Boundedness of the maximal operator in the local Morrey-Lorentz spaces

Abstract

In this paper we define a new class of functions called local Morrey-Lorentz spaces $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$, $0<p,q\le \mathrm{\infty }$ and $0\le \lambda \le 1$. These spaces generalize Lorentz spaces such that $M_{p,q;0}^{\mathrm{loc}}({\mathbb{R}}^n)=L_{p,q}({\mathbb{R}}^n)$. We show that in the case $\lambda <0$ or $\lambda >1$, the space $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ is trivial, and in the limiting case $\lambda =1$, the space $M_{p,q;1}^{\mathrm{loc}}({\mathbb{R}}^n)$ is the classical Lorentz space ${\mathrm{\Lambda }}_{\mathrm{\infty },t^{\frac{1}{p}-\frac{1}{q}}}({\mathbb{R}}^n)$. We show that for $0<q\le p<\mathrm{\infty }$ and $0<\lambda \le \frac{q}{p}$, the local Morrey-Lorentz spaces $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ are equal to weak Lebesgue spaces $W{L}_{\frac{1}{p}-\frac{\lambda }{q}}({\mathbb{R}}^n)$. We get an embedding between local Morrey-Lorentz spaces and Lorentz-Morrey spaces. Furthermore, we obtain the boundedness of the maximal operator in the local Morrey-Lorentz spaces.

MSC:42B20, 42B25, 42B35, 47G10.

1 Introduction

The aim of this paper is to define a new class of functions called local Morrey-Lorentz spaces $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ and to study the boundedness of the maximal operator in these spaces. Local Morrey-Lorentz spaces $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ are generalizations of Lorentz spaces $L_{p,q}({\mathbb{R}}^n)$. Lorentz spaces, introduced by Lorentz in the 1950s [1, 2], are generalizations of the more familiar $L_p$ spaces. Lorentz spaces, which are Banach spaces, appear to be useful in the general interpolation theory by Calderón (see [3]). Peetre [4] identified Lorentz spaces as intermediate spaces for interpolation theory by Lions and Peetre (see [5]). Riviere and Sagher [6] generalized the results of Calderón contained in [3] to include Lorentz spaces having coefficients p and q greater than zero; similarly, Kree and Peetre generalized the results of Lions and Peetre obtained in [5].

For $x\in {\mathbb{R}}^n$ and $t>0$, let $B(x,t)$ denote the open ball centered at x of radius t, and let $|B(x,t)|$ be the Lebesgue measure of the ball $B(x,t)$. Note that $|B(x,t)|={\omega }_n{t}^n$, where ${\omega }_n$ denotes the volume of the unit ball in ${\mathbb{R}}^n$. Let f be a locally integrable function on ${\mathbb{R}}^n$. The Hardy-Littlewood maximal function Mf of f is defined by

$$Mf(x)=\underset{t>0}{sup}\frac{1}{|B(x,t)|}{\int }_{B(x,t)}|f(y)|dy,x\in {\mathbb{R}}^n.$$

Maximal operators play an important role in the differentiability properties of functions, singular integrals and partial differential equations. They often provide a deeper and more simplified approach to understanding problems in these areas than other methods. It is well known that for the classical Hardy-Littlewood maximal operator the rearrangement inequality

$$c{f}^{\ast \ast }(t)\le {(Mf)}^{\ast }(t)\le C{f}^{\ast \ast }(t),t\in (0,\mathrm{\infty }),$$

holds, ([7], Chapter 3, Theorem 3.8), where $f^{\ast }(t)$ is the non-increasing rearrangement of f and

$$f^{\ast \ast }(t)=\frac{1}{t}{\int }_0^t{f}^{\ast }(t)dt.$$

Mingione [8] defined the Lorentz-Morrey spaces ${\mathcal{L}}_{p,q;\lambda }({\mathbb{R}}^n)$ as follows.

Definition 1 [8]

The Lorentz-Morrey spaces ${\mathcal{L}}_{p,q;\lambda }({\mathbb{R}}^n)$ is the set of all measurable functions f on ${\mathbb{R}}^n$: for $1\le p<\mathrm{\infty }$, $0<q<\mathrm{\infty }$ and $0\le \lambda \le n$, $f\in {\mathcal{L}}_{p,q;\lambda }({\mathbb{R}}^n)$ iff

$${\parallel f\parallel }_{{\mathcal{L}}_{p,q;\lambda }}=\underset{x\in {\mathbb{R}}^n,t>0}{sup}{t}^{-\frac{\lambda }{p}}{\parallel {\chi }_{B(x,t)}f\parallel }_{L_{p,q}}<\mathrm{\infty }.$$

Here ${\parallel \cdot \parallel }_{L_{p,q}}$ denotes the Lorentz norm of a function (see preliminaries).

In [8], Section 4.1, Mingione studied the boundedness of the restricted fractional maximal operator $M_{\beta ,B}$

$$M_{\beta ,B}f(x)=\underset{B(x,t)\subset B}{sup}{|B(x,t)|}^{\frac{\beta }{n}-1}{\int }_{B(x,t)}|f(y)|dy,x\in {\mathbb{R}}^n,$$

in the restricted Lorentz-Morrey spaces ${\mathcal{L}}_{p,q;\lambda }(B)$, where B is any ball. Mingione derived a general non-linear version, extending a priori estimates and regularity results for possibly degenerate non-linear elliptic problems to the various spaces of Lorentz and Lorentz-Morrey type considered in [9–11].

