Abstract Weighted Morrey Spaces and Applications

Abstract

We introduce the abstract weighted Morrey spaces $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )$, where $\mu$ is a general measure; investigate the properties of their predual spaces; and prove the boundedness of the Hardy–Littlewood maximal operator. Furthermore, we obtain an extrapolation theorem on $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )$, and consequently establish the norm inequalities for bounded oscillation ($\mathfrak{B}O$) operators on $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )$. As an application, we verify that $\mathfrak{B}O$ operators include the maximal operators, Littlewood–Paley square operators, and Carleson operators on classical weighted Morrey spaces, as well as Calderón–Zygmund operators on weighted Morrey spaces of homogeneous type.

1. Introduction

In 1972, Muckenhoupt [1] introduced the theory of $A_p$ weights with respect to the Lebesgue measure and established the boundedness of the Hardy–Littlewood maximal operator on weighted Lebesgue spaces $L^p(\mathbb{R},dx)$. This pioneering work laid the foundation for modern weighted theory in harmonic analysis. In recent decades, weighted theories on ${\mathbb{R}}^n$ (with the Lebesgue measure) and on spaces of homogeneous type have driven research into weighted theory on diverse measure spaces; see [2,3,4,5] and the references therein.

Morrey spaces were originally introduced by Morrey [6] in 1938 while studying the existence and differentiability of solutions to certain linear elliptic PDEs. Subsequently, significant attention has been devoted to the $A_p$ weighted theory on Morrey spaces with respect to various measures. In 2009, Komori and Shirai [7] investigated $A_p$ weights on Morrey spaces with respect to the Lebesgue measure; see [5,8,9,10,11,12,13,14] for related work. Later, in 2017, Kokilashvili and Meskhi [15] extended this study to weighted Morrey spaces of homogeneous type; see [16,17,18] for further work on homogeneous spaces.

In 1982, the classical extrapolation theory was first introduced by Rubio de Francia [19,20]. In 2023, Cao, Marín, and Martell [21] extended this extrapolation theory to the context of Banach spaces and modular spaces. For further references, see [4,5,22,23,24,25].

In 2019, Karagulyan [26] introduced the abstract Lebesgue space with a ball basis, and established a class of $\mathfrak{B}O$ operators. Karagulyan proved that, in the sense of the Lebesgue measure, $\mathfrak{B}O$ operators include maximal operators, Carleson operators and martingale transforms, and also demonstrated the fact that classical Calderón–Zygmund operators of homogeneous spaces are $\mathfrak{B}O$ operators, and established a weighted norm inequality for $\mathfrak{B}O$ operators on abstract measure spaces.

In 2026, Shan, Xue, and Zhou [27] established abstract Morrey spaces and obtained norm estimates for a class of $\mathfrak{B}O$ operators. These advances underscore the importance of $\mathfrak{B}O$ operators as a unified framework that can encompass a broader class of classical operators and establish a unified sparse domination method in abstract Morrey spaces. For further results on abstract function measure spaces, see [26,28,29,30,31].

With the deepening investigation of abstract function space theory and building upon the above research findings, this paper naturally turns to abstract weighted Morrey spaces. The specific motivations are as follows:

• Komori and Shirai [7] established weighted Morrey spaces on the Lebesgue measure, extending the related theory of Morrey spaces. Relevant results regarding weighted homogeneous Morrey spaces are provided by [15,32]. Thus, we intend to develop a new type of weighted Morrey space, which encompass the aforementioned spaces as special cases, thereby extending the classical theory.
• Shan, Xue, and Zhou [27] established abstract Morrey spaces with the ball basis, while Karagulyan [26] developed weighted norm inequalities and $\mathfrak{B}O$ operators in the abstract $L_u^p$ setting. By combining these two frameworks, this paper constructs abstract weighted Morrey spaces; this synthesis introduces the Morrey parameter $\kappa$ and the weight class ${\mathbb{W}}_{p,\kappa }$, which bring new challenges not present in either precursor.

This paper establishes the theory of abstract weighted Morrey spaces associated with a ball basis and investigates their dual spaces, extrapolation properties, and weighted norm inequalities for $\mathfrak{B}O$ operators, with the following principal contributions:

• We present the first systematic treatment of abstract weighted Morrey spaces built upon a ball basis. The resulting framework encompasses various classical spaces—including Morrey spaces on the Lebesgue measure and spaces of homogeneous type on the Lebesgue measure—and therefore possesses a high degree of generality. As a central structural result, we identify their predual as a weighted block space. We further establish an extrapolation theorem for these weighted Morrey spaces, which unifies and extends the classical Rubio de Francia method to the Morrey-type setting.
• We establish norm estimates for $\mathfrak{B}O$ operators on abstract weighted Morrey spaces. Moreover, we verify that on classical weighted Morrey spaces, the maximal operator, Littlewood–Paley square operator, and Carleson operator are all $\mathfrak{B}O$ operators, and that Calderón–Zygmund operators are $\mathfrak{B}O$ operators on weighted homogeneous Morrey spaces.

In conclusion, the organization of this paper is as follows: In Section 2, we present the necessary definitions and fundamental properties required for our main results. Specifically, we introduce the definitions of ball basis, Muckenhoupt weights, and establish abstract weighted Morrey spaces along with their predual spaces. Section 3 discusses the boundedness of the Hardy–Littlewood maximal operator on abstract weighted Morrey spaces with ball basis and weighted block spaces, and proves that the predual space of weighted Morrey spaces are precisely the weighted block space. Section 4 is devoted to developing extrapolation theory on abstract weighted Morrey spaces and presenting some applications. In Section 5, working within the framework of abstract weighted Morrey spaces with ball basis, we provide concrete examples and discuss Calderón–Zygmund operators on homogeneous spaces, Littlewood–Paley operators on ${\mathbb{R}}^n$, and Carleson operators.

2. Preliminary

2.1. Ball Basis on Abstract Measure Spaces

Within the broad context of abstract spaces, Karagulyan [26] undertakes a fundamental generalization of the traditional distance concept by replacing it with a more flexible and encompassing metric. This broader metric structure enables us to transform conventional geometric notions into a unified ball basis framework, thereby establishing a more powerful and universally applicable tool for analysis across diverse mathematical domains. We now recall the definition of the ball basis.

Definition 1 

([30]). Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{M}$ is a σ-algebra on X. A collection $\mathfrak{B}\subset \mathfrak{M}$ is called a ball basis if it satisfies the following:

(B₁)

$\mathfrak{B}$ is a family of balls. For every $D\in \mathfrak{B}$, $0<\mu \left(D\right)<\infty$.

(B₂)

For any two points $a,b\in X$, there exists ball $D\in \mathfrak{B}$ containing both a and b.

(B₃)

For any $\epsilon >0$ and any measurable set $E\in \mathfrak{M}$, there exists a finite or infinite sequence $\left\{D_k\right\}\subset \mathfrak{B}$, satisfying $\mu (E\Delta {\bigcup }_k{D}_k)<\epsilon$.

(B₄)

For each ball $D\in \mathfrak{B}$, there corresponds a ball $D^{\mathfrak{C}}\in \mathfrak{B}$ such that

$${\alpha }_1\mu \left(D\right)\le \mu \left(D^{\mathfrak{C}}\right)\le {\alpha }_2\mu \left(D\right)$$

and

$$\bigcup _{\begin{array}{c}{D}_{*}\in \mathfrak{B},D\cap D_{*}\ne \varnothing \\ \mu \left(D_{*}\right)\le 2\mu \left(D\right)\end{array}}{D}_{*}\subseteq D^{\mathfrak{C}},$$

where ${\alpha }_1,{\alpha }_2>1$ are universal constants. Moreover, if $C,D\in \mathfrak{B}$ with $C\subset D$, then $C^{\mathfrak{C}}\subset D^{\mathfrak{C}}.$

We denote by $(X,\mathfrak{M},\mu )$ the abstract measure space endowed with a ball basis.

Example 1. 

Below, we give examples that imply ball basis in the classical example.

• Euclidean space: In Euclidean space, the ball column $\mathfrak{B}=\{B_k:B_k=B(x_k,r)=\{x\in {\mathbb{R}}^n:\parallel x_k{-x\parallel }_{l^2}\le r,0<r<\infty \left\}\right\}$, can form a ball basis, where ${\parallel x-y\parallel }_{l^2}=\sqrt{{\sum }_{i=1}^n{(x_i-y_i)}^2},x=(x_1,x_2,\dots ,x_n),y=(y_1,y_2,\dots ,y_n)\in {\mathbb{R}}^n$.
• Spaces of homogeneous type: In spaces of homogeneous type $(X,\rho ,\mathfrak{M},\mu )$, ρ is a quasimetric on X. If $\mathfrak{U}\left(\mu \right)=\left\{B\right(x,r):B(x,r)=\{y\in X:\rho (x,y)<r\left\}\right\}$ satisfies the density condition; then, the enlarged family of balls ${\mathfrak{U}}^{\prime }\left(\mu \right)$ is said to constitute a ball basis [26].
• Dyadic cubes: We shall focus on the square $Q={[a,b]}^n=\{x=(x_1,x_2\dots ,x_n):x_i\in [a,b],a,b\in R,1\le i\le n\}\subset {\mathbb{R}}^n$; the collection of its dyadic sub-cubes forms a ball basis for $(Q,\mu )$.
  However, the example given below fails to constitute a valid ball basis.
• Non-example: Let the ball $B=B(x,r)=\{y=(y_1,y_2\dots ,y_n):y\in {\mathbb{R}}^n,|y-x|<r,$ $1\le i\le n, n\in \mathbb{N}, r>0\}$ be defined, for which the family of sets $\mathcal{B}=\left\{B_k:B_k=B(x,r/2^k), k=1,2,\dots .\right\}$. This $\mathcal{B}$ is not a ball basis for $(B,\mu )$.

Remark 1 

([26]). (a) Given a ball $D\in \mathfrak{B}$, its hull is defined recursively by $D^{\left[0\right]}=D$ and $D^{[n+1]}={\left(D^{\left[n\right]}\right)}^{\mathfrak{C}}$. By condition $B_4$, we have ${\alpha }_1\mu \left(D^{\left[n\right]}\right)\le \mu \left(D^{[n+1]}\right)\le {\alpha }_2\mu \left(D^{\left[n\right]}\right)$. Applying this inequality repeatedly yields ${\alpha }_1^{n+1}\mu \left(D\right)\le \mu \left(D^{[n+1]}\right)\le {\alpha }_2^{n+1}\mu \left(D\right)$.

(b) 

Let $(X,\mathfrak{M},\mu )$ denote a measure space and $\mathfrak{B}\subset \mathfrak{M}$ be a ball basis. For any $E\in \mathfrak{B}$, there holds

$$X=\underset{n\to \infty }{lim}{E}^{\left[n\right]}.$$

2.2. Muckenhoupt $A_{p,\mathfrak{B}}$ Weight

Let u be a measurable function on $(X,\mathfrak{M},\mu )$. If $u\left(x\right)\in (0,\infty )$ for $\mu$, almost every x, then u, is said to be a weight function.We begin by recalling some notations, lemmas, and relevant properties concerning weights. In particular, we recall the definition of the Muckenhoupt weight classes $A_{p,\mathfrak{B}}$ on an abstract measure space $(X,\mathfrak{M},\mu )$, as these classes form an essential component of extrapolation theory.

