Sambourou Massinanke, Sékou Coulibaly, Mamadou Traore
Departement d’Etudes et de Recherches de Mathématiques et d’Informatique, Université des Sciences, des Techniques et des Technolo-gies de Bamako, Bamako, Mali.
DOI: 10.4236/jamp.2024.122027   PDF    HTML   XML   132 Downloads   459 Views   Citations

Abstract

Wiener amalgam spaces are a class of function spaces where the function’s local and global behavior can be easily distinguished. These spaces are ex-tensively used in Harmonic analysis that originated in the work of Wiener. In this paper: we first introduce a two-variable exponent amalgam space (L^q(),l^p())(Ω). Secondly, we investigate some basic properties of these spaces, and finally, we study their dual.

Keywords

Amalgam Spaces, Variable Exponent Lebesgue Spaces, Dual of a Vector Space

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1. Introduction

In recent years, function spaces with variable exponents have been intensively studied by an important number of authors. The generalized Lebesgue spaces $L^{p\left(x\right)}$ (or variable exponent Lebesgue spaces) appeared in literature for the first time already in an article by Orlicz [1] , but the advanced development started with the paper [2] of Kovacik and Rakosnik in 1991. A survey of the history of the field with a bibliography of more than a hundred titles published up to 2004 can be found in [3] . To illustrate the importance of Wiener amalgams, let us mention one specific example, which today plays a central role in the theory of time-frequency analysis. This is the space $W\left(\mathcal{F}{L}^1\mathrm{,}{L}^1\right)$ consisting of functions that are locally the Fourier transform of an $L^1$ function and have a global behavior $L^1$ .

The motivation to study such function spaces comes from applications to fluid dynamics [4] [5] , image processing [6] , PDE (Partial Differential Equation) and the calculus of variation [7] [8] .

In the early 1980s, in a series of articles, Feichtinger provides the most general definition of Wiener Amalgam (WA) [9] [10] [11] .

For an introduction to WA on the real line and for some historical notes, we refer to [12] .

In mathematical domain, Wiener amalgams proved to be a very useful tool, for instance in time-frequency analysis [13] (e.g. the Balian-Low theorem [12] ) and sampling theory. Our interest in those spaces arose from the Wiener Amalgams of the spaces with constant exponents [14] .

F. Holland began his systematic study in 1975 [15] . Since, he has been widely studied by [16] [17] [18] .

Only some papers treat the Wiener amalgam with one variable exponent [19] [20] [21] .

It seems that Wiener amalgams with two or more variable exponents have not yet been considered in full generality. In this work, we define a two-variable exponent amalgam space $\left(L^{q()},l^{p()}\right)\left(\Omega \right)$ and give some properties and study their dual.

Some properties of variable exponent amalgam space can be derived in the same way as for usual amalgams $\left(L^q\mathrm{,}{l}^p\right)$ , where $q\mathrm{,}p$ are constant, while others are very complicate.

The following definitions and results on the amalgam spaces $\left(L^q\mathrm{,}{l}^p\right)$ with constant exponents can be found in [15] [17] [22] [23] [24] [25] [26] .

1) Classical Wiener Amalgam space $\left(L^q\mathrm{,}{l}^p\right)$ with constant exponents

We give d as a fixed positive integer and ${ℝ}^d$ as the d-dimensional Euclidean space equipped with its Lebesgue measure dx.

For $1\le p\mathrm{,}q\le \infty$ , the amalgam of $L^q$ and $l^p$ is the space $\left(L^q\mathrm{,}{l}^p\right)$ defined by:

$\left(L^q\mathrm{,}{l}^p\right)\left({ℝ}^d\right)=\left\{f\in L_{loc}^1\left({ℝ}^d\right)\mathrm{<mo>\; :\; </mo>}{}_1\Vert f\Vert {}_{q\mathrm{,}p}<\infty \right\}\mathrm{,}$

where for $r>0$

${}_r\Vert f\Vert {}_{q\mathrm{,}p}=\{\begin{array}{ll}{\left[{\displaystyle \underset{k\in {\mathbb{Z}}^d}{\sum }}{\Vert f{\chi }_{I_k^r}\Vert }_q^p\right]}^{\frac{1}{p}}\hfill & \text{if}p<\infty \hfill \\ \underset{k\in {\mathbb{Z}}^d}{\sup }{\Vert f{\chi }_{I_k^r}\Vert }_q\hfill & \text{if}p=\infty \hfill \end{array}$ (1)

with $I_k^r={\displaystyle \underset{j=1}{\overset{d}{\prod }}\left[k_j\times r,\left(k_j+1\right)\times r\right[}$ and $k={\left(k_j\right)}_{1\le j\le d}\in {\mathbb{Z}}^d$ .

The map $f\mapsto {\Vert f\Vert }_p$ denotes the usual norm on Lebesgue space $L^p\left({ℝ}^d\right)$ on ${ℝ}^d$ while ${\chi }_E$ stands for the characteristic function of the subset E of ${ℝ}^d$ .

2) Some basic facts about amalgam spaces $\left(L^q\mathrm{,}{l}^p\right)\left({ℝ}^d\right)$ with constant exponents

Let $1\le p\mathrm{,}q\le \infty$ . Amalgam spaces $\left(L^q\mathrm{,}{l}^p\right)\left({ℝ}^d\right)$ are defined in (1).

Here are the well-known results properties (see, for example, [15] [17] [22] [23] [24] [25] [26] ):

· For $0<r<\infty$ , $f\mapsto {}_r\Vert f\Vert {}_{q,p}$ is a norm on $\left(L^q\mathrm{,}{l}^p\right)\left({ℝ}^d\right)$ equivalent to ${}_1\Vert f\Vert {}_{q\mathrm{,}p}$ (the equivalence constants depend only on r).

With respect to these norms, the amalgam spaces $\left(L^q\mathrm{,}{l}^p\right)\left({ℝ}^d\right)$ are Banach spaces.

· The spaces are strictly increasing with the global exponent p and (strictly) decreasing with a growing local exponent q more precisely:

*

${}_1\Vert f\Vert {}_{q\mathrm{,}p}\le {}_1\Vert f\Vert {}_{q\mathrm{,}{p}_1}\text{if}1\le p_1<p$ (2)

that is:

*

$1\le p_1<p\Rightarrow \left(L^q\mathrm{,}{l}^{p_1}\right)\left({ℝ}^d\right)\subset \left(L^q\mathrm{,}{l}^p\right)(\; ℝ\; d\; )$

*

${}_1\Vert f\Vert {}_{q\mathrm{,}p}\le {}_1\Vert f\Vert {}_{q_1\mathrm{,}p}\text{if}q<q_1$ (3)

that is:

$q<q_1\Rightarrow \left(L^{q_1}\mathrm{,}{l}^p\right)\left({ℝ}^d\right)\subset \left(L^q\mathrm{,}{l}^p\right)(\; ℝ\; d\; )$

· For $0<r<\infty$ , Holder’s inequality is fulfilled:

${\Vert fg\Vert }_1\le {}_r\Vert f\Vert {}_{q\mathrm{,}p}\times {}_r\Vert g\Vert {}_{q^{\prime }\mathrm{,}{p}^{\prime }}\mathrm{,}f\mathrm{,}g\in L_{loc}^1\left({ℝ}^d\right)$ (4)

where $q^{\prime }\mathrm{,}{p}^{\prime }$ are conjugate exponents of $q\mathrm{,}p$ that is $\frac{1}{q}+\frac{1}{q^{\prime }}=\frac{1}{p}+\frac{1}{p^{\prime }}=1$ .

3) Duality of the wiener amalgam spaces $\left(L^q\mathrm{,}{l}^p\right)\left({ℝ}^d\right)$ with constant exponents

· When $1\le q,p<\infty$ , $\left(L^q\mathrm{,}{l}^p\right)\left({ℝ}^d\right)$ is isometrically isomorphic to the dual ${\left(L^q\mathrm{,}{l}^p\right)}^{\mathrm{<mo>\; *\; </mo>}}\left({ℝ}^d\right)$ of $\left(L^q\mathrm{,}{l}^p\right)\left({ℝ}^d\right)$ in the sense that for any element T of ${\left(L^q\mathrm{,}{l}^p\right)}^{\mathrm{<mo>\; *\; </mo>}}\left({ℝ}^d\right)$ , there is an unique element $\phi \left(T\right)$ of $\left(L^{q^{\prime }}\mathrm{,}{l}^{p^{\prime }}\right)\left({ℝ}^d\right)$ such that:

$\langle T\mathrm{,}f\rangle ={\displaystyle {\int }_{{ℝ}^d}}f\left(x\right)\phi \left(T\right)\left(x\right)\text{d}x\mathrm{,}f\in \left(L^q\mathrm{,}{l}^p\right)\left({ℝ}^d\right)$ and furthermore

${}_1\Vert \phi \left(T\right)\Vert {}_{q^{\prime }\mathrm{,}{p}^{\prime }}=\Vert T\Vert$ .

We recall that $\Vert T\Vert \mathrm{<mo>\; :\; </mo>}=\sup \left\{\left|\langle T\mathrm{,}f\rangle \right|\mathrm{<mo>\; :\; </mo>}f\in L_{loc}^q\left({ℝ}^d\right)\mathrm{,}{}_1\Vert f\Vert {}_{q\mathrm{,}p}\le 1\right\}$ .