Ragusa [12] defined the Lorentz-Morrey spaces $L_{p,q;\lambda }({\mathbb{R}}^n)$ and studied some embeddings between these spaces.

Definition 2 [12]

The Lorentz-Morrey spaces $L_{p,q;\lambda }({\mathbb{R}}^n)$ is the set of all measurable functions f on ${\mathbb{R}}^n$: for $1\le p<\mathrm{\infty }$, $0<q<\mathrm{\infty }$ and $0\le \lambda \le n$, iff

$${\parallel f\parallel }_{L_{p,q;\lambda }}=\underset{x\in {\mathbb{R}}^n,t>0}{sup}{t}^{-\frac{\lambda }{q}}{\parallel {\chi }_{B(x,t)}f\parallel }_{L_{p,q}}<\mathrm{\infty }.$$

Accordingly, f belongs to

$$L_{p,\mathrm{\infty };\lambda }({\mathbb{R}}^n)\equiv W{L}_{p,\lambda }({\mathbb{R}}^n)\text{iff}{\parallel f\parallel }_{L_{p,\mathrm{\infty };\lambda }}={\parallel f\parallel }_{W{L}_{p,\lambda }}<\mathrm{\infty }.$$

Note that the spaces ${\mathcal{L}}_{p,q;\lambda }({\mathbb{R}}^n)$ and $L_{p,q;\lambda \frac{q}{p}}({\mathbb{R}}^n)$ defined by Mingione and Ragusa respectively coincide, thus

$${\mathcal{L}}_{p,q;\lambda }\left({\mathbb{R}}^n\right)=L_{p,q;\lambda \frac{q}{p}}\left({\mathbb{R}}^n\right).$$

Recall that the local Morrey-type spaces $L{M}_{p\theta ,w}$ were introduced and the boundedness in these spaces of the fractional integral operators and singular integral operators defined on homogeneous Lie groups were proved by Guliyev [13] in the doctoral thesis (see, also [14–16]). They are given by

$${\parallel f\parallel }_{L{M}_{p\theta ,w}}={\parallel w(r){\parallel f\parallel }_{L_p(B(0,r))}\parallel }_{L_{\theta }(0,\mathrm{\infty })},$$

where w is a positive measurable function defined on $(0,\mathrm{\infty })$. Some necessary and sufficient conditions for the boundedness of the maximal, fractional maximal, Riesz potential and singular integral operators in the local Morrey-type space $L{M}_{p\theta ,w}$ were given in [17–21]. We should explain that the spaces $L{M}_{p\theta ,w}$ are closely related to the $B_{\sigma }$ spaces (see [22, 23]).

This paper is organized as follows. In Section 2 we give some notations and definitions of the Morrey, Lorentz and classical Lorentz spaces. In Section 3 we define a new class of functions called local Morrey-Lorentz spaces $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$, $0<p,q\le \mathrm{\infty }$ and $0\le \lambda \le 1$. These spaces generalize Lorentz spaces such that $M_{p,q;0}^{\mathrm{loc}}({\mathbb{R}}^n)=L_{p,q}({\mathbb{R}}^n)$. We show that in the case $\lambda <0$ or $\lambda >1$, the space $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ is trivial, and in the limiting case $\lambda =1$, the space $M_{p,q;1}^{\mathrm{loc}}({\mathbb{R}}^n)$ is the classical Lorentz space ${\mathrm{\Lambda }}_{\mathrm{\infty },t^{\frac{1}{p}-\frac{1}{q}}}({\mathbb{R}}^n)$. We also show that for $0<q\le p<\mathrm{\infty }$ and $0<\lambda \le \frac{q}{p}$, the local Morrey-Lorentz spaces $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ are equal to weak Lebesgue spaces $W{L}_{\frac{1}{p}-\frac{\lambda }{q}}({\mathbb{R}}^n)$. In Section 4 we prove the boundedness of the maximal operator in $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$.

Throughout the paper, we write $A\lesssim B$ if there exists a positive constant C, independent of appropriate quantities such as functions, satisfying $A\le CB$. If $p\in [1,\mathrm{\infty }]$, the conjugate number $p^{\prime }$ is defined by $p^{\prime }=\frac{p}{p-1}$ and if $p\in (0,1)$, the conjugate number $p^{\prime }$ is defined by $p^{\prime }=\frac{p}{1-p}$.