Definition 2 

([30]). For $p\in (1,\infty )$, the Muckenhoupt $A_{p,\mathfrak{B}}$ class on a measure space $(X,\mathfrak{M},\mu )$ with a ball basis $\mathfrak{B}$ is the collection of all weight functions $u:X\to (0,\infty )$ such that

$${\left[u\right]}_{A_{p,\mathfrak{B}}}:=\underset{A\in \mathfrak{B}}{sup}\left({\displaystyle \frac{1}{\mu \left(A\right)}}{\int }_{A}u\left(x\right)d\mu \left(x\right)\right){\left({\displaystyle \frac{1}{\mu \left(A\right)}}{\int }_{A}u{\left(x\right)}^{1-p^{\prime }}d\mu \left(x\right)\right)}^{p-1}<\infty ,$$

where $p^{\prime }=p/(p-1)$. For the endpoint $p=1$, the class $A_{1,\mathfrak{B}}$ consists of those weights u satisfying

$${\left[u\right]}_{A_{1,\mathfrak{B}}}:=\parallel u^{-1}\left(x\right)\underset{\begin{array}{c}x\in A\\ A\in \mathfrak{B}\end{array}}{sup}{\displaystyle \frac{1}{\mu \left(A\right)}}{\int }_A{|u\left(x\right)|d\mu \left(x\right)\parallel }_{L^{\infty }(X,\mathfrak{M},\mu )}<\infty .$$

We also set $A_{\infty ,\mathfrak{B}}:={\bigcup }_{p\ge 1}{A}_{p,\mathfrak{B}}$.

Lemma 1 

([30]). Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis. Given that u is a weight function with $u\in A_{\infty ,\mathfrak{B}}={\bigcup }_{p\ge 1}{A}_{p,\mathfrak{B}}$, the following holds:

(1) 

There exist constants $0<\gamma \le 1\le C_0<\infty$ such that for every $E\in \mathfrak{B}$ and measurable set $F\subset E$,

$${\displaystyle \frac{\mu \left(F\right)}{\mu \left(E\right)}}\le C_0{\left({\displaystyle \frac{u\left(F\right)}{u\left(E\right)}}\right)}^{\gamma }.$$

(2) 

For each $\zeta \in (0,1)$ there exist a number $\xi \in (0,1)$ such that, for every $E\in \mathfrak{B}$ and measurable set $E\subset F$,

$$\mu \left(F\right)\ge \zeta \mu \left(E\right)\Rightarrow u\left(F\right)\ge \xi u\left(E\right),$$

and conversely,

$$\mu \left(F\right)\le \zeta \mu \left(E\right)\Rightarrow u\left(F\right)\le \xi u\left(E\right).$$

2.3. Abstract Weighted Lebesgue Spaces

Definition 3 

([29]). Given a measure space $(X,\mathfrak{M},\mu )$, $1<p<\infty$, and $u\in A_{p,\mathfrak{B}}$. For a measurable function f, the collection

$$L_u^p(X,\mathfrak{M},\mu ):=\left\{{f\mathrm{is}\mathit{measurable}: \parallel f\parallel }_{L_u^p(X,\mathfrak{M},\mu )}<\infty \right\}$$

is called the abstract weighted Lebesgue spaces, where

$${\parallel f\parallel }_{L_u^p(X,\mathfrak{M},\mu )}:={\left({\int }_X{|f\left(x\right)|}^{p}u\left(x\right)d\mu \left(x\right)\right)}^{1/p}.$$

For convenience, we abbreviate ${\parallel f\parallel }_{L_u^p(X,\mathfrak{M},\mu )}$ as ${\parallel f\parallel }_{L_u^p}$.

Let $1<p<\infty$ and $u\in A_{p,\mathfrak{B}}$. From the definition of abstract weighted Lebesgue spaces and properties of $A_{p,\mathfrak{B}}$ weights,

(a)

For any $A\in \mathfrak{B}$, ${\chi }_A\in L_u^p$.

(b)

For any sequence of measurable functions ${\left\{h_k\right\}}_{k=1}^{\infty }$ with $0\le h_k\uparrow h$ a.e. $k\to \infty$, then

$$\underset{k\to \infty }{lim}\parallel h_k{\parallel }_{L_u^p}={\parallel h\parallel }_{L_u^p}.$$

(c)

For any $h,g\in L_u^p$, if $|h|\le |g|$ a.e., then

$${\parallel h\parallel }_{L_u^p}\le {\parallel g\parallel }_{L_u^p}.$$

(d)

For any $h,g\in L_u^p$, then

$${\parallel h+g\parallel }_{L_u^p}\le {\parallel h\parallel }_{L_u^p}+{\parallel g\parallel }_{L_u^p}.$$

From Reference [33], the above construction yields two lemmas.

Lemma 2 

([33]). Let $1<p<\infty$. Then, $L^p$ is a Banach function space with the norm

$${\parallel h\parallel }_{L^p}=sup\left\{\left|{\int }_{X}h\left(x\right)g\left(x\right)d\mu \left(x\right)\right|:g\in L^{p^{\prime }},{\parallel g\parallel }_{L^{p^{\prime }}}\le 1\right\},$$

where the dual space of $L^p$ is $L^{p^{\prime }}$.

By analogy with the above lemma, Hölder’s inequality can be obtained for abstract weighted Lebesgue spaces:

Lemma 3 

([33]). Consider a measure space $(X,\mathfrak{M},\mu )$ with a ball basis $\mathfrak{B}$. Given that $h,g$ are measurable functions, if $1<p<\infty$, $v=u^{-{\displaystyle \frac{1}{p-1}}}$ and $u\in A_{p,\mathfrak{B}}$, then

$$\begin{array}{cc}\hfill {\int }_X|h\left(x\right)g\left(x\right)|d\mu \left(x\right)& ={\int }_X\left|h\left(x\right)u{\left(x\right)}^{-1/p}\right|·\left|g\left(x\right)u{\left(x\right)}^{1/p}\right|d\mu \left(x\right)\hfill \\ \hfill & \le {C\parallel h\parallel }_{L_v^{p^{\prime }}}{\parallel g\parallel }_{L_u^p}.\hfill \end{array}$$

Furthermore, using properties of weighted Lebesgue spaces, we have

$$\begin{array}{cc}\hfill {\parallel h\parallel }_{L_u^p}& =\parallel h{u}^{1/p}{\parallel }_{L^p}\hfill \\ \hfill & \le Csup\left\{\left|{\int }_{X}h\left(x\right)u{\left(x\right)}^{1/p}g\left(x\right)u{\left(x\right)}^{-1/p}d\mu \left(x\right)\right|:g{u}^{-1/p}\in L^{p^{\prime }},{\parallel g{u}^{-1/p}\parallel }_{L^{p^{\prime }}}\le 1\right\}\hfill \\ \hfill & \le Csup\left\{\left|{\int }_{X}h\left(x\right)g\left(x\right)d\mu \left(x\right)\right|:g\in L_v^{p^{\prime }},{\parallel g\parallel }_{L_v^{p^{\prime }}}\le 1\right\},\hfill \end{array}$$

for some constant $C>0$. That is, with respect to the abstract Lebesgue spaces, the dual space of $L_u^p$ is $L_v^{p^{\prime }}$.

2.4. Abstract Weighted Morrey Spaces and Their Dual Spaces

The core of our work consists of introducing an abstract weighted Morrey space, defined as follows.

Definition 4. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis, $1<p<\infty$, $0<\kappa <1$, and $u\in A_{p,\mathfrak{B}}$. For a measurable function f, the collection

$$M_{p,u}^{\kappa }(X,\mathfrak{M},\mu ):=\{f\in L_{\mathrm{loc}}^p{\left(u\right):\parallel f\parallel }_{M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )}<\infty \}$$

is called the abstract weighted Morrey spaces, where $L_{\mathrm{loc}}^p\left(u\right)$ denotes the space of locally p-integrable functions associated with the weight u, and

$${\parallel f\parallel }_{M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )}=\underset{A\in \mathfrak{B}}{sup}{\displaystyle \frac{1}{u{\left(A\right)}^{\kappa /p}}}{\left({\int }_A{|f\left(x\right)|}^{p}u\left(x\right)d\mu \left(x\right)\right)}^{1/p}<\infty .$$

Clearly, ${\parallel f\parallel }_{M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )}$ defines a norm. For convenience, we abbreviate it as ${\parallel f\parallel }_{M_{p,u}^{\kappa }}$.

Remark 2. 

In what follows, we show that the constructed space contains certain important function spaces.

(a) 

If $(X,\mathfrak{M},\mu )$ is the Euclidean space ${\mathbb{R}}^n$ and $d\mu \left(x\right)=dx$, then $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )=L_u^{p,\kappa }\left({\mathbb{R}}^n\right)$ is the classical weighted Morrey space [7].

(b) 

If $u\left(x\right)\equiv 1$ and $\kappa =1-p/q$ with $0<q<p$, then $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )=M_p^q(X,\mathfrak{M},\mu )$ is the abstract Morrey space [27].

(c) 

If $u\left(x\right)\equiv 1$, $\kappa =1-p/q$ with $0<q<p$, and $(X,\mathfrak{M},\mu )$ is the Euclidean space with $d\mu \left(x\right)=dx$, then $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )=M_p^q\left({\mathbb{R}}^n\right)$ is the classical Morrey space [6].

(d) 

If $\kappa =0$, then $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )=L_u^p(X,\mathfrak{M},\mu )$ is the abstract weighted Lebesgue space, if $\kappa =1$, then $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )=L_u^{\infty }\left(X\right)$ [30].

(e) 

If $u\left(x\right)\equiv 1$ and $(X,\mathfrak{M},\mu )$ is the Euclidean space ${\mathbb{R}}^n$ with $d\mu \left(x\right)=dx$, if $\kappa =0$, then $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )=L_u^{\infty }\left({\mathbb{R}}^n\right)$, and if $\kappa =1$, then $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )=L^{\infty }\left({\mathbb{R}}^n\right)$ [34].

For the definition of block space on abstract measure spaces, see [27]. We now present the definition of their weighted analogues. First, we give the definition of ${\mathfrak{h}}_{p,u}^{\kappa }$.

Definition 5. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis, $1<p<\infty$, $0<\kappa <1$, and $u\in A_{p,\mathfrak{B}}$. We say Lebesgue measurable function $a\in {\mathfrak{h}}_{p,u}^{\kappa }$, if there exists $D\in \mathfrak{B}$ with $suppa\subseteq D$ and

$${\parallel a\parallel }_{L_u^p}\le {\displaystyle \frac{1}{u{\left(D\right)}^{\kappa /p}}}.$$

We now present the definition of weighted block space on abstract measure space.

Definition 6. 

