Homogeneous Grand Mixed Herz–Morrey Spaces and Their Applications

Abstract

In this paper, we introduce the homogeneous grand mixed Herz–Morrey spaces $M{\dot{K}}_{\tilde{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$ and investigate their fundamental properties. We further explore the boundedness of sublinear operators and fractional-type operators on these spaces, establishing new results that contribute to the broader understanding of their applications.

1. Introduction

The concept of Morrey spaces, $L^{p,\lambda }\left({\mathbb{R}}^n\right)$, was introduced by Morrey in 1938 [1] to address regularity issues in the calculus of variations. These spaces provide a finer characterization of local regularity compared to classical Lebesgue spaces and have found extensive applications in both harmonic analysis and partial differential equations (PDEs). In 2005, Lu and Xu [2] extended the notion of Morrey spaces ${\mathcal{M}}_{p,\lambda }\left({\mathbb{R}}^n\right)$ by introducing the homogeneous Herz–Morrey spaces $M{\dot{K}}_{p,q}^{\alpha ,\lambda }\left({\mathbb{R}}^n\right)$, defined by the norm

$${\parallel f\parallel }_{M{\dot{K}}_{p,q}^{\alpha ,\lambda }\left({\mathbb{R}}^n\right)}=\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{k\alpha p}{\parallel f{\chi }_k\parallel }_{L^q}^p\right\}}^{1/p}<\infty .$$

In 1961, Benedek and Panzone [3] introduced the concept of mixed Lebesgue spaces $L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$ for multi-index parameters $\overrightarrow{p}=(p_1,p_2,\dots ,p_n)$ with $0\le \overrightarrow{p}<\infty$ ($0\le p_i<\infty$ for all i). In recent years, mixed function spaces have gained renewed interest due to their applications in the study of PDEs, as demonstrated in [4]. Various mixed function spaces have been developed to further explore these applications, including the works in [5,6]. In 2019, Nogayama [7] introduced mixed Morrey spaces $M^{\overrightarrow{q},\lambda }\left({\mathbb{R}}^n\right)$, where $0\le \lambda <\infty$, $0\le \overrightarrow{q}<\infty$, and $\overrightarrow{q}=(q_1,q_2,\dots ,q_n)$. Subsequently, in 2021, Wei [8] proposed the homogeneous mixed Herz spaces ${\dot{K}}_{\overrightarrow{q}}^{\alpha ,p}\left({\mathbb{R}}^n\right)$ and the homogeneous mixed Herz–Morrey spaces $M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p}\left({\mathbb{R}}^n\right)$ for parameters $\alpha \in \mathbb{R}$, $0\le \lambda <\infty$, and $0\le p,\overrightarrow{q}<\infty$. The boundedness of fractional integral operators in both local and global mixed Morrey spaces was also established in [9].

Since the introduction by Iwaniec and Sbordone in [10] for studying the integrability of the Jacobian, grand Lebesgue space has been extensively studied and generalizations have been obtained [11,12,13,14].

Grand mixed Herz–Morrey spaces are commonly used in harmonic analysis for studying singular integrals, Fourier analysis, and boundary value problems. These spaces provide a framework for analyzing functions with specific regularity properties and decay rates. The study of PDEs usually involves functions with certain regularity conditions. Grand mixed Herz–Morrey spaces provide optimal regularity conditions for functions involved in PDEs, which helps analyze solutions and establish the existence and uniqueness results. Motivated by the recent advancements above, this paper aims to extend these investigations to homogeneous grand mixed Herz–Morrey spaces. We specifically examine the boundedness of sublinear operators and fractional operators in these newly defined function spaces.

This paper is structured as follows: In Section 2, we define the homogeneous grand mixed Herz–Morrey spaces and explore some of their fundamental properties. In Section 3 and Section 4, we establish the boundedness criteria for sublinear operators and fractional-type operators in these spaces.

2. Preliminaries

There are several symbols used in this context. Let $B_k=\{x\in {\mathbb{R}}^n:\left|x\right|\le 2^k\}$ and $A_k=B_k\setminus B_{k-1}$ for any $k\in \mathbb{Z}$. Denote ${\chi }_k={\chi }_{A_k}$. Throughout this paper, let $C_{({\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n)}$ denote positive constants only dependent on parameters ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n$. The letter $\overrightarrow{q}$ will denote n-tuples of numbers in the interval ${[0,\infty ]}^n$ for $n\in \mathbb{N}$. For $\overrightarrow{q}=(q_1,q_2,\dots ,q_n)\in {(1,\infty )}^n$ and $s\ne 0$, let

$${\displaystyle \frac{1}{\overrightarrow{q}}}=\left({\displaystyle \frac{1}{q_1}},{\displaystyle \frac{1}{q_2}},\dots ,{\displaystyle \frac{1}{q_n}}\right),{\displaystyle \frac{\overrightarrow{q}}{s}}=\left({\displaystyle \frac{q_1}{s}},{\displaystyle \frac{q_2}{s}},\dots ,{\displaystyle \frac{q_n}{s}}\right),{\overrightarrow{q}}^{\prime }=\left(q_1^{\prime },q_2^{\prime },\dots ,q_n^{\prime }\right),$$

where $q_j^{\prime }={\displaystyle \frac{q_j}{q_j-1}}$ is the conjugate exponent of $q_j$.