2 Preliminaries

Let E be a measurable subset of ${\mathbb{R}}^n$ and $|E|={\int }_{E}dx$. We denote by $L_p(E)$ the class of all measurable functions f defined on E for which

$$\begin{array}{c}{\parallel f\parallel }_{L_p(E)}:={\left({\int }_E{|f(y)|}^{p}dy\right)}^{\frac{1}{p}}<\mathrm{\infty },0<p<\mathrm{\infty },\hfill \\ {\parallel f\parallel }_{L_{\mathrm{\infty }}(E)}:=sup\{\alpha :\left|\{y\in E:|f(y)|\ge \alpha \}\right|>0\}.\hfill \end{array}$$

We define rearrangement of f in decreasing order by

$$f^{\ast }(t)=inf\{\lambda >0:{\mu }_f(\lambda )\le t\},\mathrm{\forall }t\in (0,\mathrm{\infty }),$$

where ${\mu }_f(\lambda )$ denotes the distribution function of f given by

$${\mu }_f(\lambda )=\left|\{y\in {\mathbb{R}}^n:|f(y)|>\lambda \}\right|.$$

We denote by $W{L}_p({\mathbb{R}}^n)$ the weak $L_p$ space of all measurable functions f with the quasi-norm

$${\parallel f\parallel }_{W{L}_p}=\underset{t>0}{sup}{t}^{1/p}{f}^{\ast }(t),1\le p<\mathrm{\infty }.$$

Now we recall the definitions of Morrey spaces, Lorentz spaces and classical Lorentz spaces.

Morrey spaces were introduced by Morrey [24] in 1938 in connection with certain problems in elliptic partial differential equations and calculus of variations. Later, Morrey spaces found important applications to Navier-Stokes and Schrödinger equations, elliptic problems with discontinuous coefficients, and potential theory.

Definition 3 [24, 25]

We denote by $L_{p,\lambda }({\mathbb{R}}^n)$ the Morrey space for $0\le \lambda \le n$, $1\le p<\mathrm{\infty }$, $f\in L_{p,\lambda }({\mathbb{R}}^n)$ if $f\in L_p^{\mathrm{loc}}({\mathbb{R}}^n)$ and

$${\parallel f\parallel }_{L_{p,\lambda }}=\underset{x\in {\mathbb{R}}^n,t>0}{sup}{t}^{-\frac{\lambda }{p}}{\parallel f\parallel }_{L_{p(B(x,t))}}<\mathrm{\infty }.$$

If $\lambda =0$, then $L_{p,0}({\mathbb{R}}^n)=L_p({\mathbb{R}}^n)$; if $\lambda =n$, then $L_{p,n}({\mathbb{R}}^n)=L_{\mathrm{\infty }}({\mathbb{R}}^n)$; if $\lambda <0$ or $\lambda >n$, then $L_{p,\lambda }({\mathbb{R}}^n)=\mathrm{\Theta }$, where Θ is the set of all functions equivalent to 0 on ${\mathbb{R}}^n$.

Also, by $W{L}_{p,\lambda }({\mathbb{R}}^n)$ we denote the weak Morrey space of all functions $f\in W{L}_p^{\mathrm{loc}}({\mathbb{R}}^n)$ for which

$${\parallel f\parallel }_{W{L}_{p,\lambda }}=\underset{x\in {\mathbb{R}}^n,t>0}{sup}{t}^{-\frac{\lambda }{p}}{\parallel f\parallel }_{W{L}_{p(B(x,t))}}<\mathrm{\infty }.$$

Lorentz spaces were introduced by Lorentz in 1950. Lorentz spaces, which are Banach spaces and generalizations of the more familiar $L_p$ spaces, appear to be useful in the general interpolation theory.

Definition 4 The Lorentz space $L_{p,q}({\mathbb{R}}^n)$, $0<p,q\le \mathrm{\infty }$, is the collection of all measurable functions f on ${\mathbb{R}}^n$ such the quantity

$${\parallel f\parallel }_{L_{p,q}}={\parallel t^{\frac{1}{p}-\frac{1}{q}}{f}^{\ast }(t)\parallel }_{L_q(0,\mathrm{\infty })}$$

(1)

is finite.

Note that $L_{p,\mathrm{\infty }}({\mathbb{R}}^n)=W{L}_p({\mathbb{R}}^n)$ (see, for example, [26]).

If $p=q=\mathrm{\infty }$, then the space $L_{\mathrm{\infty },\mathrm{\infty }}$ is denoted by $L_{\mathrm{\infty }}$.

If $1\le q\le p$ or $p=q=\mathrm{\infty }$, then the functional ${\parallel f\parallel }_{p,q}$ is a norm.

For $0<q\le p\le r\le \mathrm{\infty }$, we have, with continuous embeddings, that

$$L_{p,q}\subset L_p\subset L_{p,r}\subset W{L}_p.$$

The function $f^{\ast \ast }:(0,\mathrm{\infty })\to [0,\mathrm{\infty }]$ is defined as

$$f^{\ast \ast }(t)=\frac{1}{t}{\int }_0^t{f}^{\ast }(s)ds.$$

In the case $0<p,q\le \mathrm{\infty }$, we give a functional ${\parallel \cdot \parallel }_{Lp,q}^{\ast }$ by

$${\parallel f\parallel }_{L_{p,q}}^{\ast }:={\parallel f\parallel }_{L_{p,q}(0,\mathrm{\infty })}^{\ast }={\parallel t^{\frac{1}{p}-\frac{1}{q}}{f}^{\ast \ast }(t)\parallel }_{L_q(0,\mathrm{\infty })}$$

(with the usual modification if $0<p\le \mathrm{\infty }$, $q=\mathrm{\infty }$), which is a norm on $L_{p,q}({\mathbb{R}}^n)$ for $1<p<\mathrm{\infty }$, $1\le q\le \mathrm{\infty }$ or $p=q=\mathrm{\infty }$.