· If $1\le q,p<\infty$ , then there exist real numbers A and B such that:

$A\times {}_r\Vert f\Vert {}_{q\mathrm{,}p}\le r^{\frac{-d}{p}}{\left\{{\displaystyle {\int }_{{ℝ}^d}}{\left[{\displaystyle {\int }_{J_x^r}}{\left|f\left(y\right)\right|}^q\text{d}y\right]}^{\frac{p}{q}}\text{d}x\right\}}^{\frac{1}{p}}\le B\times {}_r\Vert f\Vert {}_{q\mathrm{,}p}$

for $f\in L_{loc}^1\left({ℝ}^d\right)$ , $r>0$ and $J_x^r={\displaystyle \underset{j=1}{\overset{d}{\prod }}\left]x_j-\frac{r}{2},x_j+\frac{r}{2}\right[}$ , $x={\left(x_j\right)}_{1\le j\le d}\in {ℝ}^d$ .

4) Denseness of some subsets in amalgam spaces $\left(L^q\mathrm{,}{l}^p\right)\left(\Omega \right)$ with constant exponents

4.1) We define $S=S\left(\Omega \right)$ to be the collection of all simple functions, that is, functions whose range is finite: $s\in S\left(\Omega \right)$ if:

$s\left(x\right)={\displaystyle \underset{j=1}{\overset{n}{\sum }}}{a}_j{\chi }_{E_j}(\; x\; )$

where the numbers $a_j$ are distinct and the sets $E_j\subset \Omega$ are pairwise disjoint.

4.2) Let Ω be an open non void set. Suppose that $1\le q,p<\infty$ , then $C_c\left(\Omega \right)$ and $S\left(\Omega \right)$ are dense in $\left(L^q\mathrm{,}{l}^p\right)\left(\Omega \right)$ .

5) Constant Lebesgue sequence spaces

a) For any real sequence ${\left(x_k\right)}_{k\in \mathbb{Z}}$ ,

${\Vert {\left(x_k\right)}_{k\in \mathbb{Z}}\Vert }_{l^p}=\{\begin{array}{l}{\left[{\displaystyle \underset{k\in \mathbb{Z}}{\sum }}{\left|x_k\right|}^p\right]}^{\frac{1}{p}}\text{if}p<\infty \\ \underset{k\in \mathbb{Z}}{\sup }\left|x_k\right|\text{if}p=\infty \end{array}$ (5)

b)

$l^p=\left\{{\left(x_k\right)}_{k\in \mathbb{Z}}\in {ℝ}^{\mathbb{Z}}:{\Vert {\left(x\right)}_{k\in \mathbb{Z}}\Vert }_{l^p}<\infty \right\}$ (6)

c) $c_0=\left\{{\left(x_k\right)}_{k\in \mathbb{Z}}\in l^{\infty }:\underset{k\to \infty }{\lim }{x}_k=0\right\}$

Therefore, we get the following proposition.

Proposition 1.

Let $1\le p\le \infty$ .

a) Endowed with the two usual operations, $l^p$ is a real vector space and the mapping ${\left(x_k\right)}_{k\in \mathbb{Z}}\mapsto {\Vert {\left(x_k\right)}_{k\in \mathbb{Z}}\Vert }_{l^p}$ makes it a Banach space:

b) Holder’s inequality: If $1\le r\mathrm{,}q\mathrm{,}p\le \infty$ and $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$ , then

${\Vert {\left(x_k{y}_k\right)}_{k\in \mathbb{Z}}\Vert }_{l^r}\le {\Vert {\left(x_k\right)}_{k\in \mathbb{Z}}\Vert }_{l^p}{\Vert {\left(y_k\right)}_{k\in \mathbb{Z}}\Vert }_{l^q}$ (7)

c) Suppose that $1\le p<\infty$ . Then the topological dual of $l^p$ is isomorphically isometric to $l^{p^{\prime }}$ and the duality bracket is defined as follows:

$\langle {\left(x_k\right)}_{k\in \mathbb{Z}},{\left(y_k\right)}_{k\in \mathbb{Z}}\rangle ={\displaystyle \underset{k\in \mathbb{Z}}{\sum }}{x}_k{y}_k,{\left(x_k\right)}_{k\in \mathbb{Z}}\in l^{p^{\prime }},{\left(y_k\right)}_{k\in \mathbb{Z}}\in l^p$

Furthermore, the following result is well-known.

Proposition 2.

a) $c_0$ is a closed sub vector space of $l^{\infty }$ whose topological dual is $l^1$

b) For any ${\left(x_k\right)}_{k\in \mathbb{Z}}\in {ℝ}^{\mathbb{Z}}$ :

$1\le p\le \tilde{p}\le \infty \Rightarrow {\Vert {\left(x_k\right)}_{k\in \mathbb{Z}}\Vert }_{l^{\tilde{p}}}\le {\Vert {\left(x_k\right)}_{k\in \mathbb{Z}}\Vert }_{l^p}$ (8)

therefore $l^p$ is continuously embedded in $l^{\tilde{p}}$ .

Given a normed vector space V, we denote by $V^{\mathrm{<mo>\; *\; </mo>}}$ the normed vector space of bounded linear functionals $V\to ℝ$ endowed with the usual operator norm. We wish to study ${\left[\left(L^{q()},l^{p()}\right)\left(\Omega \right)\right]}^{\mathrm{<mo>\; *\; </mo>}}$ . The study is motivated by norm conjugate inequality (Theorem 22). More precisely, for $g\in \left(L^{q^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)$ , we define the integral operator associated to g to be the operator $T_g\mathrm{<mo>\; :\; </mo>}\left(L^{q()},l^{p()}\right)\left(\Omega \right)\to ℝ$ given by $T_g\left(f\right)={\displaystyle {\int }_{\Omega }}f\left(x\right)g\left(x\right)\text{d}x$ Hölder’s inequality ensures that $T_g$ is a well-defined operator and that it is bounded. The linearity of the integral implies the linearity of $T_g$ whence $T_g\in {\left[\left(L^{q()},l^{p()}\right)\left(\Omega \right)\right]}^{\mathrm{<mo>\; *\; </mo>}}$ .

We have thus defined an operator $T\mathrm{<mo>\; :\; </mo>}\left(L^{q^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)\to {\left[\left(L^{q()},l^{p()}\right)\left(\Omega \right)\right]}^{\mathrm{<mo>\; *\; </mo>}}$ . Again using the linearity of the integral, we find that T is linear. What’s more, by Proposition 20, we have the identity:

$\Vert T_g\Vert \equiv \sup \left\{\left|T_g\left(f\right)\right|\mathrm{<mo>\; :\; </mo>}f\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)\mathrm{,}{}_r\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\le 1\right\}\approx {}_r\Vert g\Vert {}_{q^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}{p}^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }.$

From this, it follows that T is a bijection, bounded, linear operator from $\left(L^{q^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)$ into ${\left[\left(L^{q()},l^{p()}\right)\left(\Omega \right)\right]}^{\mathrm{<mo>\; *\; </mo>}}$ , it actually turns out that this operator is an isomorphism.

The paper is divided into four sections. Section 2 includes fundamental notations and definitions, which will be used in the subsequent sections. Section 3 contains auxiliary results and properties. Section 4 deals with the dual of $\left(L^{q()},l^{p()}\right)\left(\Omega \right)$ .

Throughout the paper, the constants are independent of the main parameters involved, with values that may differ from line to line.

2. Definitions and Notations

· d will be a fixed positive integer, Ω a non void subset of ${ℝ}^d$ , for any subset E of ${ℝ}^d$ the d-dimensional euclidean space ${ℝ}^d$ is equipped with its Lebesgue measure $E\mapsto \left|E\right|$ and ${\chi }_E$ will be the characteristic function of E, for any $x\in {ℝ}^d$ , $\left|x\right|$ will be the usual euclidean norm of x.

* $p\left(\right)\mathrm{,}q\left(\right)\mathrm{,}r\left(\right),\cdots$ in general indicate that $p\mathrm{,}q\mathrm{,}r,\cdots$ are functions used as norm indexes ( ${\Vert \cdot \Vert }_{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{\Vert \cdot \Vert }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{\Vert \cdot \Vert }_{r\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}\cdots$ ).

* $f(\cdot )\mathrm{,}g(\cdot )\mathrm{,}h(\cdot )\mathrm{,}\cdots$ in general mean that $f\mathrm{,}g\mathrm{,}h\mathrm{,}\cdots$ are functions which are applied on the elements $x\mathrm{,}y\mathrm{,}z\mathrm{,}\cdots$ of ${ℝ}^d$ , the dots between the brace refer to these elements.

Let $\mathcal{P}\left(\Omega \right)$ be the set of all Lebesgue measurable functions $p\left(\right)\mathrm{<mo>\; :\; </mo>}\Omega \to \left[\mathrm{1,}\infty \right]$ . In order to distinguish between variable and constant exponents, we will always denote exponent functions by $p\left(\right)$ .

Given $p\left(\right)$ and a set $E\subset \Omega$ , let:

$p_{-}\left(E\right)=\text{ess}{\text{inf}}_{x\in E}p\left(x\right)\mathrm{,}{p}_{+}\left(x\right)=\text{ess}{\text{sup}}_{x\in E}p\left(x\right).$

We simply write:

$p_{-}=p_{-}\left(\Omega \right)\text{and}{p}_{+}=p_{+}\left(\Omega \right)\mathrm{.}$

As in the case for the classical Lebesgue spaces, we will encounter different behaviors depending on whether:

$p\left(x\right)=1,1<p\left(x\right)<\infty ,p\left(x\right)=\infty .$

Therefore, we define three canonical subsets of Ω:

${\Omega }_{\infty }^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}={\Omega }_{\infty }=\left\{x\in \Omega \mathrm{<mo>\; :\; </mo>}p\left(x\right)=\infty \right\}$

${\Omega }_1^{p()}={\Omega }_1=\left\{x\in \Omega \mathrm{<mo>\; :\; </mo>}p\left(x\right)=1\right\}$

${\Omega }_{\mathrm{<mo>\; *\; </mo>}}^{p()}={\Omega }_{\mathrm{<mo>\; *\; </mo>}}=\left\{x\in \Omega :1<p\left(x\right)<\infty \right\}$

Below, the value of certain constants will depend on whether these sets have positive measure; if they do we will use the fact that, for instance,

${\Vert {\chi }_{{\Omega }_1^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\cdot \mathrm{<mo\; stretchy="false">\; )\; </mo>}}}\Vert }_{\infty }=1$

Given $p\left(\right)$ , we define the conjugate exponent $p^{\prime }\left(\right)$ by:

$\frac{1}{p\left(x\right)}+\frac{1}{p^{\prime }\left(x\right)}=\mathrm{1,}x\in \Omega$

with the convention $\frac{1}{\infty }=0$ .