Definition 1 

([3]). Let $\overrightarrow{p}\in {(0,\infty )}^n$. The mixed Lebesgue spaces $L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$ are defined to be the set of all measurable functions f such that

$${\parallel f\parallel }_{L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)}:={\left({\int }_{\mathbb{R}}\dots {\left({\int }_{\mathbb{R}}{\left({\int }_{\mathbb{R}}{\left|f\left(x_1,x_2,\dots ,x_n\right)\right|}^{p_1}d{x}_1\right)}^{{\displaystyle \frac{p_2}{p_1}}}d{x}_2\right)}^{{\displaystyle \frac{p_3}{p_2}}}\dots d{x}_n\right)}^{{\displaystyle \frac{1}{p_n}}}<\infty ,$$

For $p_j=\infty$, we replace the jth integral from inside to outside with an essential supremum. Take $p_1=\infty$ as an example.

$${\parallel f\parallel }_{L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)}:={\left({\int }_{\mathbb{R}}\dots {\left({\int }_{\mathbb{R}}{\left({\int }_{\mathbb{R}}\underset{x_1\in \mathbb{R}}{ess\; sup}\left|f\left(x_1,x_2,\dots ,x_n\right)\right|\right)}^{p_2}d{x}_2\right)}^{{\displaystyle \frac{p_3}{p_2}}}\dots d{x}_n\right)}^{{\displaystyle \frac{1}{p_n}}}<\infty ,$$

where

$$\underset{x\in \mathbb{R}}{\mathrm{ess}\sup } f\left(x\right):=inf\left\{\alpha :\left|\{x\in \mathbb{R}:f(x)>\alpha \}\right|=0\right\},$$

and $\left|E\right|$ denotes the Lebesgue measure of a measurable set $E\subset \mathbb{R}$.

For $\overrightarrow{p}\in {(0,\infty ]}^n$, $L_{loc}^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$ is the set consisted of all measurable functions such that for any compact set $Q\subset {\mathbb{R}}^n$, $f{\chi }_Q\in L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$.

Definition 2 

([15]). Let $1<p<\infty$ and $\theta >0$. The grand Lebesgue space $L^{p),\theta }\left({\mathbb{R}}^n\right)$ is defined as the space of measurable functions f on ${\mathbb{R}}^n$ for which the following norm is finite:

$${\parallel f\parallel }_{L^{p),\theta }\left({\mathbb{R}}^n\right)}=\underset{0<\epsilon <p-1}{sup}{\left({\epsilon }^{\theta }{\int }_{{\mathbb{R}}^n}{\left|f\left(x\right)\right|}^{p-\epsilon }dx\right)}^{{\displaystyle \frac{1}{p-\epsilon }}}.$$

Definition 3 

([16]). Let $\alpha \in \mathbb{R}$, $\theta >0$, $0\le \lambda <\infty$ and $1<p,q\le \infty$. Homogeneous grand Herz–Morrey spaces $M{\dot{K}}_{q,\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$ are defined by

$$M{\dot{K}}_{q,\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right):=\left\{f\in L_{loc}^q\left({\mathbb{R}}^n\right):{\parallel f\parallel }_{M{\dot{K}}_{q,\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}<\infty \right\},$$

where

$${\parallel f\parallel }_{M{\dot{K}}_{q,\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}=\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^q\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}.$$

Based on the above definitions, we present homogeneous grand mixed Herz–Morrey spaces $M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$.

Definition 4. 

Let $\alpha \in \mathbb{R}$, $\theta >0$, $0\le \lambda <\infty$ and $1<p,\overrightarrow{q}\le \infty$. Homogeneous grand mixed Herz–Morrey spaces $M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$ are defined by

$$M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right):=\left\{f\in L_{loc}^{\overrightarrow{q}}\left({\mathbb{R}}^n\right):{\parallel f\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}<\infty \right\},$$

where

$${\parallel f\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}=\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}.$$

Remark 1. 

It is well known that $L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)=L^p\left({\mathbb{R}}^n\right)$ for $\overrightarrow{p}=(p,p,\dots ,p)$. Therefore, it is obvious that $M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)=M{\dot{K}}_{q,\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$ for $\overrightarrow{q}=(q,q,\dots ,q)$.

The following definition is [17] (Definition 1.3.1).

Definition 5 

([17]). Let T be an operator defined on a linear space of real-valued measurable functions on ${\mathbb{R}}^n$ and taking values in the set of real-valued finite almost everywhere measurable functions on ${\mathbb{R}}^n$. T is called sublinear if for all f, g in the domain of T and all λ∈$\mathbb{R}$, we have

$$\left|T\right(f+g\left)\right|\le \left|T\right(f\left)\right|+\left|T\right(g\left)\right|\mathit{and}\left|T\right(\lambda f\left)\right|=\left|\lambda \right|\left|T\right(f\left)\right|.$$

The following are some important lemmas used in the following text.

Lemma 1 

([17]). Let $a_k\ge 0$ and $0<r\le 1$; thus,

$${\left(\sum _{k=1}^{\infty }{a}_k\right)}^r\le \sum _{k=1}^{\infty }{a}_k^r.$$

Lemma 2 

([9]). Let $1<\overrightarrow{p}<\infty$. Then, for $f\in L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$ and $g\in L^{{\overrightarrow{p}}^{\prime }}\left({\mathbb{R}}^n\right)$,

$${\int }_{{\mathbb{R}}^n}f\left(x\right)g\left(x\right)dx\le {\parallel f\parallel }_{L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)}{\parallel g\parallel }_{L^{{\overrightarrow{p}}^{\prime }}\left({\mathbb{R}}^n\right)}.$$

Lemma 3 

([9]). Let $1<\overrightarrow{p}<\infty$. If $f,g\in L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$,

$${\parallel f+g\parallel }_{L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)}\le {\parallel f\parallel }_{L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)}+{\parallel g\parallel }_{L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)}.$$

Lemma 4 

([18]). For $x_j,y_j\ge 0(j\in \mathbb{N})$, $1<p<\infty$ and ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$,

$$\sum _{j=1}^{\infty }{x}_j{y}_j\le {\left(\sum _{j=1}^{\infty }{x}_j^p\right)}^{{\displaystyle \frac{1}{p}}}{\left(\sum _{j=1}^{\infty }{y}_j^q\right)}^{{\displaystyle \frac{1}{q}}}.$$

Lemma 5 

([18]). For $x_j,y_j\ge 0(j\in \mathbb{N})$, $1<p<\infty$,

$${\left(\sum _{j=1}^{\infty }{(x_j+y_j)}^p\right)}^{{\displaystyle \frac{1}{p}}}\le {\left(\sum _{j=1}^{\infty }{x}_j^p\right)}^{{\displaystyle \frac{1}{p}}}+{\left(\sum _{j=1}^{\infty }{y}_j^p\right)}^{{\displaystyle \frac{1}{p}}}.$$

Proposition 1. 