If $1<p\le \mathrm{\infty }$, $1\le q\le \mathrm{\infty }$, then for $1\le r<p$ (see [27]),

$${\parallel f\parallel }_{L_{p,q}}\le {\parallel f\parallel }_{L_{p,q}}^{\ast }\le {\left(\frac{p}{p-r}\right)}^{1/r}{\parallel f\parallel }_{L_{p,q}}.$$

(2)

We denote by $\mathfrak{M}({\mathbb{R}}^n)$ the set of all extended real-valued measurable functions on ${\mathbb{R}}^n$ and by ${\mathfrak{M}}^{+}(0,\mathrm{\infty })$ the set of all non-negative measurable functions on $(0,\mathrm{\infty })$.

Definition 5 Let $0<p\le \mathrm{\infty }$ and $\psi \in {\mathfrak{M}}^{+}(0,\mathrm{\infty })$. We denote by ${\mathrm{\Lambda }}_{p,\psi }({\mathbb{R}}^n)$ the classical Lorentz spaces, the spaces of all measurable functions with a finite quasi-norm

$${\mathrm{\Lambda }}_{p,\psi }\left({\mathbb{R}}^n\right):=\{f\in \mathfrak{M}\left({\mathbb{R}}^n\right):{\parallel f\parallel }_{{\mathrm{\Lambda }}_{p,\psi }}:={\parallel \psi f^{\ast }\parallel }_{L_p(0,\mathrm{\infty })}\}.$$

Therefore, for $\psi (t)=t^{\frac{1}{p}-\frac{1}{q}}$, $0<p,q\le \mathrm{\infty }$, we get ${\mathrm{\Lambda }}_{p,t^{\frac{1}{p}-\frac{1}{q}}}({\mathbb{R}}^n)=L_{p,q}({\mathbb{R}}^n)$ with equality of ‘norms’.

Useful references for Lorentz spaces are, for instance, in [7, 28, 29].

Remark 1 Since ${\parallel f\parallel }_{L_{p,p}}={\parallel f\parallel }_{L_p}$, it can be easily shown that $L_{p,p;\lambda }\equiv L_{p,\lambda }$.

3 Local Morrey-Lorentz spaces $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$

In this section we define the local Morrey-Lorentz spaces $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$, $0<p,q\le \mathrm{\infty }$ and $0\le \lambda \le 1$. These spaces generalize Lorentz spaces so that $M_{p,q;0}^{\mathrm{loc}}({\mathbb{R}}^n)=L_{p,q}({\mathbb{R}}^n)$. We show that in the case $\lambda <0$ or $\lambda >1$, the space $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ is trivial, and in the limiting case $\lambda =1$, the space $M_{p,q;1}^{\mathrm{loc}}({\mathbb{R}}^n)$ is the classical Lorentz space ${\mathrm{\Lambda }}_{\mathrm{\infty },t^{\frac{1}{p}-\frac{1}{q}}}({\mathbb{R}}^n)$. We also show that for $0<q\le p<\mathrm{\infty }$ and $0<\lambda \le \frac{q}{p}$, the local Morrey-Lorentz spaces $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ are equal to weak Lebesgue spaces $W{L}_{\frac{1}{p}-\frac{\lambda }{q}}({\mathbb{R}}^n)$.

Definition 6 Let $0<p,q\le \mathrm{\infty }$ and let $0\le \lambda \le 1$. We denote by $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ the local Morrey-Lorentz spaces, the spaces of all measurable functions with a finite quasi-norm

$${\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}:=\underset{t>0}{sup}{t}^{-\frac{\lambda }{q}}{\parallel s^{\frac{1}{p}-\frac{1}{q}}{f}^{\ast }(s)\parallel }_{L_q(0,t)}.$$

If $\lambda <0$ or $\lambda >1$, then $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)=\mathrm{\Theta }$, where Θ is the set of all functions equivalent to 0 on ${\mathbb{R}}^n$. Also, $M_{p,q;0}^{\mathrm{loc}}({\mathbb{R}}^n)=L_{p,q}({\mathbb{R}}^n)$. In the case $q=p$, we denote the space $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ by $M_{p,\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$.

Lemma 1 Let $0<p,q\le \mathrm{\infty }$. Then

$$M_{p,q;1}^{\mathrm{loc}}\left({\mathbb{R}}^n\right)={\mathrm{\Lambda }}_{\mathrm{\infty },t^{\frac{1}{p}-\frac{1}{q}}}\left({\mathbb{R}}^n\right)$$

and

$${\parallel f\parallel }_{M_{p,q;1}^{\mathrm{loc}}}={\parallel f\parallel }_{{\mathrm{\Lambda }}_{\mathrm{\infty },t^{\frac{1}{p}-\frac{1}{q}}}}\equiv {\parallel {\tau }^{\frac{1}{p}-\frac{1}{q}}{f}^{\ast }(\tau )\parallel }_{L_{\mathrm{\infty }}(0,\mathrm{\infty })}.$$

Proof Let $f\in {\mathrm{\Lambda }}_{\mathrm{\infty },t^{\frac{1}{p}-\frac{1}{q}}}({\mathbb{R}}^n)$. Then

$${\left(t^{-1}{\int }_0^t{\tau }^{\frac{q}{p}-1}{f}^{\ast }{(\tau )}^{q}d\tau \right)}^{1/q}\le {\parallel f\parallel }_{{\mathrm{\Lambda }}_{\mathrm{\infty },t^{\frac{1}{p}-\frac{1}{q}}}}.$$