The weighted block space ${\mathcal{H}}_{p,u}^{\kappa }$ is defined by

$${\mathcal{H}}_{p,u}^{\kappa }=\left\{{\displaystyle \sum _{i=1}^{\infty }}{\lambda }_i{a}_i:{\displaystyle \sum _{i=1}^{\infty }}|{\lambda }_i|<\infty \mathit{and}{a}_i\in {\mathfrak{h}}_{p,u}^{\kappa }\right\}.$$

For functions in the block space ${\mathcal{H}}_{p,u}^{\kappa }$, we equip it with the norm

$${\parallel g\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}=inf\left\{{\displaystyle \sum _{i=1}^{\infty }}|{\lambda }_i|:g={\displaystyle \sum _{i=1}^{\infty }}{\lambda }_i{a}_i\mathrm{a}.\mathrm{e}.\right\}.$$

Next, we present some properties of elements in ${\mathfrak{h}}_{p,u}^{\kappa }$ and the weighted block space.

Proposition 1. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis, $1<p<\infty$, $0<\kappa <1$, and $u\in A_{p,\mathfrak{B}}$. If $g\in {\mathfrak{h}}_{p,u}^{\kappa }$, then ${\parallel g\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}\le 1$.

Proof. 

In the decomposition $g={\sum }_{i=1}^{\infty }{\lambda }_i{g}_i$, we take

$$g_1=g,g_2=g_3=\dots =0,{\lambda }_1=1,{\lambda }_2={\lambda }_3=\dots =0.$$

Application of the definition of weighted block space yields the result. □

The following proposition demonstrates how to construct an element in ${\mathfrak{h}}_{p,u}^{\kappa }$.

Proposition 2. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis, $1<p<\infty$, $0<\kappa <1$, and $u\in A_{p,\mathfrak{B}}$. For $f\in L_u^p(X,\mathfrak{M},\mu )$ with $suppf\subseteq A$ for some $A\in \mathfrak{B}$; then,

$${\parallel f\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}\le {\parallel f\parallel }_{L_u^p}{\displaystyle \frac{1}{u{\left(A\right)}^{\kappa /p}}}.$$

Proof. 

Define $g={\displaystyle \frac{1}{{\parallel f\parallel }_{L_u^p}}}{\displaystyle \frac{1}{u{\left(A\right)}^{\kappa /p}}}f$, assuming $f\ne 0$ a.e.; then, $suppg\subseteq A$ and

$${\parallel g\parallel }_{L_u^p}={\displaystyle \frac{1}{u{\left(A\right)}^{\kappa /p}}}.$$

Therefore, $g\in {\mathfrak{h}}_{p,u}^{\kappa }$. By Proposition 1, we have ${\parallel g\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}\le 1$. Combining these results yields

$${\parallel f\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}\le {\parallel f\parallel }_{L_u^p}{\displaystyle \frac{1}{u{\left(A\right)}^{\kappa /p}}}.$$

□

Proposition 3. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis. If $0<\kappa <1$, $1<p<\infty$ and $u\in A_{p,\mathfrak{B}}$, then $f\in {\mathcal{H}}_{p,u}^{\kappa }$ exactly when there exists $g\in {\mathcal{H}}_{p,u}^{\kappa }$ satisfying $|f(x)|\le g\left(x\right)$ for a.e., $x\in X$.

Proof. 

$(⟹)$ Suppose $f\in {\mathcal{H}}_{p,u}^{\kappa }$. By the properties of ${\mathcal{H}}_{p,u}^{\kappa }$, there exists a sequence ${\left\{{\lambda }_i\right\}}_{i=1}^{\infty }\in {\mathsf{\ell }}^1$ and a collection of elements $b_i$ in ${\mathfrak{h}}_{p,u}^{\kappa }$; therefore, $f={\sum }_{i=1}^{\infty }{\lambda }_i{b}_i$. Define $g={\sum }_{i=1}^{\infty }|{\lambda }_i||b_i|$. Then, we have $g\in {\mathcal{H}}_{p,u}^{\kappa }$ and $|f|\le g$.

$(⟸)$ suppose $|f(x)|\le g\left(x\right)$ for a.e. $x\in X$ and $g\in {\mathcal{H}}_{p,u}^{\kappa }$. Then, decompose g as $g={\sum }_{i=1}^{\infty }{\lambda }_i{b}_i$, where ${\left\{{\lambda }_i\right\}}_{i=1}^{\infty }\in {\mathsf{\ell }}^1\left(\mathbb{N}\right)$ and $b_i\in {\mathfrak{h}}_{p,u}^{\kappa }$. We can then see that

$${\chi }_{\{y:g(y)\ne 0\}}\left(x\right)={\displaystyle \sum _{i=1}^{\infty }}{\lambda }_i{\displaystyle \frac{1}{g\left(x\right)}}{b}_i\left(x\right),$$

then,

$$f\left(x\right)={\displaystyle \sum _{i=1}^{\infty }}{\lambda }_i{\displaystyle \frac{f\left(x\right)}{g\left(x\right)}}{b}_i\left(x\right),\mathrm{a}.\mathrm{e}.x\in X.$$

Since $|f(x)|/g\left(x\right)\le 1$ for a.e. $x\in X$, $b_{i}f/g$ are elements of ${\mathfrak{h}}_{p,u}^{\kappa }$. This proves the property. □

3. Maximal Operator on $(\mathit{X},\mathfrak{M},\mathit{\mu })$

Here, we consider the Hardy–Littlewood maximal operator and its boundedness on $M_{p,u}^{\kappa }$ and ${\mathcal{H}}_{p,u}^{\kappa }$. Below, we give the definitions of the Hardy–Littlewood maximal operator and the weighted maximal operator.

Definition 7 

([26]). Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis. For any measurable function f on the measure space $(X,\mathfrak{M},\mu )$, the Hardy–Littlewood maximal operator M associated with the family $\mathfrak{B}$ is defined by

$$Mf\left(x\right):=\underset{\begin{array}{c}x\in B\\ B\in \mathfrak{B}\end{array}}{sup}{\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_B|f\left(y\right)|d\mu \left(y\right).$$

Definition 8 

([26]). Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis. Suppose $u\in A_{p,\mathfrak{B}}$. Then, for every measurable function f, define the weighted maximal operator $M_u$ by

$$M_{u}f\left(x\right):=\underset{\begin{array}{c}x\in B\\ B\in \mathfrak{B}\end{array}}{sup}{\displaystyle \frac{1}{u\left(B\right)}}{\int }_B|f\left(y\right)|u\left(y\right)d\mu \left(y\right).$$

To prove the boundedness of the M on abstract weighted Morrey spaces with ball basis, we state the following Lemma.

Lemma 4 

([30]). Let $(X,\mathfrak{M},\mu )$ be a measure space with a ball basis $\mathfrak{B}$. For $1\le p<\infty$, if $u\in A_{p,\mathfrak{B}}$, then there exists $r>1$ such that for every $A\in \mathfrak{B}$,

$${\left({\displaystyle \frac{1}{\mu \left(A\right)}}{\int }_{A}u{\left(x\right)}^{r}d\mu \left(x\right)\right)}^{{\displaystyle \frac{1}{r}}}\le {\displaystyle \frac{\mathfrak{C}}{\mu \left(A\right)}}{\int }_{A}u\left(x\right)d\mu \left(x\right).$$

This is called the reverse Hölder inequality.

Theorem 1. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis. If $1<p<\infty$, $0<\kappa <1$ and $u\in A_{p,\mathfrak{B}}$; then, ${\parallel Mg\parallel }_{M_{p,u}^{\kappa }}\le \mathfrak{C}{\parallel g\parallel }_{M_{p,u}^{\kappa }}$.

Proof. 

(I) To establish the boundedness of M, we first prove that of $M_u$, then deduce the boundedness of the M via their relation. For any $A\in \mathfrak{B}$, let $g_1=g{\chi }_{A^{\left[2\right]}}$ and set $g_2=g-g_1$. By the sublinearity of $M_u$, then

$$\begin{array}{cc}\hfill & {\left({\int }_A{|M_{u}g\left(x\right)|}^{p}u\left(x\right)d\mu \left(x\right)\right)}^{1/p}\hfill \\ & \le {\left({\int }_A{|M_u{g}_1\left(x\right)|}^{p}u\left(x\right)d\mu \left(x\right)\right)}^{1/p}+{\left({\int }_A{|M_u{g}_2\left(x\right)|}^{p}u\left(x\right)d\mu \left(x\right)\right)}^{1/p}\hfill \\ & =:I+II.\hfill \end{array}$$

Concerning term I, by virtue of $M_u$ being bounded on abstract weighted Lebesgue spaces, it follows that

$$\begin{array}{cc}\hfill I& \le {\left({\int }_X{|M_u{g}_1\left(x\right)|}^{p}u\left(x\right)d\mu \left(x\right)\right)}^{1/p}\hfill \\ \hfill & \le \mathfrak{C}{\left({\int }_{A^{\left[2\right]}}{|g\left(x\right)|}^{p}u\left(x\right)d\mu \left(x\right)\right)}^{1/p}\hfill \\ \hfill & \le {\mathfrak{C}\parallel g\parallel }_{M_{p,u}^{\kappa }}u{\left(A\right)}^{\kappa /p}.\hfill \end{array}$$

Next, we estimate II. For each $x\in A$, then

$$M_u{g}_2\left(x\right)=\underset{\begin{array}{c}x\in A\\ A\in \mathfrak{B}\end{array}}{sup}{\displaystyle \frac{1}{u\left(A\right)}}{\int }_A|g_2\left(y\right)|u\left(y\right)d\mu \left(y\right)\le \underset{\begin{array}{c}R\in \mathfrak{B}\\ A\subseteq R^{\left[3\right]}\end{array}}{sup}{\displaystyle \frac{1}{u\left(R\right)}}{\int }_R|g\left(y\right)|u\left(y\right)d\mu \left(y\right).$$

Applying Hölder’s inequality yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{u\left(R\right)}}{\int }_R|g\left(y\right)|u\left(y\right)d\mu \left(y\right)& \le {\left({\displaystyle \frac{1}{u{\left(R\right)}^{\kappa }}}{\int }_R{|g\left(y\right)|}^{p}u\left(y\right)dy\right)}^{1/p}u{\left(R\right)}^{(\kappa -1)/p}\hfill \\ \hfill & \le {\mathfrak{C}\parallel g\parallel }_{M_{p,u}^{\kappa }}u{\left(R\right)}^{(\kappa -1)/p}\hfill \\ \hfill & \le {\mathfrak{C}\parallel g\parallel }_{M_{p,u}^{\kappa }}u{\left(A\right)}^{(\kappa -1)/p}.\hfill \end{array}$$

Thus, we obtain

$$II\le {\mathfrak{C}\parallel g\parallel }_{M_{p,u}^{\kappa }}u{\left(A\right)}^{\kappa /p}.$$

(II) Now, we prove the relationship between M and $M_u$. Let $B\in \mathfrak{B}$ contain x. By Hölder’s inequality,

$${\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_B|f\left(y\right)|d\mu \left(y\right)\le {\displaystyle \frac{1}{\mu \left(B\right)}}{\left({\int }_B{|f\left(y\right)|}^{p}u\left(y\right)d\mu \left(y\right)\right)}^{1/p}{\left({\int }_{B}u{\left(y\right)}^{-{\displaystyle \frac{1}{p-1}}}d\mu \left(y\right)\right)}^{1-1/p}.$$