If $\alpha \in \mathbb{R}$, $\theta >0$, $0\le \lambda <\infty$ and $1<p,\overrightarrow{q}\le \infty$, homogeneous grand mixed Herz–Morrey spaces $M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$ are Banach spaces.

Proof. 

We first prove that Definition 4 provides a norm. The non-negativity and homogeneity are evident. Therefore, the key aspect is to verify the Minkowski inequality. By Lemmas 3 and 5,

$$\begin{array}{cc}& {\parallel f+g\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}\hfill \\ & =\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥(f{\chi }_k+g{\chi }_k)∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ & \le \underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{\left({∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}+{∥g{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ & \le \underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }\left[{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{\left({∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\right.\hfill \\ & \left.+{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{\left({∥g{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\right]\hfill \\ & \le \underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{\left({∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ & +\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{\left({∥g{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ & =\left({\parallel f\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}+{\parallel g\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}\right).\hfill \end{array}$$

Thus, ${\parallel ·\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}$ is a norm.

Meanwhile, from the Minkowski inequality, for any $N\in \mathbb{N}$ and non-negative functions $f_1,f_2,\cdots f_N\in M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$, it is easy to obtain

$${∥\sum _{j=1}^N{f}_j∥}_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}\le \sum _{j=1}^N{\parallel f_j\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)};$$

therefore, for any $N\in \mathbb{N}$ and non-negative functions $f_1,f_2,\cdots \in M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$,

$${∥\sum _{j=1}^N{f}_j∥}_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}\le \sum _{j=1}^{\infty }{\parallel f_j\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}.$$

Then, via the monotone convergence theorem [19], for non-negative functions $f_1,f_2,\cdots \in M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$,

$${∥\sum _{j=1}^{\infty }{f}_j∥}_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}\le \sum _{j=1}^{\infty }{\parallel f_j\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}.$$

Thus, the Minkowski inequality is extended to the case of infinite terms.

Now, we proceed to demonstrate the completeness of the spaces $M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$. Consider a Cauchy sequence ${\left\{f_j\right\}}_{j=1}^{\infty }$ that satisfies

$${∥f_{j+1}-f_j∥}_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}\le 2^{-j}.$$

Let $g\left(x\right)=|f_1\left(x\right)|+{\sum }_{j=1}^{\infty }|f_{j+1}\left(x\right)-f_j\left(x\right)|$. By using the Minkowski inequality,

$$\begin{array}{cc}\hfill {\parallel g\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}& ={∥|f_1|+\sum _{j=1}^{\infty }|f_{j+1}-f_j|∥}_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}\hfill \\ & \le \parallel f_1{\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}+\sum _{j=1}^{\infty }{\parallel f_{j+1}-f_j\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}\hfill \\ & \le \parallel f_1{\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}+1\hfill \\ & <\infty ,\hfill \end{array}$$

which implies that $g\in M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$. Therefore, for a.e. $x\in {\mathbb{R}}^n$, $g\left(x\right)<\infty$. Then, $f\left(x\right)=f_1\left(x\right)+{\sum }_{j=1}^{\infty }\left(f_{j+1}\left(x\right)-f_j\left(x\right)\right)$ exists almost everywhere. Since

$$\begin{array}{cc}\hfill {∥f-f_k∥}_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}& ={∥f_k+\sum _{j=k}^{\infty }\left(f_{j+1}-f_j\right)-f_k∥}_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}\hfill \\ & \le {∥\sum _{j=k}^{\infty }\left(f_{j+1}-f_j\right)∥}_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}\hfill \\ & \le 2^{1-k},\hfill \end{array}$$

then

$$\underset{j\to \infty }{lim}{∥f-f_j∥}_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}=0;$$

thus, ${\left\{f_j\right\}}_{j=1}^{\infty }$ converges to $f\in M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$.

Thus, $M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$ are Banach spaces. □

Proposition 2. 

Let $\alpha \in \mathbb{R}$, $\theta >0$, $0\le \lambda <\infty$ and $1<p,\overrightarrow{q}\le \infty$. The following statements hold true:

(1) 

If $p_1\le p_2$, then $M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p_1),\theta }\left({\mathbb{R}}^n\right)\subset M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p_2),\theta }\left({\mathbb{R}}^n\right)$.

(2) 

If ${\alpha }_1\le {\alpha }_2$, then $M{\dot{K}}_{\overrightarrow{q},\lambda }^{{\alpha }_1,p),\theta }\left({\mathbb{R}}^n\right)\subset M{\dot{K}}_{\overrightarrow{q},\lambda }^{{\alpha }_2,p),\theta }\left({\mathbb{R}}^n\right)$.

Proof. 

(1) To begin with, with Lemma 1,

$${\left(\sum _{k=1}^{\infty }\left|a_k\right|\right)}^r\le \sum _{k=1}^{\infty }{\left|a_k\right|}^r(0<r⩽1).$$

Therefore, for any $f\in M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p_1),\theta }\left({\mathbb{R}}^n\right)$, by ${\displaystyle \frac{p_1}{p_2}}\in (0,1]$,

$$\begin{array}{cc}\hfill {\parallel f\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p_2),\theta }\left({\mathbb{R}}^n\right)}& =\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_2(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p_2(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p_2(1+\epsilon )}}}\hfill \\ & \le \underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{\left(2^{k\alpha }{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}\right)}^{p_1(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & \le {\parallel f\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p_1),\theta }\left({\mathbb{R}}^n\right)}\hfill \\ & <\infty ,\hfill \end{array}$$

and thus $f\in M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p_2),\theta }\left({\mathbb{R}}^n\right)$, which completes the proof.