Therefore $f\in M_{p,q;1}^{\mathrm{loc}}({\mathbb{R}}^n)$ and

$${\parallel f\parallel }_{M_{p,q;1}^{\mathrm{loc}}}\le {\parallel f\parallel }_{{\mathrm{\Lambda }}_{\mathrm{\infty },t^{\frac{1}{p}-\frac{1}{q}}}}.$$

Let $f\in M_{p,q;1}^{\mathrm{loc}}({\mathbb{R}}^n)$. By the Lebesgue theorem, we have

$$\underset{s\to 0}{lim}\frac{1}{2s}{\int }_{t-s}^{t+s}{\tau }^{\frac{q}{p}-1}{f}^{\ast }{(\tau )}^{q}d\tau =t^{\frac{q}{p}-1}{f}^{\ast }{(t)}^q.$$

Then

$$\begin{array}{rl}{t}^{\frac{1}{p}-\frac{1}{q}}{f}^{\ast }(t)& ={\left(\underset{s\to 0}{lim}\frac{1}{2s}{\int }_{t-s}^{t+s}{\tau }^{\frac{q}{p}-1}{f}^{\ast }{(\tau )}^{q}d\tau \right)}^{1/q}\\ \le 2^{-\frac{1}{q}}\underset{0<s<t}{sup}{s}^{-\frac{1}{q}}{\parallel {\tau }^{\frac{1}{p}-\frac{1}{q}}{f}^{\ast }(\tau )\parallel }_{L_q(t-s,t+s)}\\ ={\parallel {\tau }^{\frac{1}{p}-\frac{1}{q}}{f}^{\ast }(\tau )\parallel }_{L_{\mathrm{\infty }}(0,\mathrm{\infty })}\\ ={\parallel f\parallel }_{M_{p,q;1}^{\mathrm{loc}}}.\end{array}$$

Therefore $f\in {\mathrm{\Lambda }}_{\mathrm{\infty },t^{\frac{1}{p}-\frac{1}{q}}}({\mathbb{R}}^n)$ and

$${\parallel f\parallel }_{{\mathrm{\Lambda }}_{\mathrm{\infty },t^{\frac{1}{p}-\frac{1}{q}}}}\le {\parallel f\parallel }_{M_{p,q;1}^{\mathrm{loc}}}.$$

 □

Corollary 1 Let $0<p\le \mathrm{\infty }$. Then

$$M_{p,1}^{\mathrm{loc}}\left({\mathbb{R}}^n\right)=L_{\mathrm{\infty }}\left({\mathbb{R}}^n\right).$$

Lemma 2 ${\parallel \cdot \parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}$ is a quasi-norm on $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$.

Proof From the definition $f^{\ast }\ge 0$. This implies that ${\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}\ge 0$. Moreover, ${\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}=0$ implies that ${\parallel s^{\frac{1}{p}-\frac{1}{q}}{f}^{\ast }(s)\parallel }_{L_q(0,t)}=0$ for all $t>0$. Hence, we get $f^{\ast }=0$ a.e. Thus, $f=0$ since f is representative of an equivalence class.

Now, let $a\ne 0$ be a real constant, $f\in M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$. Noting

$${(af)}^{\ast }(t)=|a|f^{\ast }(t),t>0,$$

the homogeneity condition follows

$${\parallel af\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}=|a|{\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}.$$

Let $f,g\in M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ and $0<p,q<\mathrm{\infty }$. Since ${(f+g)}^{\ast }(t)\le f^{\ast }(\frac{t}{2})+g^{\ast }(\frac{t}{2})$, for any $t>0$, we have

$$\begin{array}{rl}{\int }_0^t{({(f+g)}^{\ast }(s)s^{1/p})}^q\frac{ds}{s}& \le max\{1,2^{q-1}\}({\int }_0^t{\left(f^{\ast }\left(\frac{s}{2}\right)s^{1/p}\right)}^q\frac{ds}{s}+{\int }_0^t{\left(g^{\ast }\left(\frac{s}{2}\right)s^{1/p}\right)}^q\frac{ds}{s})\\ \le 2^{\frac{q}{p}}max\{1,2^{q-1}\}({\int }_0^{\frac{t}{2}}{(f^{\ast }(s)s^{1/p})}^q\frac{ds}{s}+{\int }_0^{\frac{t}{2}}{(g^{\ast }(s)s^{1/p})}^q\frac{ds}{s}).\end{array}$$

Therefore we get

$${\parallel f+g\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}\le 2^{\frac{1}{p}}max\{1,2^{1-\frac{1}{q}}\}({\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}+{\parallel g\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}).$$

Let $f,g\in M_{p,\mathrm{\infty };\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$, $0<p\le \mathrm{\infty }$ and $q=\mathrm{\infty }$. Then

$$\begin{array}{rl}\underset{t>0}{sup}{t}^{1/p}{(f+g)}^{\ast }(t)& \le \underset{t>0}{sup}{t}^{1/p}(f^{\ast }\left(\frac{t}{2}\right)+g^{\ast }\left(\frac{t}{2}\right))\\ =\underset{u>0}{sup}{(2u)}^{1/p}(f^{\ast }(u)+g^{\ast }(u))\\ \le 2^{1/p}(\underset{u>0}{sup}{u}^{1/p}{f}^{\ast }(u)+\underset{u>0}{sup}{u}^{1/p}{g}^{\ast }(u))\\ =2^{1/p}({\parallel f\parallel }_{L_{p,\mathrm{\infty }}}+{\parallel g\parallel }_{L_{p,\mathrm{\infty }}}).\end{array}$$