By the definition of the weighted maximal operator,

$${\displaystyle \frac{1}{u\left(B\right)}}{\int }_B{|f\left(y\right)|}^{p}u\left(y\right)d\mu \left(y\right)\le M_u{(|f|}^p)\left(x\right);$$

hence,

$${\left({\int }_B{|f\left(y\right)|}^{p}u\left(y\right)d\mu \left(y\right)\right)}^{1/p}\le {\left(M_u{(|f|}^p)\left(x\right)\right)}^{1/p}u{\left(B\right)}^{1/p}.$$

Since $u\in A_{p,\mathfrak{B}}$, there exists a constant $\mathfrak{C}$ such that for every $B\in \mathfrak{B}$,

$$\left({\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_{B}u\left(y\right)d\mu \left(y\right)\right){\left({\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_{B}u{\left(y\right)}^{-{\displaystyle \frac{1}{p-1}}}d\mu \left(y\right)\right)}^{p-1}\le {\mathfrak{C}}^p.$$

Rearranging yields

$$u{\left(B\right)}^{1/p}{\left({\int }_{B}u{\left(y\right)}^{-{\displaystyle \frac{1}{p-1}}}d\mu \left(y\right)\right)}^{1-1/p}\le \mathfrak{C}\mu \left(B\right).$$

Combining the estimates, we obtain

$${\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_B|f\left(y\right)|d\mu \left(y\right)\le {\displaystyle \frac{1}{\mu \left(B\right)}}{\left(M_u{(|f|}^p)\left(x\right)\right)}^{1/p}u{\left(B\right)}^{1/p}\mathfrak{C}{\displaystyle \frac{\mu \left(B\right)}{u{\left(B\right)}^{1/p}}}=\mathfrak{C}{\left(M_u{(|f|}^p)\left(x\right)\right)}^{1/p}.$$

Since this holds for any $x\in B$ and $B\in \mathfrak{B}$, taking the supremum gives

$$Mf\left(x\right)=\underset{\begin{array}{c}x\in B\\ B\in \mathfrak{B}\end{array}}{sup}{\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_B|f\left(y\right)|d\mu \left(y\right)\le \mathfrak{C}{\left(M_u{(|f|}^p)\left(x\right)\right)}^{1/p}.$$

Lemma 4 implies that there exists $1<r<p$ such that $u\in A_r$. Combining (I) and (II), we have

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{1}{u{\left(A\right)}^{\kappa }}}{\int }_A{|Mg\left(x\right)|}^{p}u\left(x\right)d\mu \left(x\right)\right)}^{1/p}& \le \mathfrak{C}{\left({\displaystyle \frac{1}{u{\left(A\right)}^{\kappa }}}{\int }_A{\left[M_u{(|g|}^r)\left(x\right)\right]}^{p/r}u\left(x\right)d\mu \left(x\right)\right)}^{1/p}\hfill \\ \hfill & \le \mathfrak{C}{∥M_u{(|g|}^r)∥}_{M_{p/r,u}^{\kappa }}^{1/r}\hfill \\ \hfill & \le {\mathfrak{C}\parallel g\parallel }_{M_{p,u}^{\kappa }}.\hfill \end{array}$$

Therefore, the proof is complete. □

The following theorem identifies the ${\mathcal{H}}_{p,u}^{\kappa }$ as the predual of $M_{p^{\prime },v}^{\kappa }$.

Theorem 2. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis. Let $1<p<\infty$, $0<\kappa <1$ and $u\in A_{p,\mathfrak{B}}$, where $v=u^{-{\displaystyle \frac{1}{p-1}}}$. Then,

$$M_{p^{\prime },v}^{\kappa }={\left({\mathcal{H}}_{p,u}^{\kappa }\right)}^{*},$$

where ${\left({\mathcal{H}}_{p,u}^{\kappa }\right)}^{*}$ denotes the predual space of ${\mathcal{H}}_{p,u}^{\kappa }$.

Proof. 

(I) First, we prove that ${\left({\mathcal{H}}_{p,u}^{\kappa }\right)}^{*}\hookleftarrow M_{p^{\prime },v}^{\kappa }$. For any $B\in \mathfrak{B}$, let b be a function in ${\mathfrak{h}}_{p,u}^{\kappa }$ satisfing $suppb\subset B$. For any $f\in M_{p^{\prime },v}^{\kappa }$, by Hölder’s inequality, then

$$\begin{array}{cc}\hfill {\int }_X|f\left(x\right)b\left(x\right)|d\mu \left(x\right)& ={\int }_X\left|f\left(x\right)u{\left(x\right)}^{-1/p}\right|\left|b\left(x\right)u{\left(x\right)}^{1/p}\right|d\mu \left(x\right)\hfill \\ \hfill & \le \mathfrak{C}\parallel f{\chi }_B{\parallel }_{L_v^{p^{\prime }}}{\parallel b{\chi }_B\parallel }_{L_u^p}\hfill \\ \hfill & \le \mathfrak{C}{\displaystyle \frac{{∥f{\chi }_B∥}_{L_v^{p^{\prime }}}}{u{\left(B\right)}^{\kappa /p}}}.\hfill \end{array}$$

Therefore, for any $g={\sum }_{i=1}^{\infty }{\lambda }_i{b}_i\in {\mathcal{H}}_{p,u}^{\kappa }$. This leads to

$${\int }_X|f\left(x\right)g\left(x\right)|d\mu \left(x\right)\le C{\displaystyle \sum _{i=1}^{\infty }}|{\lambda }_i|{\int }_X|f\left(x\right)b_i{\left(x\right)|d\mu \left(x\right)\le \mathfrak{C}\parallel g\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}{\parallel f\parallel }_{M_{p^{\prime },v}^{\kappa }}.$$

Hence, $M_{p^{\prime },v}^{\kappa }\hookrightarrow {\left({\mathcal{H}}_{p,u}^{\kappa }\right)}^{*}$.

(II) Next, we prove that ${\left({\mathcal{H}}_{p,u}^{\kappa }\right)}^{*}\hookrightarrow M_{p^{\prime },v}^{\kappa }$. Let $L\in {\left({\mathcal{H}}_{p,u}^{\kappa }\right)}^{*}$. For each ball $B\in \mathfrak{B}$, consider the subspace

$$X_B=\{h\in L_u^p:supph\subset B\},$$

which is closed in $L_u^p$. For $h\in X_B$, define

$$\tilde{h}={\displaystyle \frac{h}{{\parallel h\parallel }_{L_u^p}u{\left(B\right)}^{\kappa /p}}}.$$

Then, from the definition of ${\mathfrak{h}}_{p,u}^{\kappa }$, it follows that $\tilde{h}\in {\mathfrak{h}}_{p,u}^{\kappa }$; hence, $\tilde{h}\in {\mathcal{H}}_{p,u}^{\kappa }$ and

$$|L\left(h\right)|=|L\left(\tilde{h}\right){|\parallel h\parallel }_{L_u^p}u{\left(B\right)}^{\kappa /p}\le {\parallel L\parallel \parallel h\parallel }_{L_u^p}u{\left(B\right)}^{\kappa /p}.$$

Thus, the linear functional

$$L_B:X_B\to \mathbb{C},L_B\left(h\right)=L\left(h\right)$$

satisfies $|L_B{\left(h\right)|\le \parallel L\parallel u\left(B\right)}^{\kappa /p}{\parallel h\parallel }_{L_u^p}$. According to the Hahn–Banach theorem, $L_B$ extends to a bounded linear functional on the whole space $L_u^p$ (still denoted by $L_B$) with the same norm. Since ${\left(L_u^p\right)}^{*}=L_v^{p^{\prime }}$, there exists a unique function $f_B\in L_v^{p^{\prime }}$ with $supp{f}_B\subset B$ such that

$$L_B\left(h\right)={\int }_X{f}_B\left(x\right)h\left(x\right)d\mu \left(x\right),\forall h\in L_u^p.$$

In particular, for any $h\in X_B$, we have $L\left(h\right)={\int }_X{f}_B\left(x\right)h\left(x\right)d\mu \left(x\right)$.

Now, we show consistency: if $B_1,B_2\in \mathfrak{B}$ and $B\subset B_1\cap B_2$ (with $B\in \mathfrak{B}$); then, for every $h\in X_B$,

$${\int }_X{f}_{B_1}\left(x\right)hd\mu \left(x\right)=L\left(h\right)={\int }_X{f}_{B_2}\left(x\right)h\left(x\right)d\mu \left(x\right).$$

Hence, $f_{B_1}=f_{B_2}$ almost everywhere on B. Choose an increasing sequence of balls $B_k$ such that ${\bigcup }_k{B}_k=X$. Define $f\left(x\right)=f_{B_k}\left(x\right)$ for any k with $x\in B_k$; the consistency property guarantees that f is well defined and measurable. Moreover, for any fixed $B\in \mathfrak{B}$, take a ball $B_0$ containing B; then, $f=f_{B_0}$ on B, so $f{\chi }_B=f_{B_0}{\chi }_B$ and consequently, $f{\chi }_B\in L_v^{p^{\prime }}$.

In what follows, we prove that $f\in M_{p^{\prime },v}^{\kappa }$ and that $L=L_f$. For any $B\in \mathfrak{B}$, there exists $A\in \mathfrak{B}$ with $B\subseteq A$. For any $s\in L_u^p$ and $B\in \mathfrak{B}$, let

$$H={\displaystyle \frac{s{\chi }_B}{\parallel s{\chi }_B{\parallel }_{L_u^p}u{\left(B\right)}^{\kappa /p}}}.$$

Then, H is an element of ${\mathfrak{h}}_{p,u}^{\kappa }$. Using the properties of ${\mathfrak{h}}_{p,u}^{\kappa }$, we derive ${\parallel H\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}\le 1$, i.e.,

$$\parallel s{\chi }_B{\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}\le {\parallel s{\chi }_B\parallel }_{L_u^p}u{\left(B\right)}^{\kappa /p}.$$

Since $H\in {\mathfrak{h}}_{p,u}^{\kappa }$,

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{u{\left(B\right)}^{\kappa /p}}}{\parallel f{\chi }_B\parallel }_{L_v^{p^{\prime }}}& ={\displaystyle \frac{1}{u{\left(B\right)}^{\kappa /p}}}\underset{{\parallel s\parallel }_{L_u^p}=1}{sup}\left|{\int }_{B}f\left(z\right)s\left(z\right)d\mu \left(z\right)\right|\hfill \\ & =\underset{{\parallel s\parallel }_{L_u^p}=1}{sup}\left|{\int }_A{f}_A\left(z\right){\displaystyle \frac{s\left(z\right)}{u{\left(B\right)}^{\kappa /p}}}d\mu \left(z\right)\right|\hfill \\ & \le {\mathfrak{C}\parallel L\parallel }_{{\left({\mathcal{H}}_{p,u}^{\kappa }\right)}^{*}}\underset{{\parallel s\parallel }_{L_u^p}=1}{sup}{∥{\displaystyle \frac{s{\chi }_B}{u{\left(B\right)}^{\kappa /p}}}∥}_{{\mathcal{H}}_{p,u}^{\kappa }}\le \mathfrak{C}{\parallel L\parallel }_{{\left({\mathcal{H}}_{p,u}^{\kappa }\right)}^{*}}.\hfill \end{array}$$

Taking supremum over all B gives ${\parallel f\parallel }_{M_{p^{\prime },v}^{\kappa }}\le \parallel L\parallel$. Hence, $f\in M_{p^{\prime },v}^{\kappa }$.