(2) For any $f\in M{\dot{K}}_{\overrightarrow{q},\lambda }^{{\alpha }_1,p),\theta }\left({\mathbb{R}}^n\right)$, by ${\alpha }_1\le {\alpha }_2$,

$$\begin{array}{cc}\hfill {\parallel f\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{{\alpha }_2,p),\theta }\left({\mathbb{R}}^n\right)}& =\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k{\alpha }_{2}p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ & =\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k({\alpha }_2-{\alpha }_1)p(1+\epsilon )}{2}^{k{\alpha }_{1}p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ & \le \underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k_0({\alpha }_2-{\alpha }_1)p(1+\epsilon )}{2}^{k{\alpha }_{1}p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ & =2^{k_0\left({\alpha }_2-{\alpha }_1\right)}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k{\alpha }_{1}p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ & =2^{k_0({\alpha }_2-{\alpha }_1)}{\parallel f\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{{\alpha }_1,p),\theta }\left({\mathbb{R}}^n\right)}\hfill \\ & <\infty ,\hfill \end{array}$$

so $f\in M{\dot{K}}_{\overrightarrow{q},\lambda }^{{\alpha }_2,p),\theta }\left({\mathbb{R}}^n\right)$, which completes the proof. □

Proposition 3. 

If $\alpha \in \mathbb{R}$, $\theta >0$, $0\le \lambda <\infty$ and $1<p_i,\overrightarrow{q_i}\le \infty (i=1,2)$, with ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{p_1}}+{\displaystyle \frac{1}{p_2}}$, ${\displaystyle \frac{1}{\lambda }}={\displaystyle \frac{1}{{\lambda }_1}}+{\displaystyle \frac{1}{{\lambda }_2}}$ and ${\displaystyle \frac{1}{\overrightarrow{q}}}={\displaystyle \frac{1}{{\overrightarrow{q}}_1}}+{\displaystyle \frac{1}{{\overrightarrow{q}}_2}}$. Thus, the following inequality holds:

$${\parallel fg\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}\le {\parallel f\parallel }_{M{\dot{K}}_{{\overrightarrow{q}}_1,{\lambda }_1}^{\alpha ,p_1),\theta }\left({\mathbb{R}}^n\right)}{\parallel g\parallel }_{M{\dot{K}}_{{\overrightarrow{q}}_2,{\lambda }_2}^{\alpha ,p_2),\theta }\left({\mathbb{R}}^n\right).}$$

Proof. 

By utilizing Lemmas 2 and 4,

$$\begin{array}{cc}\hfill & {\parallel fg\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & =\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_{k}g{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & \le \underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q_1}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}{∥g{\chi }_k∥}_{L^{\overrightarrow{q_2}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & \le \underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_1(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q_1}}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}{\left({\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_2(1+\epsilon )}{∥g{\chi }_k∥}_{L^{\overrightarrow{q_2}}\left({\mathbb{R}}^n\right)}^{p_2(1+\epsilon )}\right)}^{{\displaystyle \frac{1}{p_2(1+\epsilon )}}}\hfill \\ \hfill & \le {\parallel f\parallel }_{M{\dot{K}}_{{\overrightarrow{q}}_1,{\lambda }_1}^{\alpha ,p_1),\theta }\left({\mathbb{R}}^n\right)}{\parallel g\parallel }_{M{\dot{K}}_{{\overrightarrow{q}}_2,{\lambda }_2}^{\alpha ,p_2),\theta }\left({\mathbb{R}}^n\right).}\hfill \end{array}$$

□

3. Boundedness of the Sublinear Operators in Homogeneous Grand Mixed Herz–Morrey Spaces

The following theorem describes the boundedness of a class of sublinear operators.

Theorem 1. 

Let $\theta >0$, $0\le \lambda <\infty$, $1<p,q_i<\infty$ and $-{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}<\alpha <n\left(1-{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}\right)$. Let T be a sublinear operator satisfying the following conditions:

(i) 

${\parallel Tf\parallel }_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}\le C_0{\parallel f\parallel }_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}$;

(ii) 

For any $k\in \mathbb{Z}$ and function f with $suppf\subset A_k$ and $\left|x\right|\ge 2^{k+1}$,

$$\left|Tf\left(x\right)\right|\le C_1{\displaystyle \frac{{\parallel f\parallel }_{L^1\left({\mathbb{R}}^n\right)}}{{\left|x\right|}^n}};$$

(1)

(iii) 

For any $k\in \mathbb{Z}$ and function f with $suppf\subset A_k$ and $\left|x\right|\le 2^{k-2}$,

$$\left|Tf\left(x\right)\right|\le C_1{\displaystyle \frac{{\parallel f\parallel }_{L^1\left({\mathbb{R}}^n\right)}}{2^{nk}}}.$$

(2)

Thus, T is bounded on $M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$.

Proof. 

For $1<p<\infty$, decompose

$$f\left(x\right)=\sum _{l=-\infty }^{\infty }f\left(x\right){\chi }_l\left(x\right):=\sum _{l=-\infty }^{\infty }{f}_l\left(x\right).$$

Since T is a sublinear operator, by Proposition 1, we have

$$\begin{array}{cc}\hfill {\parallel Tf\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}=& \underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥\left(Tf\right){\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & \underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥\left(\sum _{l=-\infty }^{\infty }\left|T\left(f{\chi }_l\right)\right|\right){\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & \underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥\left(\sum _{l=-\infty }^{k-2}\left|T\left(f{\chi }_l\right)\right|\right){\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & +\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥\left(\sum _{l=k-1}^{k+1}\left|T\left(f{\chi }_l\right)\right|\right){\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & +\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥\left(\sum _{l=k+2}^{\infty }\left|T\left(f{\chi }_l\right)\right|\right){\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill =:& \left(E_1+E_2+E_3\right).\hfill \end{array}$$

To begin with, we make an estimation for $E_2$, by Condition (i) and Proposition 1,

$$\begin{array}{cc}\hfill E_2\le & \underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥\sum _{l=k-1}^{k+1}\left|T\left(f{\chi }_l\right)\right|∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_0\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥\sum _{l=k-1}^{k+1}\left|f{\chi }_l\right|∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_0\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_{k-1}∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & +C_0\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & +C_0\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_{k+1}∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}.\hfill \end{array}$$