Therefore we get

$${\parallel f+g\parallel }_{M_{p,\mathrm{\infty };\lambda }^{\mathrm{loc}}}\le 2^{\frac{1}{p}}({\parallel f\parallel }_{M_{p,\mathrm{\infty };\lambda }^{\mathrm{loc}}}+{\parallel g\parallel }_{M_{p,\mathrm{\infty };\lambda }^{\mathrm{loc}}}).$$

 □

The following theorem states that $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ local Morrey-Lorentz spaces are equivalent to $W{L}_q({\mathbb{R}}^n)$ weak Lebesgue spaces in the case $\frac{1}{q}=\frac{1}{p}-\frac{\lambda }{q}$ and $0<\lambda \le \frac{q}{p}$.

Theorem 1 Let $0<q\le p<\mathrm{\infty }$, $\frac{1}{r}=\frac{1}{p}-\frac{\lambda }{q}$ and $0<\lambda \le \frac{q}{p}$. Then

$$M_{p,q;\lambda }^{\mathrm{loc}}\left({\mathbb{R}}^n\right)\equiv W{L}_r\left({\mathbb{R}}^n\right)$$

and

$${\left(\frac{q}{p}\right)}^{-\frac{1}{q}}{\parallel f\parallel }_{W{L}_r}\le {\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}\le {\lambda }^{-\frac{1}{q}}{\parallel f\parallel }_{W{L}_r},f\in W{L}_r\left({\mathbb{R}}^n\right).$$

Proof Suppose $f\in M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$. Then, from the monotonicity of $f^{\ast }$ on $(0,\mathrm{\infty })$ for all $t>0$, we get

$$\begin{array}{rl}{t}^{\frac{1}{r}}{f}^{\ast }(t)& =t^{-\frac{\lambda }{q}+\frac{1}{p}}{f}^{\ast }(t)\\ ={\left(\frac{q}{p}\right)}^{\frac{1}{q}}{\left(t^{-\lambda }{f}^{\ast }{(t)}^q{\int }_0^t{s}^{\frac{q}{p}-1}ds\right)}^{\frac{1}{q}}\\ \le {\left(\frac{q}{p}\right)}^{\frac{1}{q}}{\left(t^{-\lambda }{\int }_0^t{(f^{\ast }(s)s^{\frac{1}{p}})}^q\frac{ds}{s}\right)}^{\frac{1}{q}}\le {\left(\frac{q}{p}\right)}^{\frac{1}{q}}{\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}.\end{array}$$

Thus, $f\in W{L}_r({\mathbb{R}}^n)$ and

$${\parallel f\parallel }_{W{L}_r}:=\underset{t>0}{sup}{t}^{\frac{1}{r}}{f}^{\ast }(t)\le {\left(\frac{q}{p}\right)}^{\frac{1}{q}}{\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}.$$

Let $f\in W{L}_r({\mathbb{R}}^n)$, where $\frac{1}{r}=\frac{1}{p}-\frac{\lambda }{q}$. Then, for all $t>0$,

$$t^{\frac{1}{r}}{f}^{\ast }(t)=t^{-\frac{\lambda }{q}+\frac{1}{p}}{f}^{\ast }(t)\le {\parallel f\parallel }_{W{L}_r}.$$

This implies that

$$f^{\ast }(t)\le {\parallel f\parallel }_{W{L}_r}{t}^{\frac{\lambda }{q}-\frac{1}{p}},t>0.$$

Therefore we get

$${(f^{\ast }(t)t^{\frac{1}{p}})}^q\le {\parallel f\parallel }_{W{L}_r}{t}^{\lambda },t>0.$$

Consequently,

$$\begin{array}{rl}{\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}& =\underset{t>0}{sup}{\left(t^{-\lambda }{\int }_0^t{(f^{\ast }(s)s^{\frac{1}{p}})}^q\frac{ds}{s}\right)}^{\frac{1}{q}}\\ \le {\parallel f\parallel }_{W{L}_r}\underset{t>0}{sup}{\left(t^{-\lambda }{\int }_0^t{s}^{\lambda -1}ds\right)}^{\frac{1}{q}}={\lambda }^{-\frac{1}{q}}{\parallel f\parallel }_{W{L}_r}.\end{array}$$

 □

Corollary 2 Let $0<q\le p<\mathrm{\infty }$. Then

$$M_{p,q;\frac{q}{p}}^{\mathrm{loc}}\left({\mathbb{R}}^n\right)\equiv L_{\mathrm{\infty }}\left({\mathbb{R}}^n\right).$$

The following embedding is valid.