Since $p\left(\right)$ is a function, the notation $p^{\prime }\left(\right)$ can be mistaken for the derivative of $p\left(\right)$ , but we will never use the symbol <<‘>> in this sense.

The notation $p^{\prime }$ will always denote the conjugate of a constant exponent. The operation of taking the supremum/infimum of an exponent does not commute with forming the conjugate exponent. In fact, a straightforward computation shows that:

${\left(p^{\prime }\left(\right)\right)}_{+}={\left(p_{-}\right)}^{\prime }\mathrm{,}{\left(p^{\prime }\left(\right)\right)}_{-}={\left(p_{+}\right)}^{\prime }$

For simplicity, we will omit one set of parentheses and write the left-hand side of each equality as:

$p^{\prime }{\left(\right)}_{+}={\left(p_{-}\right)}^{\prime }\mathrm{,}{p}^{\prime }{\left(\right)}_{-}={\left(p_{+}\right)}^{\prime }$

We will always avoid ambiguous expressions such as ${p^{\prime }}_{+}$ .

A function $r\left(\right)\mathrm{<mo>\; :\; </mo>}\Omega \to ℝ$ is locally log-Holder continuous and denotes this by $r\left(\right)\in L{H}_0\left(\Omega \right)$ , if there exists a constant $C_0$ , such that:

$\forall x\mathrm{,}y\in \Omega \mathrm{,}\left|x-y\right|\le \frac{1}{2}\mathrm{<mo>\; :\; </mo>}\left|r\left(x\right)-r\left(y\right)\right|\le \frac{C_0}{-\log \left(\left|x-y\right|\right)}$

We say that $r\left(\right)$ is log-Holder continuous at infinity and denote this by $r\left(\right)\in L{H}_{\infty }\left(\Omega \right)$ , if there exist $C_{\infty }$ and $r_{\infty }=r\left(\infty \right)$ such that

$\forall x\in \Omega \mathrm{<mo>\; :\; </mo>}\left|r\left(x\right)-r_{\infty }\right|\le \frac{C_{\infty }}{\log \left(e+\left|x\right|\right)}$

If $r\left(\right)$ is log-Holder continuous locally and at infinity, we will denote this by writing $r\left(\right)\in LH\left(\Omega \right)=L{H}_0\left(\Omega \right)\cap L{H}_{\infty }\left(\Omega \right)$ .

If there is no confusion about the domain we will sometimes write: $L{H}_0\mathrm{,}L{H}_{\infty }$ or $LH$ .

· Let $L^0\left(\Omega \mathrm{,}dx\right)=L^0\left(\Omega \right)$ be the vector space of equivalence modulo dx-always everywhere equality of real-valued measurable functions on Ω.

· For any $q\left(\right)\in \mathcal{P}\left(\Omega \right)$ and a Lebesgue measurable function f, we denote:

${\rho }_{L^{q()}\left(\Omega \right)}\left(f\right)={\rho }_{q()\mathrm{,}\Omega }\left(f\right)={\rho }_{q()}\left(f\right)=\rho \left(f\right)={\displaystyle {\int }_{\Omega \backslash {\Omega }_{\infty }^{q()}}}{\left|f\left(x\right)\right|}^{q\left(x\right)}\text{d}x+{\Vert f\Vert }_{L^{\infty }\left({\Omega }_{\infty }^{q()}\right)}$ (9)

where

${\Vert f\Vert }_{L^{\infty }\left({\Omega }_{\infty }^{q()}\right)}=\inf \left\{\epsilon >\mathrm{0\; <mo>\; :\; </mo>}\left|f\left(x\right)\right|\le \epsilon a.e\text{.}x\in {\Omega }_{\infty }^{q()}\right\}$

We define:

${\Vert f\Vert }_{L^{q()}\left(\Omega \right)}={\Vert f\Vert }_{q()\mathrm{,}\Omega }={\Vert f\Vert }_{q()}=\inf \left\{\lambda >\mathrm{0\; <mo>\; :\; </mo>}{\rho }_{q()}\left(\frac{f}{\lambda }\right)\le 1\right\}$ (10)

$L^{q()}\left(\Omega \right)=\left\{f\in L^0\left(\Omega \right)\mathrm{<mo>\; :\; </mo>}{\Vert f\Vert }_{L^{q()}\left(\Omega \right)}<\infty \right\}$ (11)

· If f is unbounded on ${\Omega }_{\infty }^{q()}$ or $f{\left(\right)}^{q()}\notin L^1\left({\Omega }^{q()}\backslash {\Omega }_{\infty }\right)$ , we define ${\rho }_{q()}\left(f\right)=\infty$

· If $\left|{\Omega }_{\infty }^{q()}\right|=0$ in particular when $q_{+}<\infty$ , we let ${\Vert f\Vert }_{L^{\infty }\left({\Omega }_{\infty }^{q()}\right)}=0$ .

· If $\left|\Omega \backslash {\Omega }_{\infty }^{q()}\right|=0$ then ${\rho }_{q()}\left(f\right)={\Vert f\Vert }_{L^{\infty }\left({\Omega }_{\infty }^{q()}\right)}$ .

Let I be a non void countable set, $\mathcal{P}\left(I\right)$ be the set of all Lebesgue measurable functions $p\left(\right)\mathrm{<mo>\; :\; </mo>}I\to \left[\mathrm{1,}\infty \right]$ .

· For any $p\left(\right)\in \mathcal{P}\left(I\right)$ and ${\left\{a_k\right\}}_{k\in I}\in {ℝ}^I$ , we define the modular ${\rho }_{l^{p()}\left(I\right)}$ by:

${\rho }_{l^{p()}\left(I\right)}\left({\left\{a_k\right\}}_{k\in I}\right)={\rho }_{p()}\left({\left\{a_k\right\}}_{k\in I}\right)=\rho \left({\left\{a_k\right\}}_{k\in I}\right)={\displaystyle \underset{k\in I\backslash I_{\infty }^{p()}}{\sum }}{\left|a_k\right|}^{p\left(k\right)}+\underset{k\in I_{\infty }^{p()}}{\sup }\left|a_k\right|$ (12)

or

${\rho }_{l^{p()}\left(I\right)}\left({\left\{a_k\right\}}_{k\in I}\right)={\Vert {\left\{{\left|a_k\right|}^{p\left(k\right)}\right\}}_{k\in I}\Vert }_{l^1\left(I\backslash I_{\infty }^{p()}\right)}+{\Vert {\left\{\left|a_k\right|\right\}}_{k\in I}\Vert }_{l^{\infty }(I_{\infty }^{p(\; \hspace{0.17em}\; )\; )}}$

· If ${\left\{{\left|a_k\right|}^{p\left(k\right)}\right\}}_{k\in I}\notin l^1\left(I\backslash I_{\infty }^{p()}\right)$ or ${\left\{\left|a_k\right|\right\}}_{k\in I}$ is unbounded on $I_{\infty }^{p()}$ , we define ${\rho }_{p()}\left({\left\{a_k\right\}}_{k\in I}\right)=\infty$ .

· If $I_{\infty }^{p()}=\varnothing$ , in particular when $p_{+}<\infty$ , we let $\underset{k\in I_{\infty }^{p()}}{\sup }\left|a_k\right|=0$ therefore ${\rho }_{l^{p()}\left(I\right)}\left({\left\{a_k\right\}}_{k\in I}\right)={\displaystyle \underset{k\in I}{\sum }}{\left|a_k\right|}^{p\left(k\right)}$ .

· If $I\backslash I_{\infty }^{q()}=\varnothing$ then ${\rho }_{p()}\left({\left\{a_k\right\}}_{k\in I}\right)=\underset{k\in I_{\infty }^{p()}}{\sup }\left|a_k\right|$ .

Definition 3.

Let I be a non-void countable set, $\mathcal{P}\left(I\right)$ be the set of all functions $p()\mathrm{<mo>\; :\; </mo>}I\to \left[\mathrm{1,}\infty \right]$ .