Replace $k-1$ and $k+1$ in the first and third lines with new summation indices (still referred to as k), respectively; thus, we have

$$\begin{array}{cc}\hfill E_2\le & C_0\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0-1}{2}^{(k+1)\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & +C_0\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & +C_0\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0+1}{2}^{(k-1)\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_0{2}^{\alpha }\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & +C_0\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & +C_0{2}^{\lambda -\alpha }\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_0{C}_{(\lambda ,\alpha )}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_0{C}_{(\lambda ,\alpha )}{\parallel f\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right).}\hfill \end{array}$$

Next, we will estimate $E_1$. To ensure that the function in the norm symbol is non-zero, we have $x\in A_k$, which implies $\left|x\right|\ge 2^{k-1}$. Additionally, it is given that $supp\left(f{\chi }_l\right)\subset A_{k-2}$ for $l\in (-\infty ,k-2]$. Hence, according to Condition (ii), we can deduce that

$$\left|Tf\left(x\right)\right|\le C_1{\displaystyle \frac{{\parallel f\parallel }_{L^1\left({\mathbb{R}}^n\right)}}{{\left|x\right|}^n}}\le C_1{C}_{\left(n\right)}{\displaystyle \frac{{\parallel f\parallel }_{L^1\left({\mathbb{R}}^n\right)}}{2^{nk}}}.$$

By Lemmas 2 and 4 and the estimate ${∥{\chi }_Q∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}\le {\left|Q\right|}^{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{p_i}}}$ in [7], one has

$$\begin{array}{cc}\hfill E_1\le & C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{\left(\sum _{l=-\infty }^{k-2}{2}^{-kn}{∥f{\chi }_l∥}_{L^1\left({\mathbb{R}}^n\right)}\right)}^{p(1+\epsilon )}{∥{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{\left(\sum _{l=-\infty }^{k-2}{∥f{\chi }_l∥}_{L^1\left({\mathbb{R}}^n\right)}\right)}^{p(1+\epsilon )}{2}^{-knp(1+\epsilon )}{2}^{knp(1+\epsilon )\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}\right)}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{\left(\sum _{l=-\infty }^{k-2}{∥f{\chi }_l∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}{2}^{(k-l)n\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1\right)}\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill =& C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{\left(\sum _{l=-\infty }^{k-2}{∥f{\chi }_l∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}{2}^{(k-l)n\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1\right){\displaystyle \frac{1}{2}}\times 2}\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{\left(\sum _{l=-\infty }^{k-2}{∥f{\chi }_l∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^p{2}^{(k-l)np{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1\right)}\right)}^{(1+\epsilon )}\right.\hfill \\ \hfill & {\left.\times {\left(\sum _{l=-\infty }^{k-2}{2}^{(k-l)n{p}^{\prime }{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1\right)}\right)}^{{\displaystyle \frac{p(1+\epsilon )}{p^{\prime }}}}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}.\hfill \end{array}$$

Since $1<q_i<\infty$, ${\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1<0$, by the formula of sum of the geometrical series,

$$\begin{array}{cc}\hfill {\left(\sum _{l=-\infty }^{k-2}{2}^{(k-l)n{p}^{\prime }{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1\right)}\right)}^{{\displaystyle \frac{p(1+\epsilon )}{p^{\prime }}}}& ={\left({\displaystyle \frac{1}{1-2^{n{p}^{\prime }\frac{1}{2}\left(\frac{1}{n}{\sum }_{i=1}^n\frac{1}{q_i}-1\right)}}}{2}^{n{p}^{\prime }\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1\right)}\right)}^{{\displaystyle \frac{p(1+\epsilon )}{p^{\prime }}}}\hfill \\ \hfill & =C_{(n,p,\overrightarrow{q})}^{1+\epsilon }{2}^{np(1+\epsilon )\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1\right)};\hfill \end{array}$$

thus,

$$E_1\le C_1{C}_{(n,p,\overrightarrow{q})}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{\left(\sum _{l=-\infty }^{k-2}{2}^{k\alpha p+np\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1\right)}{∥f{\chi }_l∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^p{2}^{(k-l)np{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1\right)}\right)}^{(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}.$$

By using the generalized Minkowski inequality (see [17] (Exercise 1.1.6)) to exchange the order of summation, and by the formula of sum of the geometrical series (the condition $\alpha <n\left(1-{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}\right)$ ensures the convergence; that is why we present it), one has

$$\begin{array}{cc}\hfill E_1& \le C_1{C}_{(n,p,\overrightarrow{q})}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{{\displaystyle \frac{\theta }{1+\epsilon }}}\sum _{l=-\infty }^{k_0-2}{\left(\sum _{k=l+2}^{\infty }{\left(2^{k\alpha p+np\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1\right)}{∥f{\chi }_l∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^p{2}^{(k-l)np{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1\right)}\right)}^{1+ϵ}\right)}^{{\displaystyle \frac{1}{(1+\epsilon )}}}\right\}}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le C_1{C}_{(n,p,\overrightarrow{q})}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{l=-\infty }^{k_0-2}{2}^{l\alpha p(1+\epsilon )}{∥f{\chi }_l∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\sum _{k=l+2}^{\infty }{2}^{(k-l)\left(\alpha p(1+\epsilon )+np(1+\epsilon )\left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}-1\right)\right)}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & \le C_1{C}_{(n,p,\overrightarrow{q},\alpha )}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{l=-\infty }^{k_0-2}{2}^{l\alpha p(1+\epsilon )}{∥f{\chi }_l∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & \le C_1{C}_{(n,p,\overrightarrow{q},\alpha )}{\parallel f\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

To estimate $E_3$. According to $supp\left(f{\chi }_l\right)\subset A_{k+2}$ and $x\in A_k$, we can deduce that

$$\left|Tf\left(x\right)\right|\le C_1{C}_{\left(n\right)}{\displaystyle \frac{{\parallel f\parallel }_{L^1\left({\mathbb{R}}^n\right)}}{2^{nk}}}.$$