Lemma 3 Let $0<p,q<\mathrm{\infty }$ and $0\le \lambda \le 1$. Then

$$M_{p,q;\lambda }^{\mathrm{loc}}\left({\mathbb{R}}^n\right)\hookrightarrow L_{p,q;n\lambda }\left({\mathbb{R}}^n\right)$$

and for $f\in M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$,

$${\parallel f\parallel }_{L_{p,q;n\lambda }}\lesssim {\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}.$$

Proof For $x\in {\mathbb{R}}^n$ and $t>0$, from the monotonicity of $f^{\ast }$ on $(0,\mathrm{\infty })$, we get

$$\begin{array}{rl}{\parallel f{\chi }_{B(x,t)}\parallel }_{L_{p,q}}& ={\left({\int }_0^{\mathrm{\infty }}{s}^{\frac{q}{p}-1}{({(f{\chi }_{B(x,t)})}^{\ast }(s))}^{q}ds\right)}^{1/q}\\ \le {\left({\int }_0^{\mathrm{\infty }}{s}^{\frac{q}{p}-1}{({(f{\chi }_{B(x,t)})}^{\ast \ast }(s))}^{q}ds\right)}^{1/q}\\ ={\left({\int }_0^{\mathrm{\infty }}{s}^{\frac{q}{p}-1}{(\frac{1}{s}{\int }_0^s{(f{\chi }_{B(x,t)})}^{\ast }(\tau )d\tau )}^{q}ds\right)}^{1/q}.\end{array}$$

From the inequality ${(fg)}^{\ast }(s)\le f^{\ast }(s)g^{\ast }(s)$ and the equality ${({\chi }_B)}^{\ast }(s)={\chi }_{[0,|B|)}(s)$, we get

$${(f{\chi }_{B(x,t)})}^{\ast }(s)\le f^{\ast }(s){\chi }_{[0,{\omega }_n{t}^n)}(s).$$

For $x\in {\mathbb{R}}^n$ and $t>0$, we have

$$\begin{array}{r}{\parallel f{\chi }_{B(x,t)}\parallel }_{L_{p,q}}\\ \le {\left({\int }_0^{\mathrm{\infty }}{\tau }^{\frac{q}{p}-1}{(\frac{1}{\tau }{\int }_0^{\tau }{f}^{\ast }(s){\chi }_{[0,{\omega }_n{t}^n)}(s)ds)}^{q}d\tau \right)}^{1/q}\\ \lesssim {\left({\int }_0^{{\omega }_n{t}^n}{\tau }^{\frac{q}{p}-1}{(\frac{1}{\tau }{\int }_0^{\tau }{f}^{\ast }(s)ds)}^{q}d\tau \right)}^{1/q}+{\int }_0^{{\omega }_n{t}^n}{f}^{\ast }(s)ds{\left({\int }_{{\omega }_n{t}^n}^{\mathrm{\infty }}{\tau }^{\frac{q}{p}-1-q}d\tau \right)}^{1/q}\\ ={\left({\int }_0^{{\omega }_n{t}^n}{\tau }^{\frac{q}{p}-1}{(\frac{1}{\tau }{\int }_0^{\tau }{f}^{\ast }(s)ds)}^{q}d\tau \right)}^{1/q}+{\left(\frac{p^{\prime }}{q}\right)}^{\frac{1}{q}}{\omega }_n^{-\frac{1}{p^{\prime }}}{t}^{-\frac{n}{p^{\prime }}}{({\int }_0^{{\omega }_n{t}^n}{f}^{\ast }(s)ds)}^{1/q}.\end{array}$$

From the Hölder inequality we obtain

$$\begin{array}{rl}{\int }_0^{{\omega }_n{t}^n}{f}^{\ast }(s)ds& \le {\left({\int }_0^{{\omega }_n{t}^n}{f}^{\ast }{(s)}^q{s}^{\frac{q}{p}-1}ds\right)}^{1/q}{\left({\int }_0^{{\omega }_n{t}^n}{s}^{(\frac{1}{q}-\frac{1}{p})q^{\prime }}ds\right)}^{1/q^{\prime }}\\ ={((\frac{1}{q}-\frac{1}{p})q^{\prime }+1)}^{-\frac{1}{q^{\prime }}}{\left({\omega }_n{t}^n\right)}^{\frac{1}{p^{\prime }}}{\left({\int }_0^{{\omega }_n{t}^n}{f}^{\ast }{(s)}^q{s}^{\frac{q}{p}-1}ds\right)}^{1/q}.\end{array}$$

Then

$${\left({\int }_0^{\mathrm{\infty }}{s}^{\frac{q}{p}-1}{(\frac{1}{s}{\int }_0^{min\{s,{\omega }_n{t}^n\}}{f}^{\ast }(\tau )d\tau )}^{q}ds\right)}^{1/q}\lesssim {\left({\int }_0^{{\omega }_n{t}^n}{s}^{\frac{q}{p}-1}{f}^{\ast }{(s)}^{q}ds\right)}^{1/q}.$$

Therefore

$${\parallel f{\chi }_{B(x,t)}\parallel }_{L_{p,q}}\lesssim {\parallel s^{\frac{1}{p}-\frac{1}{q}}{f}^{\ast }(s)\parallel }_{L_q(0,{\omega }_n{t}^n)}.$$