For any $p\left(\right)\in \mathcal{P}\left(I\right)$ , we define the variable sequence spaces $l^{p()}\left(I\right)$ by:

$l^{p()}\left(I\right)=\left\{{\left\{a_k\right\}}_{k\in I}\in {ℝ}^I\mathrm{<mo>\; :\; </mo>}{\Vert {\left\{a_k\right\}}_{k\in I}\Vert }_{l^{p()}\left(I\right)}<\infty \right\}$ (13)

where

${\Vert {\left\{a_k\right\}}_{k\in I}\Vert }_{l^{p()}\left(I\right)}=\inf \left\{\lambda >\mathrm{0\; <mo>\; :\; </mo>}{\rho }_{l^{p()}\left(I\right)}\left(\frac{{\left\{a_k\right\}}_{k\in I}}{\lambda }\right)\le 1\right\}$ (14)

${\rho }_{l^{p()}\left(I\right)}\left({\left\{a_k\right\}}_{k\in I}\right)={\displaystyle \underset{k\in I\backslash I_{\infty }^{p()}}{\sum }}{\left|a_k\right|}^{p\left(k\right)}+\underset{k\in I_{\infty }^{p()}}{\sup }\left|a_k\right|\mathrm{.}$ (15)

Then, for any $p\left(\right)\in \mathcal{P}\left(I\right)$ , $\forall \lambda >0$ :

${\rho }_{l^{p()}\left(I\right)}\left(\frac{{\left\{a_k\right\}}_{k\in I}}{\lambda }\right)={\displaystyle \underset{k\in I\backslash I_{\infty }^{p()}}{\sum }}{\left(\frac{\left|a_k\right|}{\lambda }\right)}^{p\left(k\right)}+\underset{k\in I_{\infty }^{p()}}{\sup }\frac{\left|a_k\right|}{\lambda }\mathrm{.}$ (16)

We define on $l^{p()}\left(I\right)$ some operations as follows:

For any ${\left\{a_k\right\}}_{k\in I}\in l^{p()}\left(I\right)$ , ${\left\{b_k\right\}}_{k\in I}\in l^{p()}\left(I\right)$ , $\alpha \in ℝ$ , $\beta \in ℝ\backslash \left\{0\right\}$ : ${\left\{a_k\right\}}_{k\in I}+{\left\{b_k\right\}}_{k\in I}={\left\{a_k+b_k\right\}}_{k\in I}$ ; $\alpha \cdot {\left\{a_k\right\}}_{k\in I}={\left\{\alpha \cdot a_k\right\}}_{k\in I}$ ; ${\left\{a_k\right\}}_{k\in I}\cdot {\left\{b_k\right\}}_{k\in I}={\left\{a_k\cdot b_k\right\}}_{k\in I}$ ; $\frac{{\left\{a_k\right\}}_{k\in I}}{\beta }={\left\{\frac{a_k}{\beta }\right\}}_{k\in I}$ .

We also define the absolute value of any element ${\left\{a_k\right\}}_{k\in I}$ of $l^{p()}\left(I\right)$ by:

$\left|{\left\{a_k\right\}}_{k\in I}\right|={\left\{\left|a_k\right|\right\}}_{k\in I}$

the s-power of ${\left\{a_k\right\}}_{k\in I}$ of $l^{p()}\left(I\right)$ (with $1\le s<\infty$ ) is defined by:

${\left|{\left\{a_k\right\}}_{k\in I}\right|}^s={\left\{{\left|a_k\right|}^s\right\}}_{k\in I}$

Remark that:

· If $p\left(\right)<\infty$ ,

then ${\rho }_{p(),I}\left({\left\{a_k\right\}}_{k\in I}\right)={\displaystyle \underset{k\in I\backslash I_{\infty }^{p()}}{\sum }}{\left|a_k\right|}^{p\left(k\right)}={\displaystyle \underset{k\in I}{\sum }}{\left|a_k\right|}^{p\left(k\right)}$ .

· If $p\left(\right)=\infty$ ,

then ${\rho }_{p()\mathrm{,}I}\left({\left\{a_k\right\}}_{k\in I}\right)=\underset{k\in I}{\sup }\left|a_k\right|$ .

Properties 4.

1) Let’s prove that:

$1\le p\left(\right)<\tilde{p}\left(\right)\le \infty \Rightarrow {\Vert {\left\{a_k\right\}}_{k\in I}\Vert }_{l^{p()}\left(I\right)}\le {\Vert {\left\{a_k\right\}}_{k\in I}\Vert }_{l^{p()}\left(I\right)}.$ (17)

Remark that (17) generalizes (8).

· Case 1: $p\left(\right)<\tilde{p}\left(\right)<\infty$

$p\left(\right)<\tilde{p}\left(\right)<\infty \Rightarrow I_{\infty }^{p()}=I_{\infty }^{\tilde{p}()}=\varnothing$

Let’s prove that: $p\left(\right)<\tilde{p}\left(\right)<\infty$ a.e. on I $\begin{array}{l}\Rightarrow l^{p()}\left(I\right)=\left\{{\left\{a_k\right\}}_{k\in I}\in {\left(ℝ\right)}^I:\inf \left\{\lambda >0:{\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{\lambda }\right)}^{p\left(k\right)}\le 1\right\}<\infty \right\}\\ \subset l^{\tilde{p}()}\left(I\right)=\left\{{\left\{a_k\right\}}_{k\in I}\in {\left(ℝ\right)}^I:\inf \left\{\lambda >0:{\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{\lambda }\right)}^{\tilde{p}\left(k\right)}\le 1\right\}<\infty \right\}\end{array}$ .

If ${\left(a_k\right)}_{k\in I}$ belongs to the left-hand side set, then:

$\inf \left\{\lambda >0:{\displaystyle \underset{k}{\sum }}{\left(\frac{\left|a_k\right|}{\lambda }\right)}^{p\left(k\right)}\le 1\right\}<\infty \Rightarrow \exists 0<{\lambda }_0<\infty :$

${\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{{\lambda }_0}\right)}^{p\left(k\right)}\le 1$ (18)

this implies that $\forall k\in I\mathrm{<mo>\; :\; </mo>}{\left(\frac{\left|a_k\right|}{{\lambda }_0}\right)}^{p\left(k\right)}\le 1\Rightarrow$

$\forall k\in I\mathrm{<mo>\; :\; </mo>}\frac{\left|a_k\right|}{{\lambda }_0}\le 1.$ (19)

Since $\forall k\in I:1\le p\left(k\right)<\tilde{p}\left(k\right)<\infty \Rightarrow \forall k\in I:\tilde{p}\left(k\right)-p\left(k\right)>0$ , this inequality with (19) $\Rightarrow {\left(\frac{\left|a_k\right|}{{\lambda }_0}\right)}^{\tilde{p}\left(k\right)-p\left(k\right)}\le 1$ then ${\left(\frac{\left|a_k\right|}{{\lambda }_0}\right)}^{\tilde{p}\left(k\right)}\le {\left(\frac{\left|a_k\right|}{{\lambda }_0}\right)}^{p\left(k\right)}$ , therefore ${\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{{\lambda }_0}\right)}^{\tilde{p}\left(k\right)}\le {\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{{\lambda }_0}\right)}^{p\left(k\right)}\stackrel{\mathrm{<mo\; stretchy="false">\; (\; </mo>18\; <mo\; stretchy="false">\; )\; </mo>}}{\le }1$ , this implies that ${\left(a_k\right)}_{k\in I}\in l^{\tilde{p}\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)$ then $l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)\subset l^{\tilde{p}\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)$ .

· Case 2: $p\left(\right)<\infty$ ; $\tilde{p}\left(\right)=\infty$ .

$p\left(\right)<\infty \Rightarrow I_{\infty }^{p()}=\varnothing$ and $\tilde{p}\left(\right)=\infty \Rightarrow I_{\infty }^{\tilde{p}()}=I$

In this case: ${\rho }_{l^{p()}\left(I\right)}\left({\left\{a_k\right\}}_{k\in I}\right)={\displaystyle \underset{k\in I\backslash I_{\infty }^{p()}}{\sum }}{\left|a_k\right|}^{p\left(k\right)}+\underset{k\in I_{\infty }^{p()}}{\sup }\left|a_k\right|={\displaystyle \underset{k\in I}{\sum }}{\left|a_k\right|}^{p(\; k\; )}$

${\rho }_{l^{\tilde{p}\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)}\left({\left\{a_k\right\}}_{k\in I}\right)={\displaystyle \underset{k\in I\backslash I_{\infty }^{\tilde{p}\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}}{\sum }}{\left|a_k\right|}^{p\left(k\right)}+\underset{k\in I_{\infty }^{\tilde{p}\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}}{\sup }\left|a_k\right|=\underset{k\in I}{\sup }\left|a_k\right|$

$l^{p()}\left(I\right)=\left\{{\left\{a_k\right\}}_{k\in I}\in {\left(ℝ\right)}^I:\inf \left\{\lambda >0:{\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{\lambda }\right)}^{p\left(k\right)}\le 1\right\}<\infty \right\},$

$l^{\tilde{p}()}\left(I\right)=\left\{{\left\{a_k\right\}}_{k\in I}\in {\left(ℝ\right)}^I:\inf \left\{\lambda >0:\underset{k\in I}{\sup }\frac{\left|a_k\right|}{\lambda }\le 1\right\}<\infty \right\}.$

Let’s compare $\left\{\lambda >0:{\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{\lambda }\right)}^{p\left(k\right)}\le 1\right\}$ and $\left\{\lambda >0:\underset{k\in I}{\sup }\frac{\left|a_k\right|}{\lambda }\le 1\right\}$

Take ${\lambda }_0$ in the first (left-hand side) set, then ${\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{{\lambda }_0}\right)}^{p\left(k\right)}\le 1$ , then $\forall k\in I\mathrm{<mo>\; :\; </mo>}{\left(\frac{\left|a_k\right|}{{\lambda }_0}\right)}^{p\left(k\right)}\le 1\Rightarrow \forall k\in I\mathrm{<mo>\; :\; </mo>}\frac{\left|a_k\right|}{{\lambda }_0}\le 1\Rightarrow \underset{k\in I}{\sup }\frac{\left|a_k\right|}{{\lambda }_0}\le 1$ this implies that ${\lambda }_0$ belongs to the right-hand side set, therefore:

We have:

$\begin{array}{l}\left\{\lambda >0:{\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{\lambda }\right)}^{p\left(k\right)}\le 1\right\}\subset \left\{\lambda >0:\underset{k\in I}{\sup }\frac{\left|a_k\right|}{\lambda }\le 1\right\}\\ \Rightarrow \inf \left\{\lambda >0:\underset{k\in I}{\sup }\frac{\left|a_k\right|}{\lambda }\le 1\right\}\le \inf \left\{\lambda >0:{\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{\lambda }\right)}^{p\left(k\right)}\le 1\right\}\\ \Rightarrow l^{p()}\left(I\right)\subset l^{\tilde{p}()}(\; I\; )\end{array}$

2) Given a non-void countable set I and $p\left(\right)\in \mathcal{P}\left(I\right)$ such that $I_{\infty }^{p()}=\varnothing$ , then for all s such that $\frac{1}{p_{-}}\le s<\infty$ , we have:

${\Vert {\left\{{\left|a_k\right|}^s\right\}}_{k\in I}\Vert }_{l^{p()}\left(I\right)}={\Vert {\left\{a_k\right\}}_{k\in I}\Vert }_{l^{sp()}\left(I\right)}^s.$

To prove this, let $\mu ={\lambda }^{\frac{1}{s}}$

$\begin{array}{c}{\Vert {\left\{{\left|a_k\right|}^s\right\}}_{k\in I}\Vert }_{l^{p()}\left(I\right)}=\inf \left\{\lambda >0:{\displaystyle \underset{k\in I}{\sum }}{\left(\frac{{\left|a_k\right|}^s}{\lambda }\right)}^{p\left(k\right)}\le 1\right\}\\ =\inf \left\{{\mu }^s>0:{\displaystyle \underset{k\in I}{\sum }}{\left(\frac{{\left|a_k\right|}^s}{{\mu }^s}\right)}^{p\left(k\right)}\le 1\right\}\\ =\inf \left\{{\mu }^s>0:{\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{\mu }\right)}^{sp\left(k\right)}\le 1\right\}\\ ={\Vert {\left\{a_k\right\}}_{k\in I}\Vert }_{l^{sp()}\left(I\right)}^s\end{array}$

3) When $p\left(\right)=p$ , $1\le p\le \infty$ , the definition (13) is equivalent to the classical norm of $l^p$ seen in (6), let’s prove it:

For $p\left(\right)=p<\infty$ ,

then $I_{\infty }^{p()}=\varnothing$ and ${\rho }_{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}I}\left({\left\{a_k\right\}}_{k\in I}\right)={\displaystyle \underset{k\in I}{\sum }}{\left|a_k\right|}^{p\left(k\right)}$ ,

$\begin{array}{c}{\Vert {\left\{a_k\right\}}_{k\in I}\Vert }_{l^{p()}\left(I\right)}=\inf \left\{\lambda >0:{\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{\lambda }\right)}^{p\left(k\right)}\le 1\right\}=\inf \left\{\lambda >0:{\displaystyle \underset{k\in I}{\sum }}{\left(\frac{\left|a_k\right|}{\lambda }\right)}^p\le 1\right\}\\ =\inf \left\{\lambda >0:\lambda \ge {\left({\displaystyle \underset{k\in I}{\sum }}{\left|a_k\right|}^p\right)}^{\frac{1}{p}}\right\}={\left({\displaystyle \underset{k\in I}{\sum }}{\left|a_k\right|}^p\right)}^{\frac{1}{p}}={\Vert {\left\{a_k\right\}}_{k\in I}\Vert }_{l^p\left(I\right)}.\end{array}$

For $p\left(\right)=p=\infty$ ,

then

$I_{\infty }^{p()}=I$ and ${\rho }_{l^{p()}\left(I\right)}\left({\left\{a_k\right\}}_{k\in I}\right)=\underset{k\in I}{\sup }\left|a_k\right|$ , therefore

$\begin{array}{c}{\Vert {\left\{a_k\right\}}_{k\in I}\Vert }_{l^{p()}\left(I\right)}=\inf \left\{\lambda >0:\underset{k\in I}{\sup }\frac{\left|a_k\right|}{\lambda }\le 1\right\}=\inf \left\{\lambda >0:\lambda \ge \underset{k\in I}{\sup }\left|a_k\right|\right\}\\ =\underset{k\in I}{\sup }\left|a_k\right|={\Vert {\left\{a_k\right\}}_{k\in I}\Vert }_{l^{+\infty }\left(I\right)}.\end{array}$

4) Given a countable and non-void set I and $p\left(\right)\in \mathcal{P}\left(I\right)$ , for all:

${\left\{a_k\right\}}_{k\in I}\in l^{p()}\left(I\right)$ and ${\left\{b_k\right\}}_{k\in I}\in l^{p^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)$ , if ${\left\{a_k{b}_k\right\}}_{k\in I}\in l^1\left(I\right)$ ,

then

${\displaystyle \underset{k\in I}{\sum }}{a}_k{b}_k\le K_{p()}{\Vert {\left\{a_k\right\}}_{k\in I}\Vert }_{l^{p()}\left(I\right)}\times {\Vert {\left\{b_k\right\}}_{k\in I}\Vert }_{l^{p^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)}\mathrm{.}$ (20)

This inequality can be generalized in the following way:

5) Given a countable and non-void set I and $q\left(\right)\mathrm{,}r\left(\right)\in \mathcal{P}\left(I\right)$ , define $p\left(\right)$ by:

$\frac{1}{p\left(\right)}=\frac{1}{q\left(\right)}+\frac{1}{r\left(\right)}\text{on}I$

Then, there exists a constant K such that for all:

${\left\{a_k\right\}}_{k\in I}\in l^{q()}\left(I\right)\mathrm{,}{\left\{b_k\right\}}_{k\in I}\in l^{r\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)\mathrm{<mo>\; :\; </mo>}{\left\{a_k{b}_k\right\}}_{k\in I}\in l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)$ and

${\Vert {\left\{a_k{b}_k\right\}}_{k\in I}\Vert }_{l^{p()}\left(I\right)}\le K_{p()}{\Vert {\left\{a_k\right\}}_{k\in I}\Vert }_{l^{q()}\left(I\right)}\times {\Vert {\left\{b_k\right\}}_{k\in I}\Vert }_{l^{r()}\left(I\right)}\mathrm{.}$ (21)

In fact, we have the following result.

Proposition 5.

Given a non void countable set I and $p\left(\right)\in \mathcal{P}\left(I\right)$ .

a) Suppose that ${\left\{b_i\right\}}_{i\in I}\in l^{p^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)$ . Then:

•) $a\mapsto T_b\left(a\right)={\displaystyle \underset{i\in I}{\sum }}{a}_i{b}_i$ is a continuous linear functional on $l^{p()}\left(I\right)$ and

$\Vert T_b\Vert \equiv \sup \left\{\left|{\displaystyle \underset{i\in I}{\sum }}{a}_i{b}_i\right|\mathrm{<mo>\; :\; </mo>}a={\left\{a_i\right\}}_{i\in I}\in l^{p()}\left(I\right)\mathrm{,}{\Vert a\Vert }_{l^{p()}\left(I\right)}\le 1\right\}={\Vert b\Vert }_{l^{p^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}(\; I\; )}$

••) If $p^{\prime }\left(\right)<\infty$ on I, then

$\Vert T_b\Vert =\max \left\{\left|{\displaystyle \underset{i\in I}{\sum }}{a}_i{b}_i\right|:a={\left\{a_i\right\}}_{i\in I}\in l^{p()}\left(I\right),{\Vert a\Vert }_{l^{p()}\left(I\right)}=1\right\}.$

b) Suppose that $b={\left\{b_i\right\}}_{i\in I}\in {ℂ}^I$ such that

$N_{p^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(b\right)=\sup \left\{\left|{\displaystyle \underset{i\in I}{\sum }}{a}_i{b}_i\right|:a={\left\{a_i\right\}}_{i\in I}\in {ℂ}^I,Card\left(\left\{i\in I:a_i\ne 0\right\}\right)<\infty ,{\Vert a\Vert }_{l^{p()}\left(I\right)}=1\right\}<\infty \mathrm{.}$

Then,

$b\in l^{p^{\prime }(\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)\text{and}{\Vert b\Vert }_{l^{p^{\prime }(\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)}=N_{p^{\prime }(\mathrm{<mo\; stretchy="false">\; )\; </mo>}}(\; b\; )$

c) Suppose that $p\left(\right)\in \mathcal{P}\left(I\right)\mathrm{,}p\left(\right)<\infty$ on I and T belongs to the dual ${\left(l^{p()}\left(I\right)\right)}^{\mathrm{<mo>\; *\; </mo>}}$ of $l^{p()}\left(I\right)$ .

Then,

there exists $b={\left(b_i\right)}_{i\in I}\in l^{p^{\prime }\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)$ such that:

$\forall a={\left\{a_i\right\}}_{i\in I}\in l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(I\right)\mathrm{<mo>\; :\; </mo>}T\left(a\right)={\displaystyle \underset{i\in I}{\sum }}{a}_i{b}_i$

For the proof of (20) and (21) and Proposition 5, consult Theorem 2.26 and Corollary 2.28 of [27] take account of the fact that $\left(l^{p()}\mathrm{,}{\Vert \cdot \Vert }_{l^{p()}}\right)$ is in fact the Lebesgue space $L^{p()}\left(E\mathrm{,}\mathcal{A}\mathrm{,}\mu \right)$ where $E=\mathbb{Z}$ , $\mathcal{A}=\left\{X\mathrm{<mo>\; :\; </mo>}X\subset E\right\}=2^X$ the power of the set X, $\mu$ is defined as: $\forall k\in \mathbb{Z}\mathrm{<mo>\; :\; </mo>}\mu \left\{k\right\}=1$ , $\mu$ is a counting measure.