For $-{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}<\alpha$, by the Hölder inequality, one gets

$$\begin{array}{cc}\hfill E_3\le & C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥\left(\sum _{l=k+2}^{\infty }\left|T\left(f{\chi }_l\right)\right|\right){\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{\left(\sum _{l=k+2}^{\infty }{2}^{(k-l){\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}}{∥f{\chi }_l∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p}{\left(\sum _{l=-\infty }^{k-2}{∥f{\chi }_l∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}{2}^{(k-l)p{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}}\right)}^{(1+\epsilon )}\right.\hfill \\ & {\left.\times {\left(\sum _{l=-\infty }^{k-2}{2}^{(k-l)p^{\prime }{\displaystyle \frac{1}{2}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}}\right)}^{{\displaystyle \frac{p(1+\epsilon )}{p^{\prime }}}}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_1{C}_{(n,p,\overrightarrow{q})}\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{l=-\infty }^{k_0}{2}^{l\alpha p(1+\epsilon )}{\parallel f{\chi }_l\parallel }_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\sum _{k=-\infty }^{l-2}{2}^{(k-l)p(1+\epsilon )\left(\alpha +{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}\right)}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_1{C}_{(n,p,\overrightarrow{q},\alpha )}{\parallel f\parallel }_{M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right).}\hfill \end{array}$$

This proof is complete. □

The following corollary can be derived from Theorem 1.

Corollary 1. 

Let $\theta >0$, $0\le \lambda <\infty$, $1<p,\overrightarrow{q}\le \infty$ and $-{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}<\alpha <n\left(1-{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{1}{q_i}}\right)$. Assuming a sublinear operator T satisfies

$$\left|Tf\left(x\right)\right|\le C{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{\left|f\right(y\left)\right|}{{|x-y|}^n}}dy$$

for $f\in L^1\left({\mathbb{R}}^n\right)$ with compact support, $x\notin \mathrm{supp}f$, and T is bounded on $L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)$, then T is bounded on $M{\dot{K}}_{\overrightarrow{q},\lambda }^{\alpha ,p),\theta }\left({\mathbb{R}}^n\right)$.

Proof. 

We only need to verify Conditions (ii) and (iii).

For $y\in A_k$ and $\left|x\right|\ge 2^{k+1}$, there holds $\left|x\right|\ge 2\left|y\right|$; thus, $|x-y|\ge \left|x\right|-\left|y\right|\ge {\displaystyle \frac{\left|x\right|}{2}}$, which implies that, for $\left|x\right|\ge 2^{k+1}$ and $\mathrm{supp}f\subset A_k$,

$$\left|Tf\left(x\right)\right|\le C{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{\left|f\right(y\left)\right|}{{|x-y|}^n}}dy\le 2^{n}C{\displaystyle \frac{1}{{\left|x\right|}^n}}{\int }_{A_k}\left|f\left(y\right)\right|dy\le 2^{n}C{\displaystyle \frac{{\parallel f\parallel }_{L^1\left({\mathbb{R}}^n\right)}}{{\left|x\right|}^n}}.$$

Therefore, the operator T satisfies Condition (ii).

Meanwhile, for $y\in A_k$ and $\left|x\right|\le 2^{k-2}$, there holds $\left|y\right|\ge 2\left|x\right|$; thus, $|x-y|\ge \left|y\right|-\left|x\right|\ge {\displaystyle \frac{\left|y\right|}{2}}\ge 2^{k-2}$. Therefore, for $\left|x\right|\le 2^{k-2}$ and $\mathrm{supp}f\subset A_k$,

$$\left|Tf\left(x\right)\right|\le C{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{\left|f\right(y\left)\right|}{{|x-y|}^n}}dy\le 2^{2n}C{\displaystyle \frac{1}{2^{nk}}}{\int }_{A_k}\left|f\left(y\right)\right|dy\le 2^{2n}C{\displaystyle \frac{{\parallel f\parallel }_{L^1\left({\mathbb{R}}^n\right)}}{2^{nk}}},$$

which completes the proof. □

4. Boundedness of the Fractional Type Operators in Homogeneous Grand Mixed Herz–Morrey Spaces

The following theorem provides boundedness for another type of sublinear operator, which is also known as a fractional operator due to the denominator being fractional in Conditions (ii) and (iii).

Theorem 2. 

Let $0<l<n$, $0\le \lambda <\infty$, $1<p_1\le p_2\le \infty$, $1<\overrightarrow{q_1}<{\displaystyle \frac{1}{l}}$, $l={\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{1i}}}-{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{2i}}}$, where $\overrightarrow{q_2}$ is another n-tuple that satisfies $\lambda -{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{2i}}}<\alpha <n-{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{1i}}}.$ Let $I_l$ be a sublinear operator satisfying the following conditions:

(i)

$\parallel I_l{f\parallel }_{L^{\overrightarrow{q_2}}\left({\mathbb{R}}^n\right)}\le C_0{\parallel f\parallel }_{L^{\overrightarrow{q_1}}\left({\mathbb{R}}^n\right)}$;

(ii)

For any $j\in \mathbb{Z}$ and function f with $suppf\subset A_j$ and $\left|x\right|\ge 2^{j+1}$,

$$\mid I_{l}f\left(x\right)\mid \le C_1{\displaystyle \frac{{\parallel f\parallel }_{L^1\left({\mathbb{R}}^n\right)}}{{\left|x\right|}^{n-l}}};$$

(3)

(iii)

For any $j\in \mathbb{Z}$ and function f with $suppf\subset A_j$ and $\left|x\right|\le 2^{j+1}$,

$$\mid I_{l}f\left(x\right)\mid \le C_1{\displaystyle \frac{{\parallel f\parallel }_{L^1\left({\mathbb{R}}^n\right)}}{2^{j(n-l)}}}.$$

(4)

Thus, $I_l$ is also bounded from $M{\dot{K}}_{{\overrightarrow{q}}_1,\lambda }^{\alpha ,p_1),\theta }\left({\mathbb{R}}^n\right)$ to $M{\dot{K}}_{{\overrightarrow{q}}_2,\lambda }^{\alpha ,p_2),\theta }\left({\mathbb{R}}^n\right)$.