Consequently,

$$\begin{array}{rl}{\parallel f\parallel }_{L_{p,q;n\lambda }}& =\underset{x\in {\mathbb{R}}^n}{sup}\underset{t>0}{sup}{t}^{-\frac{n\lambda }{q}}{\parallel f{\chi }_{B(x,t)}\parallel }_{L_{p,q}}\\ \lesssim \underset{t>0}{sup}{t}^{-\frac{n\lambda }{q}}{\parallel s^{\frac{1}{p}-\frac{1}{q}}{f}^{\ast }(s)\parallel }_{L_q(0,{\omega }_n{t}^n)}\\ \approx \underset{t>0}{sup}{t}^{-\frac{\lambda }{q}}{\parallel s^{\frac{1}{p}-\frac{1}{q}}{f}^{\ast }(s)\parallel }_{L_q(0,t)}\\ ={\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}.\end{array}$$

 □

4 Boundedness of the maximal operator in the local Morrey-Lorentz spaces

In this section, the boundedness of the maximal operator M in local Morrey-Lorentz spaces $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$ is proved.

Theorem 2 Let $0<p<\mathrm{\infty }$, $1\le q\le \mathrm{\infty }$, $0<\lambda \le 1$ or $1<p<\mathrm{\infty }$, $1\le q\le \mathrm{\infty }$, $\lambda =0$. Then the maximal operator M is bounded on the local Morrey-Lorentz spaces $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$.

Proof By the definition of a norm in local Morrey-Lorentz spaces,

$$\begin{array}{rl}{\parallel Mf\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}& =\underset{r>0}{sup}{r}^{-\frac{\lambda }{q}}{\parallel t^{\frac{1}{p}-\frac{1}{q}}{(Mf)}^{\ast }(t)\parallel }_{L_q(0,r)}\\ \le C\underset{r>0}{sup}{r}^{-\frac{\lambda }{q}}{\parallel t^{\frac{1}{p}-\frac{1}{q}}{f}^{\ast \ast }(t)\parallel }_{L_q(0,r)}\\ =C\underset{r>0}{sup}{r}^{-\frac{\lambda }{q}}{\parallel t^{\frac{1}{p}-\frac{1}{q}-1}{\int }_0^t{f}^{\ast }(s)ds\parallel }_{L_q(0,r)}.\end{array}$$

Having applied the generalized Minkowski inequality, we get

$$\begin{array}{rl}{\parallel Mf\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}& \le C\underset{r>0}{sup}{r}^{-\frac{\lambda }{q}}{\parallel t^{\frac{1}{p}-\frac{1}{q}-1}{\int }_0^t{f}^{\ast }(s)ds\parallel }_{L_q(0,r)}\\ =C\underset{r>0}{sup}{r}^{-\frac{\lambda }{q}}{\left({\int }_0^r{({\int }_0^t{f}^{\ast }(s)ds)}^q{t}^{\frac{q}{p}-1-q}dt\right)}^{\frac{1}{q}}\\ =C\underset{r>0}{sup}{r}^{-\frac{\lambda }{q}}{\left({\int }_0^r{({\int }_0^1{f}^{\ast }(ts)ds)}^q{t}^{\frac{q}{p}-1}dt\right)}^{\frac{1}{q}}\\ \le C\underset{r>0}{sup}{r}^{-\frac{\lambda }{q}}{\int }_0^1{\left({\int }_0^r{f}^{\ast }{(ts)}^q{t}^{\frac{q}{p}-1}dt\right)}^{\frac{1}{q}}ds\\ \le C{\int }_0^1{\left(\underset{r>0}{sup}{r}^{-\lambda }{\int }_0^r{f}^{\ast }{(ts)}^q{t}^{\frac{q}{p}-1}dt\right)}^{\frac{1}{q}}ds\\ =C{\int }_0^1{\left(\underset{r>0}{sup}{r}^{-\lambda }{\int }_0^{sr}{f}^{\ast }{(t)}^q{t}^{\frac{q}{p}-1}dt\right)}^{\frac{1}{q}}{s}^{-\frac{1}{p}}ds\\ =C{\int }_0^1{\left(\underset{r>0}{sup}{r}^{-\lambda }{\int }_0^r{f}^{\ast }{(t)}^q{t}^{\frac{q}{p}-1}dt\right)}^{\frac{1}{q}}{s}^{-\frac{1}{p}+\frac{\lambda }{q}}ds\\ ={\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}C{\int }_0^1{s}^{-\frac{1}{p}+\frac{\lambda }{q}}ds\\ =C{(\frac{1}{p^{\prime }}+\frac{\lambda }{q})}^{-1}{\parallel f\parallel }_{M_{p,q;\lambda }^{\mathrm{loc}}}\end{array}$$

since $-\frac{1}{p}+\frac{\lambda }{q}>-1$.

Therefore, the maximal operator M is bounded on $M_{p,q;\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$. □

Corollary 3 Let $1\le p<\mathrm{\infty }$, $0<\lambda \le 1$. Then the maximal operator M is bounded in $M_{p,\lambda }^{\mathrm{loc}}({\mathbb{R}}^n)$.

Corollary 4 Let $1<p<\mathrm{\infty }$, $1\le q\le \mathrm{\infty }$. Then the maximal operator M is bounded in $L_{p,q}({\mathbb{R}}^n)$.

Corollary 5 [29]

Let $1<q<\mathrm{\infty }$. Then the maximal operator M is bounded in $W{L}_q({\mathbb{R}}^n)$.