In this work, we will need the following lemma called Norm-modular unit ball property.

Lemma 6. (Norm-modular unit ball property)

Let Ω be a non void set $q\left(\right)\in \mathcal{P}\left(\Omega \right)$ , suppose that $q_{+}<\infty$ . For any sequence $\left\{f_n\right\}\subset L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)$ and $f\in L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)$ , ${\Vert f-f_n\Vert }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\to 0$ if only if ${\rho }_{L^{q()}\left(\Omega \right)}\left(f-f_n\right)\to 0$ .

Remark that the discrete version of this lemma is also valid.

Historic of the definition

Recall that: For $1\le p\mathrm{,}q\le \infty$ , $r>0$ ,

$I_k^r={\displaystyle \underset{j=1}{\overset{d}{\prod }}}\left[k_j\times r\mathrm{,}\left(k_j+1\right)\times r\right[\mathrm{,}\text{with}k={\left(k_j\right)}_{1\le j\le d}\in {\mathbb{Z}}^d\mathrm{.}$

The amalgam of $L^q$ and $l^p$ is the space $\left(L^q\mathrm{,}{l}^p\right)$ defined (see (1)) by:

$\left(L^q\mathrm{,}{l}^p\right)\left({ℝ}^d\right)=\left\{f\in L_{loc}^1\left({ℝ}^d\right)\mathrm{<mo>\; :\; </mo>}{}_1\Vert f\Vert {}_{q\mathrm{,}p}<\infty \right\}\mathrm{,}$

where

${}_r\Vert f\Vert {}_{q\mathrm{,}p}=\{\begin{array}{ll}{\left[{\displaystyle \underset{k\in {\mathbb{Z}}^d}{\sum }}{\Vert f{\chi }_{I_k^r}\Vert }_q^p\right]}^{\frac{1}{p}}\hfill & \text{if}p<\infty \hfill \\ \underset{k\in {\mathbb{Z}}^d}{\sup }{\Vert f{\chi }_{I_k^r}\Vert }_q\hfill & \text{if}p=\infty \hfill \end{array}$

Suppose that q is a function $q\left(\right)$ and p a constant and taking account of (10), we get:

${}_r\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p}=\{\begin{array}{ll}{\left[{\displaystyle \underset{k\in {\mathbb{Z}}^d}{\sum }}{\Vert f{\chi }_{I_k^r}\Vert }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}^p\right]}^{\frac{1}{p}}\hfill & \text{if}p<\infty \hfill \\ \underset{k\in {\mathbb{Z}}^d}{\sup }{\Vert f{\chi }_{I_k^r}\Vert }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\hfill & \text{if}p=\infty \hfill \end{array}$ (22)

${\left\{{\Vert f{\chi }_{I_k^r}\Vert }_{q()}\right\}}_{k\in {\mathbb{Z}}^d}$ is real sequence indexed by a countable set ${\mathbb{Z}}^d$ .

Suppose that q and p are both functions $q\left(\right)$ and $p\left(\right)$ and taking account of (14), (22) becomes:

${}_r\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}={\Vert {\left\{{\Vert f{\chi }_{I_k^r}\Vert }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}(\; \mathbb{Z}\; d\; )}$

which may be rewritten with more information under the form:

${}_r\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }={\Vert {\left\{{\Vert f{\chi }_{I_k^r}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}$ (23)

where

$L^{q()}\left(\Omega \right)=\left\{f\in L^0\left(\Omega \right)\mathrm{<mo>\; :\; </mo>}{\Vert f\Vert }_{L^{q()}\left(\Omega \right)}<\infty \right\}$

${\Vert f\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}=\inf \left\{\lambda >\mathrm{0\; <mo>\; :\; </mo>}{\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{f}{\lambda }\right)\le 1\right\}$

${\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(f\right)={\displaystyle {\int }_{\Omega \backslash {\Omega }_{\infty }^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}}}{\left|f\left(x\right)\right|}^{q\left(x\right)}\text{d}x+{\Vert f\Vert }_{L^{\infty }({\Omega }_{\infty }^{q\mathrm{<mo\; stretchy="false">\; (\; \hspace{0.17em}\; )\; )\; </mo>}}}$

$l^{p()}\left({\mathbb{Z}}^d\right)=\left\{{\left\{a_k\right\}}_{k\in I}\in {\left(ℝ\right)}^{{\mathbb{Z}}^d}\mathrm{<mo>\; :\; </mo>}{\Vert {\left\{a_k\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p()}\left({\mathbb{Z}}^d\right)}<\infty \right\}$

${\Vert {\left\{a_k\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p()}\left({\mathbb{Z}}^d\right)}=\inf \left\{\lambda >\mathrm{0\; <mo>\; :\; </mo>}{\rho }_{l^{p()}\left({\mathbb{Z}}^d\right)}\left(\frac{{\left\{a_k\right\}}_{k\in {\mathbb{Z}}^d}}{\lambda }\right)\le 1\right\}$

${\rho }_{l^{p()}\left({\mathbb{Z}}^d\right)}\left({\left\{a_k\right\}}_{k\in {\mathbb{Z}}^d}\right)={\displaystyle \underset{k\in {\mathbb{Z}}^d\backslash {\left({\mathbb{Z}}^d\right)}_{\infty }^{p()}}{\sum }}{\left|a_k\right|}^{p\left(k\right)}+\underset{k\in {\left({\mathbb{Z}}^d\right)}_{\infty }^{p()}}{\sup }\left|a_k\right|$

Definition 7.

Let Ω be a set such that ${\mathbb{Z}}^d\subset \Omega$ , for any $q\left(\right)\in \mathcal{P}\left(\Omega \right)$ , $p\left(\right)\in \mathcal{P}\left({\mathbb{Z}}^d\right)$ , let $f\in L_{loc}^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)$ , $I_k^r={\displaystyle \underset{j=1}{\overset{d}{\prod }}\left[k_j\times r\mathrm{,}\left(k_j+1\right)\times r\right[}$ , with $k={\left(k_j\right)}_{1\le j\le d}\in {\mathbb{Z}}^d$ , $0<r<\infty$ , for any Lebesgue measurable function f we define the non negative real number:

${}_r\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }={\Vert {\left\{{\Vert f{\chi }_{I_k^r}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}={\Vert {\left\{F\left(k\mathrm{,}r\right)\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}$ (24)

where

$F\left(k,r\right)={\Vert f{\chi }_{I_k^r}\Vert }_{L^{q()}\left(\Omega \right)},k\in {\mathbb{Z}}^d,0<r<\infty .$

If we take $r=1$ in (24), we get:

${}_1\Vert f\Vert {}_{q(),p(),\Omega }={\Vert {\left\{F\left(k,1\right)\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p()}\left({\mathbb{Z}}^d\right)}={\Vert {\left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{q()}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p()}\left({\mathbb{Z}}^d\right)}.$ (25)

We define the two-variable exponential amalgam spaces $\left(L^{q()},l^{p()}\right)\left(\Omega \right)$ by:

$\left(L^{q()},l^{p()}\right)\left(\Omega \right)=\left\{f\in L_{loc}^{q()}\left(\Omega \right)\mathrm{<mo>\; :\; </mo>}{}_1\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }<\infty \right\}$ (26)

If there is no confusion:

${}_r\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}{ℝ}^d}\mathrm{,}\left(L^{q()},l^{p()}\right)\left({ℝ}^d\right)$ will be smply ${}_r\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}\left(L^{q()},l^{p()}\right)$ .

Explanation

To compute ${}_r\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }={\Vert {\left\{{\Vert f{\chi }_{I_k^r}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}$ , we first calculate:

${\Vert f{\chi }_{I_k^r}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}=\inf \left\{\lambda >\mathrm{0\; <mo>\; :\; </mo>}{\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{f{\chi }_{I_k^r}}{\lambda }\right)\le 1\right\}$ ,

(where ${\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{f{\chi }_{I_k^r}}{\lambda }\right)={\displaystyle {\int }_{\Omega \backslash {\Omega }_{\infty }^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}}}{\left|\left(\frac{f{\chi }_{I_k^r}\left(y\right)}{\lambda }\right)\right|}^{q\left(y\right)}\text{d}y+{\Vert \frac{f{\chi }_{I_k^r}}{\lambda }\Vert }_{L^{\infty }\left({\Omega }_{\infty }^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)}$ ), this result depends at least on k and r, we denote it by $\beta \left(k\mathrm{,}r\right)$ , after that we consider ${\left\{\beta \left(k\mathrm{,}r\right)\right\}}_{k\in {\mathbb{Z}}^d}$ and determine:

${\Vert {\left\{\beta \left(k\mathrm{,}r\right)\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p()}\left({\mathbb{Z}}^d\right)}=\inf \left\{\lambda >\mathrm{0\; <mo>\; :\; </mo>}{\rho }_{l^{p()}\left({\mathbb{Z}}^d\right)}\left(\frac{{\left\{\beta \left(k\mathrm{,}r\right)\right\}}_{k\in {\mathbb{Z}}^d}}{\lambda }\right)\le 1\right\}$ .

where

${\rho }_{l^{p()}\left({\mathbb{Z}}^d\right)}\left(\frac{{\left\{\beta \left(k\mathrm{,}r\right)\right\}}_{k\in {\mathbb{Z}}^d}}{\lambda }\right)={\displaystyle \underset{k\in {\mathbb{Z}}^d\backslash {\left({\mathbb{Z}}^d\right)}_{\infty }^{p()}}{\sum }}{\left(\frac{\left|\beta \left(k\mathrm{,}r\right)\right|}{\lambda }\right)}^{p\left(k\right)}+{\Vert {\left\{\frac{\left|\beta \left(k\mathrm{,}r\right)\right|}{\lambda }\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{L^{+\infty }\left({\left({\mathbb{Z}}^d\right)}_{\infty }^{p()}\right)}$ , this result depends at least on r.