Proof. 

Since the proof for $p_2=\infty$ is straightforward, we will focus on the case where $1<p_2<\infty$. Assuming $f\in M{\dot{K}}_{{\overrightarrow{q}}_1,\lambda }^{\alpha ,p_1),\theta }\left({\mathbb{R}}^n\right)$, decompose

$$f\left(x\right)=\sum _{j=-\infty }^{\infty }f\left(x\right){\chi }_j\left(x\right):=\sum _{j=-\infty }^{\infty }{f}_j\left(x\right),$$

and $\parallel I_l{f\parallel }_{M{\dot{K}}_{{\overrightarrow{q}}_2,\lambda }^{\alpha ,p_2),\theta }\left({\mathbb{R}}^n\right)}\le {\parallel I_{l}f\parallel }_{M{\dot{K}}_{{\overrightarrow{q}}_2,\lambda }^{\alpha ,p_1),\theta }\left({\mathbb{R}}^n\right)}$ is gained through Proposition 2.

Furthermore,

$$\begin{array}{cc}\hfill \parallel I_l{f\parallel }_{M{\dot{K}}_{{\overrightarrow{q}}_2,\lambda }^{\alpha ,p_2),\theta }\left({\mathbb{R}}^n\right)}& \le \underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_1(1+\epsilon )}{∥|I_{l}f|{\chi }_k∥}_{L^{{\overrightarrow{q}}_2}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & \le \underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_1(1+\epsilon )}{∥\left[\sum _{j=-\infty }^{k-2}\left|I_l\left(f_j\right)\right|\right]{\chi }_k∥}_{L^{{\overrightarrow{q}}_2}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & +\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_1(1+\epsilon )}{∥\left[\sum _{j=k-1}^{k+1}\left|I_l\left(f_j\right)\right|\right]{\chi }_k∥}_{L^{{\overrightarrow{q}}_2}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & +\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_1(1+\epsilon )}{∥\left[\sum _{j=k+2}^{\infty }\left|I_l\left(f_j\right)\right|\right]{\chi }_k∥}_{L^{{\overrightarrow{q}}_2}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & =:\left(J_1+J_2+J_3\right).\hfill \end{array}$$

For $J_2$, $I_l$ is bounded from $L^{\overrightarrow{q_1}}\left({\mathbb{R}}^n\right)$ to $L^{\overrightarrow{q_2}}\left({\mathbb{R}}^n\right)$,

$$\begin{array}{cc}\hfill J_2& \le C_0\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_1}{\left[\sum _{j=k-1}^{k+1}{∥I_l\left(f_j\right)∥}_{L^{{\overrightarrow{q}}_2}\left({\mathbb{R}}^n\right)}\right]}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ \hfill & \le C_0\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_1(1+\epsilon )}{\left[\sum _{j=k-1}^{k+1}{∥f_j∥}_{L^{{\overrightarrow{q}}_2}\left({\mathbb{R}}^n\right)}\right]}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ \hfill & \le C_0\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_1(1+\epsilon )}{∥f_{k-1}∥}_{L^{{\overrightarrow{q}}_2}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ \hfill & +C_0\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_1(1+\epsilon )}{∥f_k∥}_{L^{{\overrightarrow{q}}_2}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ \hfill & +C_0\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_1(1+\epsilon )}{∥f_{k+1}∥}_{L^{{\overrightarrow{q}}_2}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ \hfill =& C_0\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0-1}{2}^{(k+1)\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & +C_0\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & +C_0\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0+1}{2}^{(k-1)\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill \le & C_0{2}^{\alpha }\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & +C_0\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & +C_0{2}^{\lambda -\alpha }\underset{\epsilon >0}{sup}\underset{k_0>0}{sup}{2}^{-k_0\lambda }{\left\{{\epsilon }^{\theta }\sum _{k=-\infty }^{k_0}{2}^{k\alpha p(1+\epsilon )}{∥f{\chi }_k∥}_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p(1+\epsilon )}}}\hfill \\ \hfill & \le C_0{C}_{(\lambda ,\alpha )}\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_1}{∥f_j∥}_{L^{{\overrightarrow{q}}_1}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ \hfill & \le C_0{C}_{(\lambda ,\alpha )}{\parallel f\parallel }_{M{\dot{K}}_{{\overrightarrow{q}}_1,\lambda }^{\alpha ,p_1),\theta }\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

Next, we consider $J_1$, $\mathrm{supp}\left(f{\chi }_l\right)\subset A_{k-2}$, $x\in A_k;$ thus, $2^{k-1}\le \left|x\right|<2^k$ through Condition (ii), as follows:

$$\begin{array}{c}\hfill \mid I_l\left(f{\chi }_j\right)\left(x\right)\mid \le C_1{\displaystyle \frac{\parallel f{\chi }_j{\parallel }_{L^1\left({\mathbb{R}}^n\right)}}{{\left|x\right|}^{n-l}}}\le C_1{C}_{\left(n\right)}{\parallel f{\chi }_j\parallel }_{L^1\left({\mathbb{R}}^n\right)}.\end{array}$$