Finally, for any finite linear combination of blocks $g=\sum {\lambda }_i{b}_i$, we have

$$L\left(g\right)=\sum {\lambda }_{i}L\left(b_i\right)=\sum {\lambda }_i\int f{b}_{i}d\mu =\int fgd\mu =L_f\left(g\right).$$

Since such combinations are dense in ${\mathcal{H}}_{p,u}^{\kappa }$, the equality extends to all $g\in {\mathcal{H}}_{p,u}^{\kappa }$. Therefore, $L=L_f$, and the map $L\mapsto f$ establishes an isometric isomorphism between ${\left({\mathcal{H}}_{p,u}^{\kappa }\right)}^{*}$ and $M_{p^{\prime },v}^{\kappa }$.

Combining (I) and (II), this completes the proof of the theorem. □

In fact, from the above theorem, we can derive a corollary concerning abstract weighted Morrey spaces.

Corollary 1. 

Let $1<p<\infty$ and let u be a weight function. Define the associated weight $v:=u^{-{\displaystyle \frac{1}{p-1}}}$. For a measurable function h, the following statements are equivalent:

1. 

$h\in M_{p^{\prime },v}^{\kappa }$;

2. 

For every g in the weighted block space ${\mathcal{H}}_{p,u}^{\kappa }$, the integral ${\int }_{X}h\left(x\right)g\left(x\right)d\mu \left(x\right)$ is finite.

Moreover, under these equivalent conditions, the norm of h in $M_{p^{\prime },v}^{\kappa }$ admits the dual representation

$${\parallel h\parallel }_{M_{p^{\prime },v}^{\kappa }}=sup\left\{{\int }_{X}h\left(x\right)g\left(x\right)d\mu \left(x\right):g\in {\mathcal{H}}_{p,u}^{\kappa },{\parallel g\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}\le 1\right\}.$$

Before we show that the maximal operator remains bounded on ${\mathcal{H}}_{p,u}^{\kappa }$, we first introduce a definition.

Definition 9. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis, and let $1<p<\infty$, $0<\kappa <p-1$. We define ${\mathbb{W}}_{p,\kappa }$ as the collection of all functions u satisfies: there exists $\mathfrak{C}>0$ such that for every $A\in \mathfrak{B}$,

$${\displaystyle \sum _{i=0}^{\infty }}{\displaystyle \frac{\parallel {\chi }_A{\parallel }_{L_v^{p^{\prime }}}}{\parallel {\chi }_{A^{[i+1]}}{\parallel }_{L_v^{p^{\prime }}}}}u{\left(A^{[i+1]}\right)}^{\kappa /p}\le \mathfrak{C}u{\left(A\right)}^{\kappa /p}.$$

Remark 3. 

The collection ${\mathbb{W}}_{p,\kappa }$ defined in Definition 9 is non-empty and non-trivial. For instance, consider the constant weight function $u\left(x\right)\equiv 1$ on $(X,\mathfrak{M},\mu )$. Since $u\equiv 1$, the associated weight $v=u^{-1/(p-1)}$ is also identically 1. It follows that for any $A\in \mathfrak{B}$, $u\left(A\right)=\mu \left(A\right)$ and $\parallel {\chi }_A{\parallel }_{L_v^{p^{\prime }}}=\mu {\left(A\right)}^{1/p^{\prime }}$. Substituting these into the series in Definition 9, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \sum _{i=0}^{\infty }}{\displaystyle \frac{\parallel {\chi }_A{\parallel }_{L_v^{p^{\prime }}}}{\parallel {\chi }_{A^{[i+1]}}{\parallel }_{L_v^{p^{\prime }}}}}u{\left(A^{[i+1]}\right)}^{\kappa /p}=& {\displaystyle \sum _{i=0}^{\infty }}{\displaystyle \frac{\mu {\left(A\right)}^{1/p^{\prime }}}{\mu {\left(A^{[i+1]}\right)}^{1/p^{\prime }}}}\mu {\left(A^{[i+1]}\right)}^{\kappa /p}\hfill \\ \hfill =& \mu {\left(A\right)}^{\kappa /p}{\displaystyle \sum _{i=0}^{\infty }}{\left({\displaystyle \frac{\mu \left(A^{[i+1]}\right)}{\mu \left(A\right)}}\right)}^{{\displaystyle \frac{\kappa }{p}}-{\displaystyle \frac{1}{p^{\prime }}}}.\hfill \end{array}$$

Let $\varrho ={\displaystyle \frac{\kappa }{p}}-{\displaystyle \frac{1}{p^{\prime }}}={\displaystyle \frac{\kappa -(p-1)}{p}}$. According to the property derived from condition $B_4$, we have $\mu \left(A^{[i+1]}\right)\ge {\alpha }_1^{i+1}\mu \left(A\right)$ with ${\alpha }_1>1$; then,

$${\displaystyle \sum _{i=0}^{\infty }}{\left({\displaystyle \frac{\mu \left(A^{[i+1]}\right)}{\mu \left(A\right)}}\right)}^{\varrho }\le {\displaystyle \sum _{i=0}^{\infty }}{\left({\alpha }_1^{\varrho }\right)}^{i+1}={\displaystyle \frac{{\alpha }_1^{\varrho }}{1-{\alpha }_1^{\varrho }}}=\mathfrak{C}<\infty .$$

This demonstrates that $u\equiv 1\in {\mathbb{W}}_{p,\kappa }$ for a wide range of parameters, ensuring that the function spaces ${\mathbb{W}}_{p,\kappa }$ are well defined and contain non-trivial weights.

Theorem 3. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis. Suppose $1<p<\infty$, $0<\kappa <1$, and $u\in A_{p,\mathfrak{B}}$, $u\in {\mathbb{W}}_{p,\kappa }$. Then, M is bounded on ${\mathcal{H}}_{p,u}^{\kappa }$.

Proof. 

(I) Let $h\in {\mathfrak{h}}_{p,u}^{\kappa }$, with $supph\subseteq A$, where $A\in \mathfrak{B}$. For any $i\in {\mathbb{N}}^{+}$, let

$$A^{\left[0\right]}=A,d_0={\chi }_{A^{\left[1\right]}}M\left(h\right)\mathrm{and}{d}_i={\chi }_{A^{[i+1]}\setminus A^{\left[i\right]}}M\left(h\right),i\in {\mathbb{N}}^{+}.$$

Then, $supp{d}_i\subseteq A^{[i+1]}\setminus A^{\left[i\right]}$ and $M\left(h\right)={\sum }_{i=0}^{\infty }{d}_i$. For $d_0$, by the boundedness of the maximal operator, we have

$$\parallel d_0{\parallel }_{L_u^p}\le {\parallel M\left(h\right)\parallel }_{L_u^p}\le \mathfrak{C}{\parallel h\parallel }_{L_u^p}\le \mathfrak{C}{\displaystyle \frac{1}{u{\left(A\right)}^{\kappa /p}}}$$

for some $\mathfrak{C}>0$. Therefore, there exists $a_0\in {\mathfrak{h}}_{p,u}^{\kappa }$ and $\mathfrak{C}>0$ such that $d_0=\mathfrak{C}{a}_0$. Following this, by Hölder’s inequality,

$$\begin{array}{cc}\hfill \parallel d_i{\parallel }_{L_u^p}& \lesssim {∥{\chi }_{A^{[i+1]}\setminus A^{\left[i\right]}}∥}_{L_u^p}{\displaystyle \frac{1}{\mu \left(A^{\left[i\right]}\right)}}{\parallel h\parallel }_{L_u^p}{\parallel {\chi }_A\parallel }_{L_v^{p^{\prime }}}\hfill \\ \hfill & \lesssim {∥{\chi }_{A^{[i+1]}}∥}_{L_u^p}{\displaystyle \frac{1}{\mu \left(A^{\left[i\right]}\right)}}{\parallel h\parallel }_{L_u^p}{\parallel {\chi }_A\parallel }_{L_v^{p^{\prime }}}\hfill \\ \hfill & \lesssim {\displaystyle \frac{{∥{\chi }_A∥}_{L_v^{p^{\prime }}}u{\left(A^{[i+1]}\right)}^{\kappa \setminus p}}{{∥{\chi }_{A^{[i+1]}}∥}_{L_v^{p^{\prime }}}u{\left(A\right)}^{\kappa \setminus p}}}{\displaystyle \frac{1}{u{\left(A^{[i+1]}\right)}^{\kappa \setminus p}}}.\hfill \end{array}$$

We define $a_i={\displaystyle \frac{1}{{\gamma }_i}}{d}_i$. Then, there exists a constant ${\mathfrak{C}}_0>0$ such that

$${\gamma }_i={\mathfrak{C}}_0{\displaystyle \frac{{∥{\chi }_A∥}_{L_v^{p^{\prime }}}u{\left(A^{[i+1]}\right)}^{\kappa \setminus p}}{{∥{\chi }_{A^{[i+1]}}∥}_{L_v^{p^{\prime }}}u{\left(A\right)}^{\kappa \setminus p}}}.$$

Then, by the properties of ${\mathfrak{h}}_{p,u}^{\kappa }$, we know that $a_i\in {\mathfrak{h}}_{p,u}^{\kappa }$ and $supp{a}_i\subseteq A^{[i+1]}$. Since $u\in {\mathbb{W}}_{p,\kappa }$, it follows from Definition 9 that

$${\displaystyle \sum _{i=1}^{\infty }}{\gamma }_i={\mathfrak{C}}_0{\displaystyle \sum _{i=1}^{\infty }}{\displaystyle \frac{{∥{\chi }_A∥}_{L_v^{p^{\prime }}}u{\left(A^{[i+1]}\right)}^{\kappa \setminus p}}{{∥{\chi }_{A^{[i+1]}}∥}_{L_v^{p^{\prime }}}u{\left(A\right)}^{\kappa \setminus p}}}<\infty .$$

Therefore,

$$M\left(h\right)={\displaystyle \sum _{i=1}^{\infty }}{\gamma }_i{a}_i\in {\mathcal{H}}_{p,u}^{\kappa }\mathrm{and}{\parallel M\left(h\right)\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}\le \mathfrak{C}$$

for some $C>0$ independent of h.