Finally, the result of the calculation of ${}_r\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}$ depends at least on r.

In other hand,

$f\in \left(L^{q\left(\right)},l^{p\left(\right)}\right)\left(\Omega \right)\iff$

$\{\begin{array}{l}{\Vert f{\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}<\infty \mathrm{,}\text{for}\text{any}\text{cube}{I}_k^1\subset \Omega \mathrm{,}\text{and}\\ {\Vert {\left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p()}\left({\mathbb{Z}}^d\right)}<\infty \end{array}$ (27)

· A sequence ${\left(f_n\right)}_n$ of $\left(\left(L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)\mathrm{,}{}_1\Vert \cdot \Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)$ is said to converge in norm to f, we note:

$f_n\to f\mathrm{,}\text{if}{}_1\Vert f-f_n\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\to 0\text{when}n\to \infty \mathrm{.}$

· For two functions (eventually constants) $q\left(\right)\mathrm{,}p\left(\right)$ on Ω such that $0<p\left(\right)<q\left(\right)<\infty$ , we define:

$L^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)+L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)=\left\{f=g+h\mathrm{<mo>\; :\; </mo>}g\in L^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)\mathrm{,}h\in L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)\right\}\mathrm{.}$

This is a Banach space with the norm:

${\Vert f\Vert }_{L^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)+L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}=\inf \left\{{\Vert g\Vert }_{L^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}+{\Vert h\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\mathrm{<mo>\; :\; </mo>}f=g+h\mathrm{,}g\in L^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)\mathrm{,}h\in L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)\right\}\mathrm{.}$

3. Properties

In this section, we will use a method to show that $\left(\left(L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)\mathrm{,}{}_1\Vert \cdot \Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\right)$ is a Banach space either Ω is bounded or unbounded.

Proposition 8.

Let Ω be a set such that ${\mathbb{Z}}^d\subset \Omega$ , given $q\left(\right)\in \mathcal{P}\left(\Omega \right)$ , $p\left(\right)\in \mathcal{P}\left({\mathbb{Z}}^d\right)$ and $f\in L_{loc}^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)$ .

1) Then $\left(L^{q()},l^{p()}\right)\left(\Omega \right)$ is a vector space and

${}_1\Vert f\Vert {}_{q(),p(),\Omega }<\infty \Rightarrow f<\infty .$

2) The function $f\mapsto {}_1\Vert f\Vert {}_{q(),p(),\Omega }$ defines a norm on $\left(L^{q()},l^{p()}\right)\left(\Omega \right)$ .

Proof.

1) $0\in \left(L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)\subset L_{loc}^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)$ which is a vector space, then it will suffice to show that for all $\alpha \mathrm{,}\beta \in ℝ$ not both 0; and $f\mathrm{,}g\in \left(L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)$ :

$\alpha f+\beta g\in \left(L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)(\; \Omega \; )$

From triangle inequality of ${\Vert \cdot \Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}$ , we have:

${\Vert \left(\alpha f+\beta g\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\le {\Vert \left(\alpha f\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}+{\Vert \left(\beta g\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}(\; \Omega \; )}$

${\Vert \cdot \Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}$ is order preserving, then:

$\begin{array}{l}{\Vert {\left\{{\Vert \left(\alpha f+\beta g\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>\; <mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}\\ \le {\Vert {\left\{{\Vert \left(\alpha f\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}+{\Vert \left(\beta g\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}\mathrm{.}\end{array}$

From triangle inequality of ${\Vert \cdot \Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}$ :

$\begin{array}{l}{\Vert {\left\{{\Vert \left(\alpha f+\beta g\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}\\ \le {\Vert {\left\{{\Vert \left(\alpha f\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}+{\Vert {\left\{{\Vert \left(\beta g\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}\mathrm{.}\end{array}$

From the homogeneity of ${\Vert \cdot \Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}$ and ${\Vert \cdot \Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}$ , we have:

$\begin{array}{l}{\Vert {\left\{{\Vert \left(\alpha f+\beta g\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}\\ \le \left|\alpha \right|{\Vert {\left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}+\left|\beta \right|{\Vert {\left\{{\Vert g{\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}(\; \mathbb{Z}\; d\; )}\end{array}$

that is

${}_1\Vert \alpha f+\beta g\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\le {\left|\alpha \right|}_1{\Vert f\Vert }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }+{\left|\beta \right|}_1{\Vert g\Vert }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }$ (28)

It is obvious that ${}_1\Vert f\Vert {}_{q(),p(),\Omega }<\infty \Rightarrow f<\infty$ .

2) Let $f\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ .

It is easy to see that:

${}_1\Vert f\Vert {}_{q(),p(),\Omega }\ge 0$ , let ${}_1\Vert f\Vert {}_{q(),p(),\Omega }=0$ .

${}_1\Vert f\Vert {}_{q(),p(),\Omega }=0\Rightarrow {\Vert {\left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p()}\left({\mathbb{Z}}^d\right)}=0$

$\Rightarrow {\left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{q()}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}={\left\{0\right\}}_{k\in {\mathbb{Z}}^d}$

$\Rightarrow {\Vert f{\chi }_{I_k^1}\Vert }_{L^{q()}\left(\Omega \right)}=\mathrm{0,}k\in {\mathbb{Z}}^d$

$\Rightarrow f{\chi }_{I_k^1}=\mathrm{0,}k\in {\mathbb{Z}}^d$

$\Rightarrow f=0$

Homogeneity of ${}_1\Vert \cdot \Vert {}_{q(),p(),\Omega }$ :

Let $f\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ , and $\alpha \in ℝ$ , by homogeneity of ${\Vert \cdot \Vert }_{L^{q()}\left(\Omega \right)}$ and ${\Vert \cdot \Vert }_{l^{p()}\left({\mathbb{Z}}^d\right)}$ , we have:

$\begin{array}{c}{}_1\Vert \alpha f\Vert {}_{q(),p(),\Omega }={\Vert {\left\{{\Vert \left(\alpha f\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p()}\left({\mathbb{Z}}^d\right)}\\ ={\Vert \left|\alpha \right|{\left\{{\Vert \left(f\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p()}\left({\mathbb{Z}}^d\right)}\\ =\left|\alpha \right|{\Vert {\left\{{\Vert \left(f\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p()}\left({\mathbb{Z}}^d\right)}\\ =\left|\alpha \right|\times {}_1\Vert f\Vert {}_{q(),p(),\Omega }\mathrm{.}\end{array}$

Triangle inequality of ${}_1\Vert \cdot \Vert {}_{q(),p(),\Omega }$ :

In (28), if we take $\alpha =\beta =1$ , we will get:

${}_1\Vert f+g\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\le {}_1\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }+{}_1\Vert g\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\mathrm{.}$

¨

We will need the following lemmas.

Lemma 9. [28]

Let $\left(X\mathrm{,}\mathcal{A}\mathrm{,}\mu \right)$ be a measure space such that $\mu \left(X\right)<\infty$ .

Then, $L^q\left(X\mathrm{,}\mathcal{A}\mathrm{,}\mu \right)\subset L^p\left(X\mathrm{,}\mathcal{A}\mathrm{,}\mu \right)$ that is ${\Vert f\Vert }_p\le C\left(q\mathrm{,}p\right)\times {\Vert f\Vert }_q$ for any $1\le p\le q\le \infty$ .

Where

$C\left(q\mathrm{,}p\right)=\{\begin{array}{l}{\left[\mu \left(X\right)\right]}^{\frac{1}{p}-\frac{1}{q}}\text{if}q<\infty \\ {\left[\mu \left(X\right)\right]}^{\frac{1}{p}}\text{if}q=\infty \end{array}$

Lemma 10. (Monotone Convergence)

Let Ω be a set such that ${\mathbb{Z}}^d\subset \Omega$ , given $q\left(\right)\in \mathcal{P}\left(\Omega \right)$ , $p\left(\right)\in \mathcal{P}\left({\mathbb{Z}}^d\right)$ and $f\in L_{loc}^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)$ .

If ${\left\{f^{\left[n\right]}\right\}}_{n\ge 1}\subset \left(\left(L^{q()},l^{p()}\right)\left(\Omega \right)\mathrm{,}{}_1\Vert \cdot \Vert {}_{q(),p(),\Omega }\right)$ is a sequence of non negative functions such that $f^{\left[n\right]}$ increases to a function f pointwise always everywhere (a.e.).

Then:

either $f\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ and

${}_1\Vert f^{\left[n\right]}\Vert {}_{q(),p(),\Omega }\to {}_1\Vert f\Vert {}_{q(),p(),\Omega }$

or

$\begin{array}{l}f\notin \left(L^{q()},l^{p()}\right)\left(\Omega \right)\text{and}\\ {}_1\Vert f^{\left[n\right]}\Vert {}_{q(),p(),\Omega }\to \infty ={}_1\Vert f\Vert {}_{q(),p(),\Omega }\mathrm{.}\end{array}$

Proof.

First:

$\begin{array}{c}{\rho }_{L^{q()}\left(\Omega \right)}\left(f\right)={\displaystyle {\int }_{\Omega \backslash {\Omega }_{\infty }^{q()}}}{\left|f\left(x\right)\right|}^{q\left(x\right)}\text{d}x+{\Vert f\Vert }_{L^{\infty }\left({\Omega }_{\infty }^{q()}\right)}\\ ={\displaystyle {\int }_{\Omega \backslash {\Omega }_{\infty }}}{\left|f\left(x\right)\right|}^{q\left(x\right)}\text{d}x+{\Vert f\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\mathrm{.}\end{array}$

Let’s decompose f as:

$f=f_1+f_2$ (29)

where $f_1=f{\chi }_{\left\{x\in \Omega :\left|f\left(x\right)\right|<1\right\}}$ , $f_2=f{\chi }_{\left\{x\in \Omega :\left|f\left(x\right)\right|\ge 1\right\}}$ .

