > 0$. Then applying Hölder's inequality, Fubini's theorem for series and $2^{-p(1+\varepsilon)} < 2^{-p}$, we obtain that

On the other hand, applying Hölder's inequality, (5.4), and note that $h: = n\delta_{1}+\alpha_{\infty} > 0$, for $\rm A_{32}$ we get

This combine with the estimate of $\rm A_{31}$ to obtained that ${\rm A_{3}}\lesssim\|\boldsymbol{b}\|_{*}\|f\|_{M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega)}.$ Furthermore, we can get ${\rm A}\lesssim \; \|\boldsymbol{b}\|_{*}\|f\|_{M\dot{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega)}.$

Next, we consider $\rm S_{1}$. By the notion of $\rm S_{1}$, splitting $\rm S_{1}$ as follows

To show $\rm S_{11}$, using (5.2) and note that $e: = n\delta_{2}-\alpha_{\infty} > 0$, we have

We can obtained $\rm S_{11}$ further caculation by Hölder inequality and use the fact that $v: = n\delta_{2}-\alpha(0) > 0$, we get

The estimate for $\rm S_{12}$ follows from a similar method to $\rm A_{1}$ and note that $e: = n\delta_{2}-\alpha_{\infty} > 0, k > 0$. So we cancel the proof of $\rm S_{12}$.

For $\rm S_{2}$, in the view of the boundedness of $T^{\boldsymbol{b}}$ on $L^{q(\cdot)}(\omega)$, we obtain that

We cancel the proof of $\rm S_{3}$. Since the estimate for $\rm S_{3}$ can be obtained by similar way to $\rm A_{31}$ and using the fact that $h: = n\delta_{1}+\alpha_{\infty} > 0, \; k > 0.$

Therefore, combining the estimates for $\rm A$ and $\rm S$ to deduce that

which ends the proof.

Corollary. Let $p, \alpha(\cdot)$ and $q(\cdot)$ as in Theorem 5.1. If a sublinear operator $T$ satisfies the condition $(4.7)$, for any integrable function $f$ with compact support and $T^{\boldsymbol{b}}$ is bounded on $L^{q(\cdot)}(\omega)$, then $T^{\boldsymbol{b}}$ is bounded on ${M{K}_{p), q(\cdot)}^{\alpha(\cdot), \lambda, \theta}(\omega)}$.

6.   Conclusions

In this article, we introduced the concept of weighted grand Herz-Morrey spaces and investigate the relationship between weighted grand Herz-Morrey spaces and weighted Herz-Morrey spaces. In addition, we proved the boundedness of sublinear operators with certain weak size conditions on weighted grand Herz-Morrey spaces. As an application, we obtained the boundedness estimation for multilinear commutators of sublinear operators on weighted grand Herz-Morrey spaces. These results are new even in unweighted setting.

Use of AI tools declaration

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

Acknowledgments

We would like to thank the editors and reviewers for their helpful suggestions.

This research is supported by the National Natural Science Foundation of Xinjiang Province of China (Nos: 2021D01C463, 2022D01C734), Yili Normal University High-level Talents Program of Academic Integrity (No: YSXSJS22001) and Scientific Research Project (No: 22XKZY11).

Conflict of interest

The authors declare that there is no conflict of interest.