(II) Now, we consider an arbitrary $g\in {\mathcal{H}}_{p,u}^{\kappa }$. We may write $g={\sum }_{j=1}^{\infty }{\lambda }_j{h}_j$, where $h_j\in {\mathfrak{h}}_{p,u}^{\kappa }$ and ${\sum }_{j=1}^{\infty }|{\lambda }_j{|\le 2\parallel g\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}$. Let $\left\{a_{j,i}\right\}$ and $\left\{{\gamma }_{j,i}\right\}$ be defined as in part (I), with h replaced by $h_j$. Then, $M\left(h_j\right)={\sum }_{i=1}^{\infty }{\gamma }_{j,i}{a}_{j,i}$. By the sublinearity of M, we have

$$M\left(g\right)\le {\displaystyle \sum _{j=1}^{\infty }}|{\lambda }_j|M\left(h_j\right)={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \sum _{i=1}^{\infty }}|{\lambda }_j|{\gamma }_{j,i}{a}_{j,i}.$$

Since

$${\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \sum _{i=1}^{\infty }}|{\lambda }_j|{\gamma }_{j,i}\le \mathfrak{C}{\displaystyle \sum _{j=1}^{\infty }}|{\lambda }_j{|\le \mathfrak{C}\parallel g\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }},$$

it follows that ${\sum }_{j=1}^{\infty }{\sum }_{i=1}^{\infty }|{\lambda }_j|{\gamma }_{j,i}{a}_{j,i}\in {\mathcal{H}}_{p,u}^{\kappa }$. Let

$$G=M\left(g\right)/\left({\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \sum _{i=1}^{\infty }}|{\lambda }_j|{\gamma }_{j,i}{a}_{j,i}\right).$$

Clearly,

$$M\left(g\right)={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \sum _{i=1}^{\infty }}|{\lambda }_j|{\gamma }_{j,i}{g}_{j,i},$$

where $g_{j,i}=G{a}_{j,i}$. Since $|G|\le 1$, we have $g_{j,i}\in {\mathfrak{h}}_{p,u}^{\kappa }$ and $supp{g}_{j,i}\subseteq supp{a}_{j,i}$. Therefore,

$${\parallel M\left(g\right)\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}\le {\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \sum _{i=1}^{\infty }}|{\lambda }_j|{\gamma }_{j,i}\le \mathfrak{C}{\parallel g\parallel }_{{\mathcal{H}}_{p,u}^{\kappa }}.$$

Combining (I) and (II), we verify the boundedness of M on ${\mathcal{H}}_{p,u}^{\kappa }$. □

4. Extrapolation on Abstract Weighted Morrey Spaces

In this section, we extend the extrapolation theory to weighted abstract Morrey spaces. The boundedness of the maximal operator M on the ${\mathcal{H}}_{p,u}^{\kappa }$ is crucial; it guarantees that the series defining $\mathfrak{R}$ converges. The class ${\mathbb{W}}_{p,\kappa }$ consists of those functions u for which M is bounded on ${\mathcal{H}}_{p,u}^{\kappa }$; this property ensures the iteration is well defined and the resulting operator $\mathfrak{R}$ has the desired $A_{1,\mathfrak{B}}$ property.

Definition 10. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis. Suppose that $0<p_0<\infty$, $1<p<\infty$, $0<\kappa <1$, and $u^{p_0}\in A_{p/p_0,\mathfrak{B}}\cap {\mathbb{W}}_{p/p_0,\kappa p_0}$. Define $v^{\prime }=u^{-1/({\displaystyle \frac{p}{p_0}}-1)}$. For a locally integrable function h on X, set

$$\mathfrak{R}h={\displaystyle \sum _{t=0}^{\infty }}{\displaystyle \frac{M^t\left(h\right)}{2^t{\parallel M\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}\to {\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}^t}},$$

where $M^0$ is the identity operator, $M^t$ denotes the t-th iteration of M, and ${\parallel M\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}\to {\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}^t$ represents the operator norm of M on ${\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}$.

Proposition 4. 

Under the assumptions of Definition 10, the operator $\mathfrak{R}$ is well defined on ${\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}$ and satisfies

1. 

$|h(x)|\le \mathfrak{R}h\left(x\right)$ for almost every x;

2. 

${\parallel \mathfrak{R}h\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}\le 2{\parallel h\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}$;

3. 

$\mathfrak{R}h$ is an $A_{1,\mathfrak{B}}$ weight with ${\left[\mathfrak{R}h\right]}_{A_{1,\mathfrak{B}}}\le 2{\parallel M\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}\to {\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}$, i.e.,

$$M\left(\mathfrak{R}h\right)\left(x\right)\le {2\parallel M\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}\mathfrak{R}h\left(x\right)a.e.$$

Proof. 

Because $u^{p_0}\in {\mathbb{W}}_{p/p_0,\kappa p_0}$, the maximal operator M is bounded on ${\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}$ with norm ${\parallel M\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}\to {\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}$. Hence, for any h in this block space, the series defining $\mathfrak{R}h$ converges in norm and pointwise almost everywhere; moreover,

$${\parallel \mathfrak{R}h\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}\le {\displaystyle \sum _{t=0}^{\infty }}{\displaystyle \frac{{\parallel M\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}\to {\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}^t{\parallel h\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}}{2^t{\parallel M\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}\to {\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}^t}}=2{\parallel h\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}.$$

Clearly, $|h|\le \mathfrak{R}h$, because the term $t=0$ equals h. Moreover, since M is a sublinear operator, for any $h\in {\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}$, then

$$\begin{array}{cc}\hfill M\left(\mathfrak{R}h\right)& \le {\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{M^{k+1}h}{2^k{\parallel M\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}^k}}\hfill \\ \hfill & \le {2\parallel M\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{M^{k}h}{2^k{\parallel M_S\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}^k}}\hfill \\ \hfill & \le 2\parallel M_S{\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}\mathfrak{R}h.\hfill \end{array}$$

Therefore, $\mathfrak{R}h\in A_{1,\mathfrak{B}}$ and the last inequality holds. □

Following the approach established in [25], we now prove an extrapolation theorem for abstract weighted Morrey spaces on abstract measure spaces. The idea is as follows: assume that for every $A_{1,\mathfrak{B}}$ weight $\psi$ of the form $\mathfrak{R}h$, we have a weighted $L^{p_0}$ estimate for a family $\mathcal{F}$ of function pairs. Then, by duality between the weighted Morrey spaces $M_{p,u}^{\kappa }$ and the block space ${\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}$, we can “lift” the $L^{p_0}$ estimate to the full Morrey norm.

Theorem 4. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis. $0<p_0<p<\infty$, $0<\kappa <1$, and $u\in A_{p/p_0,\mathfrak{B}}$, $u^{p_0}\in {\mathbb{W}}_{p/p_0,\kappa p_0}$. Given an extrapolation family $\mathcal{F}$ such that for every $\psi \in \{\mathfrak{R}h:h\in {\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}\}$, and ${\parallel h\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}\le 1$; then,

$${\int }_X{|f\left(x\right)|}^{p_0}\psi \left(x\right)d\mu \left(x\right)\le \mathfrak{C}{\int }_X{|g\left(x\right)|}^{p_0}\psi \left(x\right)d\mu \left(x\right)<\infty ,(f,g)\in \mathcal{F},$$

where $\mathfrak{C}$ is a constant independent of f and g. Then, for every $(f,g)\in \mathcal{F}$ with $f,g\in M_{p,u}^{\kappa }$; then,

$${\parallel f\parallel }_{M_{p,u}^{\kappa }}\le \mathfrak{C}{\parallel g\parallel }_{M_{p,u}^{\kappa }}.$$

Proof. 

Let $(f,g)\in \mathcal{F}$ satisfy that f is measurable and $g\in M_{p,u}^{\kappa }$. For any $h\in {\mathcal{H}}_{{(p/p_0)}^{\prime },v}^{\kappa ,p_0}$ satisfying ${\parallel h\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}\le 1$, define $\psi =\mathfrak{R}h$. Then, by the mapping properties of $\mathfrak{R}$, there exists a constant $\mathfrak{C}>0$ such that

$$\begin{array}{cc}\hfill {\int }_X{|f\left(x\right)|}^{p_0}|h\left(x\right)|d\mu \left(x\right)& \le {\int }_X{|f\left(x\right)|}^{p_0}\mathfrak{R}h\left(x\right)d\mu \left(x\right)\le \mathfrak{C}{\int }_X{|g\left(x\right)|}^{p_0}\mathfrak{R}h\left(x\right)d\mu \left(x\right)\hfill \\ \hfill & \le {\mathfrak{C}\parallel |g|}^{p_0}{\parallel }_{M_{(p/p_0),u}^{\kappa p_0}}{\parallel \mathfrak{R}h\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}}\le {\mathfrak{C}\parallel g\parallel }_{M_{p,u}^{\kappa }}^{p_0}{\parallel h\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}.\hfill \end{array}$$

Furthermore, since

$$sup\left\{{\int }_X{|f\left(x\right)|}^{p_0}|h\left(x\right)|d\mu \left(x\right):h\in {\mathfrak{h}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}\right\}\le \mathfrak{C}{\parallel g\parallel }_{M_{p,u}^{\kappa }}^{p_0},$$

the predual property yields, for some $\mathfrak{C}>0$,

$$\begin{array}{cc}\hfill {\parallel f\parallel }_{M_{p,u}^{\kappa }}^{p_0}& ={\parallel |f|}^{p_0}{\parallel }_{M_{(p/p_0),u}^{\kappa p_0}}\hfill \\ \hfill & =sup\left\{{\int }_X{|f\left(x\right)|}^{p_0}|h\left(x\right)|d\mu \left(x\right):h\in {\mathfrak{h}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}\right\}\hfill \\ \hfill & \le {\mathfrak{C}\parallel g\parallel }_{M_{p,u}^{\kappa }}^{p_0}.\hfill \end{array}$$

This completes the proof of the theorem. □

5. Applications

Definition 11 

([33]). Let $L^0(X,\mathfrak{M},\mu )$ be the set of measurable functions. Suppose $C(X,\mathfrak{M},\mu )$ is a suitable linear subspace of $L^0(X,\mathfrak{M},\mu )$. $T_M:C(X,\mathfrak{M},\mu )\to L^0(X,\mathfrak{M},\mu )$ is an operator. $T_M$ is called a sublinear operator if it satisfies the following properties for all $h,g\in C(X,\mathfrak{M},\mu )$ and all scalars $\lambda \in \mathbb{R}$:

$$|T_M\left(\lambda h\right)|=|\lambda ||T_M\left(h\right)|,$$

$$|T_M(h+g)|\le |T_M\left(h\right)|+|T_M\left(g\right)|.$$

A sublinear operator $T_M$ is said to be linearizable if there exist a Banach space $\mathcal{N}$ and a linear operator $\mathcal{T}$ taking values in $\mathcal{N}$ such that $|T_M{\left(f\right)\left(x\right)|=\parallel \mathcal{T}\left(f\right)\left(x\right)\parallel }_{\mathcal{N}}$.