Thus, by the similar method to that in the proof of Theorem 1, one has

$$\begin{array}{cc}\hfill J_1& \le C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{k\alpha p_1}{∥\left[\sum _{j=-\infty }^{k-2}{2}^{-k(n-l)}{∥f{\chi }_j∥}_{L^1\left({\mathbb{R}}^n\right)}\right]{\chi }_k∥}_{L^{{\overrightarrow{q}}_2}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & \le C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{\left(k(\alpha -n+l)+{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{2i}}}\right)p_1(1+\epsilon )}{\left(\sum _{j=-\infty }^{k-2}{\parallel f{\chi }_j\parallel }_{L^1\left({\mathbb{R}}^n\right)}\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & \le C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{2}^{\left(k(\alpha -n+l)+{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{2i}}}\right)p_1(1+\epsilon )}{\left(\sum _{j=-\infty }^{k-2}\parallel f{\chi }_j{\parallel }_{L^{\overrightarrow{q_1}}\left({\mathbb{R}}^n\right)}{\parallel {\chi }_j\parallel }_{L^{{\overrightarrow{q_1}}^{\prime }}\left({\mathbb{R}}^n\right)}\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & \le C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{k=-\infty }^{k_0}{\left[{\displaystyle \sum _{j=-\infty }^{k-2}}{2}^{j\alpha }{\parallel f{\chi }_j\parallel }_{L^{\overrightarrow{q_1}}\left({\mathbb{R}}^n\right)}{2}^{(j-k)\left(n-{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{1i}}}-\alpha \right)}\right]}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & \le C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }\left\{\sum _{k=-\infty }^{k_0}\right(\sum _{j=-\infty }^{k-2}{2}^{j\alpha }{\parallel f{\chi }_j\parallel }_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}{2}^{(j-k){\displaystyle \frac{1}{2}}\left(n-{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{1i}}}-\alpha \right)}\hfill \\ & \times \sum _{j=-\infty }^{k-2}{2}^{(j-k){\displaystyle \frac{1}{2}}\left(n-{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{1i}}}-\alpha \right)}{)}^{p(1+\epsilon )}{\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & \le C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }\{\sum _{k=-\infty }^{k_0}\left(\sum _{j=-\infty }^{k-2}{2}^{j{p}_1(1+\epsilon )\alpha }{\parallel f{\chi }_j\parallel }_{L^{\overrightarrow{q}}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}{2}^{(j-k){\displaystyle \frac{p_1(1+\epsilon )}{2}}\left(n-{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{1i}}}-\alpha \right)}\right)\hfill \\ & \times {\left(\sum _{j=-\infty }^{k-2}{2}^{(j-k){\displaystyle \frac{{p_1(1+\epsilon )}^{\prime }}{2}}\left(n-{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{1i}}}-\alpha \right)}\right)}^{{\displaystyle \frac{p_1}{p_1^{\prime }}}}{\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & \le C_1{C}_{(n,p,\overrightarrow{q},\alpha )}\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{j=-\infty }^{k_0}{2}^{j\alpha p_1(1+\epsilon )}{∥f_j∥}_{L^{{\overrightarrow{q}}_1}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & \le C_1{C}_{(n,p,\overrightarrow{q},\alpha )}{\parallel f\parallel }_{M{\dot{K}}_{{\overrightarrow{q}}_1,\lambda }^{\alpha ,p_1),\theta }\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

$J_3$ is similar to $J_1$ through Condition (iii), as follows:

$$\begin{array}{c}\hfill \left|I_l\left(f{\chi }_j\right)\left(x\right)\right|\le C_1{\displaystyle \frac{\parallel f{\chi }_j{\parallel }_{L^1\left({\mathbb{R}}^n\right)}}{2^{-j(n-l)}}}\le C_1{C}_{\left(n\right)}{\parallel f{\chi }_j\parallel }_{L^{\overrightarrow{q_1}}\left({\mathbb{R}}^n\right)}.\end{array}$$

Then, with the use of a similar method to that in the proof of Theorem 1,

$$\begin{array}{cc}\hfill J_3& \le C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{j=-\infty }^{k_0}{\left(\sum _{j=k+2}^{\infty }{2}^{j\alpha }{\parallel f{\chi }_j\parallel }_{L^{\overrightarrow{q_1}}\left({\mathbb{R}}^n\right)}{2}^{(k-j)\left({\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{2i}}}+\alpha \right)}\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & \le C_1{C}_{\left(n\right)}\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }\{\sum _{j=-\infty }^{k_0}{2}^{j\alpha p_1}{\parallel f{\chi }_j\parallel }_{L^{\overrightarrow{q_1}}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}{2}^{(k-j)\left({\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{2i}}}+\alpha \right){\displaystyle \frac{p_1(1+\epsilon )}{2}}}\hfill \\ & \times {\left(\sum _{j=k+2}^{\infty }{2}^{(k-j)\left({\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{2i}}}+\alpha \right){\displaystyle \frac{p_1{(1+\epsilon )}^{\prime }}{2}}}\right)}^{{\displaystyle \frac{p_1}{p_1^{\prime }}}}{\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ \hfill & \le C_1{C}_{(n,p,\overrightarrow{q},\alpha )}\underset{\epsilon >0}{sup}\underset{k_0\in \mathbb{Z}}{sup}{2}^{-k_0\lambda }{\left\{\sum _{j\in \mathbb{Z}}{2}^{j\alpha p_1(1+\epsilon )}{\parallel f{\chi }_j\parallel }_{L^{\overrightarrow{q_1}}\left({\mathbb{R}}^n\right)}^{p_1(1+\epsilon )}\right\}}^{{\displaystyle \frac{1}{p_1(1+\epsilon )}}}\hfill \\ & \le C_1{C}_{(n,p,\overrightarrow{q},\alpha )}{\parallel f\parallel }_{M{\dot{K}}_{{\overrightarrow{q}}_1,\lambda }^{\alpha ,p_1),\theta }\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

That is also why $\lambda -{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{q_{2i}}}<\alpha$. Therefore,

$$\parallel I_l{f\parallel }_{M{\dot{K}}_{{\overrightarrow{q}}_2,\lambda }^{\alpha ,p_2),\theta }\left({\mathbb{R}}^n\right)}\le (C_0{C}_{(\lambda ,\alpha )}+2C_1{C}_{(n,p,\overrightarrow{q},\alpha )}){\parallel f\parallel }_{M{\dot{K}}_{{\overrightarrow{q}}_1,\lambda }^{\alpha ,p_1),\theta }\left({\mathbb{R}}^n\right)}.$$

This completes the proof of the theorem. □

5. Conclusions

In this article, we define homogeneous grand mixed Herz–Morrey spaces and explore some of their fundamental properties. Furthermore, we obtain the boundedness of sublinear operators and fractional-type operators in homogeneous grand mixed Herz–Morrey spaces.