Definition 12 

([30]). Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis. Given Banach spaces $\mathcal{N}$ and $r\in [1,\infty )$, suppose that there exist linear operators $\mathcal{T}$ taking values in $\mathcal{N}$ such that for each $x\in X$, $|T_M{\left(f\right)\left(x\right)|=\parallel \mathcal{T}\left(f\right)\left(x\right)\parallel }_{\mathcal{N}}$, or in the case $\mathcal{N}=\mathbb{R}$, $\mathcal{T}:=\left\{T_M\right\}$ is a real-valued linear or sublinear operator, satisfying for all $f\in M_{p,u}^{\kappa }$:

(BI) For any $C\in \mathfrak{B}$ with $C^{\mathfrak{C}}\subset X$, there exists $D\in \mathfrak{B}$ with $D\supseteq C$ such that

$$\underset{x\in C}{sup}{∥\mathcal{T}\left(f{\mathcal{X}}_{D^{\mathfrak{C}}}\right)\left(x\right)-\mathcal{T}\left(f{\mathcal{X}}_{C^{\mathfrak{C}}}\right)\left(x\right)∥}_{\mathcal{N}}\lesssim {⟨f⟩}_{D^{\mathfrak{C}},r}.$$

(BII) For any $D\in \mathfrak{B}$.

$$\underset{x,x^{\prime }\in D}{sup}{∥(\mathcal{T}\left(f\right)-\mathcal{T}\left(f{\mathcal{X}}_{D^{\mathfrak{C}}}\right))\left(x\right)-(\mathcal{T}\left(f\right)-\mathcal{T}\left(f{\mathcal{X}}_{D^{\mathfrak{C}}}\right))\left(x^{\prime }\right)∥}_{\mathcal{N}}\lesssim {⟨f⟩}_{D,r}^{\mathfrak{C}}.$$

Then, the operator $T_M$ is called an $\mathcal{N}$-valued $\mathfrak{B}O$ operator. When $\mathcal{N}=\mathbb{R}$, that is, $T_M$ is a real-valued operator, we omit $\mathcal{N}$-valued and the ${\parallel ·\parallel }_{\mathcal{N}}$. Here,

$${⟨f⟩}_{D,r}={\left({\displaystyle \frac{1}{\mu \left(D\right)}}{\int }_D{|f\left(x\right)|}^{r}d\mu \left(x\right)\right)}^{{\displaystyle \frac{1}{r}}},{⟨f⟩}_{D,r}^{\mathfrak{C}}=\underset{C\in \mathfrak{B},C\supset D}{sup}{⟨f⟩}_{C,r}.$$

In the case $r=1$, the subscripts r are omitted, i.e., ${⟨f⟩}_{D,1}={⟨f⟩}_D$ and ${⟨f⟩}_{D,1}^{\mathfrak{C}}={⟨f⟩}_D^{\mathfrak{C}}$.

In what follows, we prove that the maximal operator is a $\mathfrak{B}O$ operator.

Theorem 5. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis. Then, the M is a $\mathfrak{B}O$ operator.

Proof. 

To prove that M is a $\mathfrak{B}O$ operator, it suffices to verify conditions (BI) and (BII) for the case $r=1$.

(I) First, we show that condition (BI) holds. For fix ball $B\in \mathfrak{B}$, consider two points $x,x^{\prime }\in B$ and a non-zero function $f\in M_{p,u}^{\kappa }$. Then, by definition of the maximal operator, we have

$$Mf\left(x\right)\le {⟨f⟩}_A+{⟨f⟩}_B^{\mathfrak{C}}$$

for some ball $A\in \mathfrak{B}$ containing x.

If $\mu \left(A\right)\le \mu \left(B\right)$, then by the relation between two balls, we have $A\subset B^{\mathfrak{C}}$, and hence,

$$M\left(f{\chi }_{B^{\mathfrak{C}}}\right)\ge {⟨f⟩}_A.$$

Combining the two inequalities above gives

$$|Mf\left(x\right)-M\left(f{\chi }_{B^{\mathfrak{C}}}\right)|=Mf\left(x\right)-M\left(f{\chi }_{B^{\mathfrak{C}}}\right)\le {⟨f⟩}_B^{\mathfrak{C}}.$$

If $\mu \left(A\right)>\mu \left(B\right)$, then we have $B\subset A^{\mathfrak{C}}$ and ${⟨f⟩}_A\lesssim {⟨f⟩}_B^{\mathfrak{C}}$, so that

$$Mf\left(x\right)\le {⟨f⟩}_A+{⟨f⟩}_B^{\mathfrak{C}}\lesssim {⟨f⟩}_B^{\mathfrak{C}}.$$

In either case, for all $x\in B$,

$$|Mf\left(x\right)-M\left(f{\chi }_{B^{\mathfrak{C}}}\right)|\lesssim {⟨f⟩}_B^{\mathfrak{C}}.$$

Consequently,

$$|(Mf\left(x\right)-M\left(f{\chi }_{B^{\mathfrak{C}}}\right)\left(x\right))-(Mf\left(x^{\prime }\right)-M\left(f{\chi }_{B^{\mathfrak{C}}}\right)\left(x^{\prime }\right))|\lesssim {⟨f⟩}_B^{\mathfrak{C}}.$$

Thus, condition (BI) is satisfied.

(II) In the following, we verify that condition (BII) holds. Fix a ball $B\in \mathfrak{B}$. Define

$$\mathcal{G}=\{A\in \mathfrak{B}:A\cap B\ne \varnothing ,\mu \left(A\right)>\mu \left(B\right)\}and\beta =\underset{A\in \mathcal{G}}{inf}\mu \left(A\right).$$

Take a ball $A\in \mathcal{G}$ such that $\beta \le \mu \left(A\right)<2\beta$ and set $B_t=A^{\mathfrak{C}}$. By condition $B_4$, we have

$$B⊊A^{\mathfrak{C}}=B_t.$$

Given any $x\in B$, any $f\in M_{p,u}^{\kappa }$, and any $\epsilon >0$, there exists a ball $D\in \mathfrak{B}$ containing x such that

$$M\left(f{\chi }_{B_t^{\mathfrak{C}}\setminus B^{\mathfrak{C}}}\right)\left(x\right)=⟨|f|{\chi }_{B_t^{\mathfrak{C}}\setminus B^{\mathfrak{C}}}{⟩}_D+\epsilon .$$

If $\mu \left(D\right)<\mu \left(B\right)$, then $D\subset B^{\mathfrak{C}}$ and consequently,

$$M\left(f{\chi }_{B_t^{\mathfrak{C}}\setminus B^{\mathfrak{C}}}\right)\left(x\right)\le \epsilon \le {⟨f⟩}_{B_t^{\mathfrak{C}}}.$$

If $\mu \left(D\right)>\mu \left(B\right)$, the fact that $D\cap B\ne \varnothing$ implies $D\in \mathcal{G}$; hence, $\mu \left(D\right)\ge \beta$. Using $\mu \left(A\right)<2\beta$ and condition $B_4$, we obtain

$$2{\alpha }_2\mu \left(D\right)>\mu \left(A\right)\ge \mu \left(A^{\left[2\right]}\right)=\mu \left(B^{\mathfrak{C}}\right),$$

which yields

$$M\left(f{\chi }_{B_t^{\mathfrak{C}}\setminus B^{\mathfrak{C}}}\right)\left(x\right)\le ⟨|f|{\chi }_{B_t^{\mathfrak{C}}\setminus B^{\mathfrak{C}}}{⟩}_D+\epsilon \lesssim {⟨f⟩}_{B_t^{\mathfrak{C}}}.$$

Since the choice of B was arbitrary, condition (BII) follows. □

Lemma 5 

([33]). Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis. Assume $r\in [1,\infty )$. If an $\mathcal{N}$-valued linear bounded oscillation operator $T_M$ with respect to $\mathfrak{B}$ satisfies the estimate $\parallel T_M{\parallel }_{L^r\to L^{r,\infty }}<\infty$, then for every $p>r$ and every $u\in A_{p/r,\mathfrak{B}}$,

$${\int }_X|T_M{f\left(x\right)|}^{p}u\left(x\right)d\mu \left(x\right)\le \mathfrak{C}{\int }_X{|f\left(x\right)|}^{p}u\left(x\right)d\mu \left(x\right).$$

Using the Lemma 5 and Theorem 4, we obtain the following boundedness result for $\mathcal{N}$-valued bounded oscillation operators.

Theorem 6. 

Let $(X,\mathfrak{M},\mu )$ be a measure space and $\mathfrak{B}$ be a ball basis, $1<p_0<p<\infty$, $0<\kappa <1$ and $u^{p_0}\in A_{p/p_0,\mathfrak{B}}$, $u^{p_0}\in {\mathbb{W}}_{p/p_0,\kappa p_0}$. Suppose for any

$$\psi \in \{\mathfrak{R}h:h\in {\mathcal{H}}_{{(p/p_0)}^{\prime },v}^{\kappa p_0},\parallel h{\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}\le 1\},$$

$T_M$ is an $\mathcal{N}$-valued linear $\mathfrak{B}O$ operator that is bounded from $L^1(X,\mathfrak{M},\mu )$ to $L^{1,\infty }(X,\mathfrak{M},\mu )$; then, $T_M$ is bounded on $M_{p,u}^{\kappa }$. Specifically, there exists a constant $\mathfrak{C}$ such that for all $f\in M_{p,u}^{\kappa }$,

$$\parallel T_M{\left(f\right)\parallel }_{M_{p,u}^{\kappa }}\le \mathfrak{C}{\parallel f\parallel }_{M_{p,u}^{\kappa }}.$$

Proof. 

For any $g\in M_{p,u}^{\kappa }$ and $h\in {\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}$ satisfying ${\parallel h\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}\le 1$, by the properties of $\mathfrak{R}$ and Corollary 1, we have

$$\begin{array}{cc}\hfill {\int }_{X}g{\left(z\right)}^p\psi \left(z\right)d\mu \left(z\right)& \le \mathfrak{C}\parallel g^p{\parallel }_{M_{(p/p_0),u}^{\kappa p}}{\parallel \mathfrak{R}h\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}\hfill \\ \hfill & \le \mathfrak{C}\parallel g^p{\parallel }_{M_{p,u}^{\kappa }}^{p_0}{\parallel h\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}.\hfill \end{array}$$

Consequently, we obtain the embedding $M_{p,u}^{\kappa }\hookrightarrow L_u^p$. Let $\mathcal{F}=\left\{\right(|T_{M}g|,|g|):g\in M_{p,u}^{\kappa }\}$. For any

$$\psi \in \{\mathfrak{R}h:h\in {\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0},\parallel h{\parallel }_{{\mathcal{H}}_{{(p/p_0)}^{\prime },v^{\prime }}^{\kappa p_0}}\le 1\},$$

because $T_M$ is bounded from $L^1(X,\mathfrak{M},\mu )$ to $L^{1,\infty }(X,\mathfrak{M},\mu )$, by Lemma 5, we obtain

$${\int }_X|T_M{g\left(z\right)|}^{p_0}\psi \left(z\right)d\mu \left(z\right)\le \mathfrak{C}{\int }_X{|g\left(z\right)|}^{p_0}\psi \left(z\right)d\mu \left(z\right).$$

Combining this with Theorem 4 yields $\parallel T_M{\left(g\right)\parallel }_{M_{p,u}^{\kappa }}\le \mathfrak{C}{\parallel g\parallel }_{M_{p,u}^{\kappa }}.$ □

Below, we provide examples of some classical operators commonly encountered in analysis, all of which are important instances of bounded oscillation operators.

5.1. Calderón–Zygmund Operators

In this subsection, we discuss the fact that Calderón–Zygmund operators are bounded oscillation operators. Since this result was established in the context of spaces of homogeneous type, we first recall their definition.

Definition 13 

([26]). A function ρ on a set X is called quasimetric if $\rho :X\times X\to [0,\infty )$ satisfies

• Positive definiteness: $\rho (a,b)\ge 0$, and $\rho (a,b)=0$ if and only if $a=b$;
• Symmetry: $\rho (a,b)=\rho (b,a)$;
• Quasi-triangle inequality: Exists a constant $\varrho >1$ such that

$$\rho (a,b)\le \varrho \left(\rho \right(a,c)+\rho (c,b\left)\right),\forall a,b,c\in X.$$

Let $(X,\mathfrak{M},\mu )$ be a measure space equipped with metric ρ. For any $a\in X$ and any radius $r\in (0,\infty )$, define a ρ-ball as

$$D(a,r)=\{b\in X:\rho (a,b)<r\},$$

where a and r are called the center and radius of the ball, respectively. Denote $D(a,r)=D\left(c\right(D),r(D\left)\right)$. The collection of all such balls is denoted by

$$\mathcal{P}\left(\rho \right)=\left\{D\right(c\left(D\right),r\left(D\right)):c(D)\in X,0<r(D)<\infty \}.$$

For any $t>0$, we define the scaled ball $mD:=D\left(c\right(D),mr(D\left)\right)$.

From the ball family $\mathcal{P}\left(\rho \right)$, we define the extended ball family $\mathcal{Q}\left(\rho \right)$ as follows:

• If $\mu \left(X\right)=\infty$, then $\mathcal{Q}\left(\rho \right)=\mathcal{P}\left(\rho \right)$;
• If $\mu \left(X\right)<\infty$, then $\mathcal{Q}\left(\rho \right)=\mathcal{P}\left(\rho \right)\cup \left\{X\right\}$.

A measure space $(X,\mathfrak{M},\mu )$ satisfies the doubling condition if for every $D\in \mathcal{P}\left(\rho \right)$, there exists a constant $\beta >0$ such that

$$\mu \left(2D\right)\le \beta \mu \left(D\right),\forall D\in \mathcal{P}\left(\rho \right).$$

Remark 4. 

For any $\gamma >0$, we have

$$\mu \left(\gamma D\right)\le \beta \left(\gamma \right)\mu \left(D\right).$$

From $\mu \left(2D\right)\le \beta \mu \left(D\right)$ for all $D\in \mathcal{P}\left(\rho \right)$, it follows that

$$\mu \left(\gamma D\right)=\mu \left(2^{\lceil {log}_2\gamma \rceil }D\right)\le {\beta }^{\lceil {log}_2\gamma \rceil }\mu \left(D\right),$$

where $\lceil x\rceil$ denotes the ceiling function, and $\beta \left(\gamma \right)$ can be written as ${\beta }^{\lceil {log}_2\gamma \rceil }$.

Definition 14 

([26]). Let $(X,\mathfrak{M},\mu )$ be a measure space and ρ be a quasimetric on X. Then, $(X,\rho ,\mathfrak{M},\mu )$ is a space of homogeneous type, if μ is a doubling measure with a respect to ρ.

For the properties of the ball families $\mathcal{P}\left(\rho \right)$ and $\mathcal{Q}\left(\rho \right)$ defined above, one may refer to [26].

Lemma 6. 

Let $(X,\rho ,\mathfrak{M},\mu )$ be a space of homogeneous type. Assume $\mathcal{P}\left(\rho \right)$ satisfies the density condition. Then, $\mathcal{Q}\left(\rho \right)$ is a ball basis with the following doubling property: for each $D=D(x_0,r)\in \mathcal{P}\left(\rho \right)$, one can find $D^{\left[1\right]}=D(x_0,R)$ satisfying $2r\le R\le \infty$.

Definition 15 

([26]). Let $(X,\rho ,\mathfrak{M},\mu )$ be a space of homogeneous type with ball basis $\mathcal{Q}\left(\rho \right)$. An operator T is called a Calderón–Zygmund operator if

(1) There exists a kernel K such that for any ball $D\in \mathcal{Q}\left(\rho \right)$, whenever $suppf\subset X\setminus 2D$, we have

$$Tf\left(x\right)={\int }_{X}K(x,y)f\left(y\right)d\mu \left(y\right).$$

(2) T is a bounded linear operator from $L^1\left(X\right)$ to $L^{1,\infty }\left(X\right)$.

(3) The kernel satisfies the following estimates (for any $t>2r\left(D\right)$):

$$\underset{x\in D,y\in X\setminus D\left(c\right(D),t)}{sup}|K(x,y)|\le {\displaystyle \frac{C_k}{\mu \left(D\right(c\left(D\right),t\left)\right)}},$$

$$\underset{x,z\in D,y\in X\setminus D\left(c\right(D),t)}{sup}|K(x,y)-K(z,y)|\le {\displaystyle \frac{C_k}{\mu \left(D\right(c\left(D\right),t\left)\right)}},$$

$$\underset{y,m\in D,x\in X\setminus D\left(c\right(D),t)}{sup}|K(x,y)-K(x,m)|\le {\displaystyle \frac{C_k}{\mu \left(D\right(c\left(D\right),t\left)\right)}}.$$

It is known that the above Calderón–Zygmund operators are bounded on weighted Morrey spaces over spaces of homogeneous type [35,36]. By showing that these Calderón–Zygmund operators coincide with $\mathfrak{B}O$ operators under an abstract measure, we obtain the weighted boundedness of Calderón–Zygmund operators on Morrey spaces based on spaces of homogeneous type.

Lemma 7 

([26]). Let $(X,\rho ,\mathfrak{M},\mu )$ be a space of homogeneous type with ball basis $\mathcal{Q}\left(\rho \right)$. If $1<p<\infty$, $0<\kappa <1$, and $u\in A_{p,\mathfrak{B}}$, then T is a $\mathfrak{B}O$ operator.

Theorem 7. 

Let $(X,\rho ,\mathfrak{M},\mu )$ be a space of homogeneous type with ball basis $\mathcal{Q}\left(\rho \right)$, $1<p<\infty$, $0<\kappa <1$, and $u\in A_{p,\mathfrak{B}}$. If $f\in M_{p,u}^{\kappa }$, then T is bounded on the homogeneous weighted Morrey spaces, i.e., there exists a constant $\mathfrak{C}>0$ such that

$${\parallel Tf\parallel }_{M_{p,u}^{\kappa }}\le \mathfrak{C}{\parallel f\parallel }_{M_{p,u}^{\kappa }}.$$

Proof. 

By Theorem 6 and Lemma 7, the conclusion follows at once. □

5.2. Littlewood–Paley Square Operators

Let the parameter $\lambda >2$. For any $x\in {\mathbb{R}}^n$, define the cone

$$N\left(x\right):=\{(y,t)\in {\mathbb{R}}_{+}^{n+1}:|y-x|<\theta t\},$$

where $\theta \in (0,\infty )$ is a fixed constant, and ${\mathbb{R}}_{+}^{n+1}={\mathbb{R}}^n\times (0,+\infty )$.

Definition 16 

([30]). Define the Littlewood–Paley square operators:

$$\begin{array}{cc}\hfill & S^1\left(f\right)\left(x\right):={\left({\int }_0^{\infty }{\left|{\int }_{{\mathbb{R}}^n}{K}_t(x,y)g\left(y\right)dy\right|}^2{\displaystyle \frac{dt}{t}}\right)}^{1/2},\hfill \\ \hfill & S^2\left(f\right)\left(x\right):={\left(\underset{N\left(x\right)}{\int \int }{\left|{\int }_{{\mathbb{R}}^n}{K}_t(x,y)g\left(y\right)dy\right|}^2{\displaystyle \frac{dxdt}{t^{n+1}}}\right)}^{1/2},\hfill \\ \hfill & S^3\left(f\right)\left(x\right):={\left(\underset{{\mathbb{R}}_{+}^{n+1}}{\int \int }{\left({\displaystyle \frac{t}{t+|x-y|}}\right)}^{\lambda n}{\left|{\int }_{{\mathbb{R}}^n}{K}_t(z,y)g\left(z\right)dz\right|}^2{\displaystyle \frac{dydt}{t^{n+1}}}\right)}^{1/2},\lambda >2,\hfill \end{array}$$

where $K_t:{\mathbb{R}}^n\setminus \{x=y\}\to \mathbb{R}.$

Lemma 8 

([30]). If $1<p<\infty$, $0<\kappa <1$, $u\in A_{p,\mathfrak{B}}$, then for any $k\in \{1,2,3\}$, when $S^k$ is a Littlewood–Paley square operator, $S^k$ is a ${\mathbb{B}}_k$-valued (as defined in (p. 3659, [30])) bounded oscillation operator.

Theorem 8. 

If $1<p<\infty$, $0<\kappa <1$, $u\in A_{p,\mathfrak{B}}$, and $f\in M_{p,u}^{\kappa }$, then for any $k\in \{1,2,3\}$, the Littlewood–Paley square operator $S^k$ is bounded on the homogeneous $M_{p,u}^{\kappa }$, i.e., there exists constant $\mathfrak{C}>0$ such that

$$\parallel S^k{f\parallel }_{M_{p,u}^{\kappa }}\le \mathfrak{C}{\parallel f\parallel }_{M_{p,u}^{\kappa }}.$$

Proof. 

The conclusion follows by applying Theorem 6 and Lemma 8. □

5.3. Carleson Operators

Definition 17 

([30]). Consider ${\left\{T_{\beta }\right\}}_{\beta \in \mathfrak{Y}}$ as a family of $\mathcal{N}$-valued linear $\mathfrak{B}O$ operators on a measure space $(X,\mathfrak{M},\mu )$. Define the Carleson operator by

$$T^{\mathfrak{Y}}f\left(x\right)=\underset{\beta \in \mathfrak{Y}}{sup}|T_{\beta }f\left(x\right)|.$$

Lemma 9 

([26]). Let $(X,\mathfrak{M},\mu )$ be a measure space with ball basis $\mathfrak{B}$. If $1<p<\infty$, $0<\kappa <1$, and $u\in A_{p,\mathfrak{B}}$, and if the set of operators ${\left\{T_{\beta }\right\}}_{\beta \in \mathfrak{Y}}$ satisfies

$$\parallel T_{\beta }{\parallel }_{L^r\left(X\right)\to L^{r,\infty }\left(X\right)}\le \mathfrak{C},$$

then $T^{\mathfrak{Y}}$ is a $\mathfrak{B}O$ operator.

Theorem 9. 

Let $(X,\mathfrak{M},\mu )$ be a measure space with ball basis $\mathfrak{B}$, $1<p<\infty$, $0<\kappa <1$, and $u\in A_{p,\mathfrak{B}}$. Then, the $T^{\mathfrak{Y}}$ is bounded on the abstract weighted Morrey spaces:

$$\parallel T^{\mathfrak{Y}}{f\parallel }_{M_{p,u}^{\kappa }}\le \mathfrak{C}{\parallel f\parallel }_{M_{p,u}^{\kappa }}.$$

Proof. 

The statement follows at once from Theorem 6 together with Lemma 9. □