Therefore,

$\begin{array}{l}{\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(f\right)\le {\displaystyle {\int }_{\Omega \backslash {\Omega }_{\infty }}}{\left(\left|f_1\left(x\right)\right|+\left|f_2\left(x\right)\right|\right)}^{q\left(x\right)}\text{d}x\\ +{\Vert \left|f_1\right|+\left|f_2\right|\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\mathrm{.}\end{array}$

We have that:

$\forall q\left(\right)\in \mathcal{P}\left(\Omega \right)\mathrm{<mo>\; :\; </mo>}1\le q_{-}\le q\left(x\right)\le q_{+}\le \infty \mathrm{,}$

in other hand, for any non negative real numbers $a\mathrm{,}b$ :

${\left(a+b\right)}^{q\left(x\right)}\le 2^{q\left(x\right)-1}\left(a^{q\left(x\right)}+b^{q\left(x\right)}\right)\le 2^{q_{+}-1}\left(a^{q\left(x\right)}+b^{q\left(x\right)}\right)\mathrm{.}$

Then,

$\begin{array}{l}{\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(f\right)\le 2^{q_{+}-1}{\displaystyle {\int }_{\Omega \backslash {\Omega }_{\infty }}}\left({\left|f_1\left(x\right)\right|}^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}x\mathrm{<mo\; stretchy="false">\; )\; </mo>}}+{\left|f_2\left(x\right)\right|}^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}x\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\text{d}x+{\Vert \left|f_1\right|+\left|f_2\right|\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\\ \le 2^{q_{+}-1}{\displaystyle {\int }_{\Omega \backslash {\Omega }_{\infty }}}\left({\left|f_1\left(x\right)\right|}^{q_{-}}+{\left|f_2\left(x\right)\right|}^{q_{+}}\right)\text{d}x+{\Vert \left|f_1\right|+\left|f_2\right|\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\\ \le 2^{q_{+}-1}{\displaystyle {\int }_{\Omega \backslash {\Omega }_{\infty }}}\left({\left|f_1\left(x\right)\right|}^{q_{-}}\right)\text{d}x+{\Vert f_1\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}+2^{q_{+}-1}{\displaystyle {\int }_{\Omega \backslash {\Omega }_{\infty }}}\left({\left|f_2\left(x\right)\right|}^{q_{+}}\right)\text{d}x+{\Vert f_2\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\\ \le \left[2^{q_{+}-1}{\Vert {\left(f_1{\chi }_{\Omega \backslash {\Omega }_{\infty }}\right)}^{q_{-}}\Vert }_1+{\Vert f_1\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]+\left[2^{q_{+}-1}{\Vert {\left(f_2{\chi }_{\Omega \backslash {\Omega }_{\infty }}\right)}^{q_{+}}\Vert }_1+{\Vert f_2\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]\\ \le \left[2^{q_{+}-1}{\Vert \left(f_1{\chi }_{\Omega \backslash {\Omega }_{\infty }}\right)\Vert }_{{}^{{}^{q_{-}}}}^{{}^{q_{-}}}+{\Vert f_1\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]+\left[2^{q_{+}-1}{\Vert \left(f_2{\chi }_{\Omega \backslash {\Omega }_{\infty }}\right)\Vert }_{{}^{q_{+}}}^{q_{+}}+{\Vert f_2\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]\mathrm{.}\end{array}$

Therefore, for any $f\in L_{loc}^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)$ , we can decompose it as $f=f_1+f_2$ such that:

$\begin{array}{c}{\rho }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(f\right)\le \left[C\left(q(\cdot )\right){\Vert \left(f_1{\chi }_{\Omega \backslash {\Omega }_{\infty }}\right)\Vert }_{{}^{q_{-}}}^{q_{-}}+{\Vert f_1\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]\\ +\left[C\left(q(\cdot )\right){\Vert \left(f_2{\chi }_{\Omega \backslash {\Omega }_{\infty }}\right)\Vert }_{{}^{q_{+}}}^{q_{+}}+{\Vert f_2\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]\end{array}$

that is:

$\begin{array}{c}{\rho }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(f\right)\le \left[C\left(q\left(\right)\right){\Vert f_1\Vert }_{L^{q_{-}}\left(\Omega \backslash {\Omega }_{\infty }\right)}^{q_{-}}+{\Vert f_1\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]\\ +\left[C\left(q\left(\right)\right){\Vert f_2\Vert }_{L^{q_{+}}\left(\Omega \backslash {\Omega }_{\infty }\right)}^{q_{+}}+{\Vert f_2\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]\end{array}$ (30)

where $C\left(q\left(\right)\right)=2^{q_{+}-1}$ .

Now, we begin the proof:

$f^{\left[n\right]}$ increases to the function f a.e. (by hypothesis), we can estimate ${}_1\Vert f-f^{\left[n\right]}\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }$ , for any $k\in {\mathbb{Z}}^d$ and $0<r<\infty$ :

$\begin{array}{l}{}_1\Vert f-f^{\left[n\right]}\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\\ ={\Vert {\left\{{\Vert \left(f-f^{\left[n\right]}\right){\chi }_{I_k^r}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p()}\left({\mathbb{Z}}^d\right)}\end{array}$ (31)

$\begin{array}{l}{\Vert \left(f-f^{\left[n\right]}\right){\chi }_{I_k^r}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\\ =\inf \left\{\lambda >\mathrm{0\; <mo>\; :\; </mo>}{\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\right)\le 1\right\}\mathrm{.}\end{array}$ (32)

To estimate ${\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\right)$ , we use (29) and (31), to get:

$f=f_1+f_2$ and $f^{\left[n\right]}=f_1^{\left[n\right]}+f_2^{\left[n\right]}$

$\begin{array}{l}{\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\right)\\ \le \left[C\left(q\left(\right)\right)\times {\Vert \frac{\left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\Vert }_{L^{q_{-}}\left(\Omega \backslash {\Omega }_{\infty }\right)}^{q_{-}}+{\Vert \frac{\left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]\\ +\left[C\left(q\left(\right)\right)\times {\Vert \frac{\left(f_2-f_2^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\Vert }_{L^{q_{+}}\left(\Omega \backslash {\Omega }_{\infty }\right)}^{q_{+}}+{\Vert \frac{\left(f_2-f_2^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]\\ \le \left[C\left(q\left(\right)\right)\times {\lambda }^{-q_{-}}{\Vert \left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{q_{-}}\left(\Omega \backslash {\Omega }_{\infty }\right)}^{q_{-}}+{\lambda }^{-1}{\Vert \left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]\\ +\left[C\left(q\left(\right)\right)\times {\lambda }^{-q_{+}}{\Vert \left(f_2-f_2^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{q_{+}}\left(\Omega \backslash {\Omega }_{\infty }\right)}^{q_{+}}+{\lambda }^{-1}{\Vert \left(f_2-f_2^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]\end{array}$

that is

$\begin{array}{l}{\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\right)\\ \le \left[C\left(q\left(\right)\right)\times {\lambda }^{-q_{-}}{\Vert \left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{q_{-}}\left(\Omega \backslash {\Omega }_{\infty }\right)}^{q_{-}}+{\lambda }^{-1}{\Vert \left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]\\ +\left[C\left(q\left(\right)\right)\times {\lambda }^{-q_{+}}{\Vert \left(f_2-f_2^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{q_{+}}\left(\Omega \backslash {\Omega }_{\infty }\right)}^{q_{+}}+{\lambda }^{-1}{\Vert \left(f_2-f_2^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right].\end{array}$

Now, if we use Lemma 9, since

$1\le q_{-}\le q_{+}\le \infty$ and $I_k^1\subset \Omega$ , we get:

$\begin{array}{c}{\Vert \left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{q_{-}}\left(\Omega \backslash {\Omega }_{\infty }\right)}\le {\Vert \left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{q_{-}}\left(\Omega \right)}\\ ={\Vert f_1-f_1^{\left[n\right]}\Vert }_{L^{q_{-}}\left(I_k^1\right)}\\ \le {\left[\left|I_k^1\right|\right]}^{\frac{1}{q_{-}}}\times {\Vert f_1-f_1^{\left[n\right]}\Vert }_{L^{+\infty }\left(I_k^1\right)}.\end{array}$

But $\left|I_k^1\right|=1$ , therefore

${\Vert \left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{q_{-}}\left(\Omega \backslash {\Omega }_{\infty }\right)}\le {\Vert f_1-f_1^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\right)}$ (33)

by the same way, we also have that:

${\Vert \left(f_2-f_2^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{q_{+}}\left(\Omega \backslash {\Omega }_{\infty }\right)}\le {\Vert f_2-f_2^{\left[n\right]}\Vert }_{L^{+\infty }\left(I_k^1\right)}$ (34)

Substituting (33) and (34) in ${\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\right)$ , we get:

$\begin{array}{l}{\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\right)\\ \le \left[C\left(q\left(\right)\right)\times {\lambda }^{-q_{-}}{\Vert f_1-f_1^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\right)}^{q_{-}}+{\lambda }^{-1}{\Vert \left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right]\\ +\left[C\left(q\left(\right)\right)\times {\lambda }^{-q_{+}}{\Vert f_2-f_2^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\right)}^{q_{+}}+{\lambda }^{-1}{\Vert \left(f_2-f_2^{\left[n\right]}\right){\chi }_{I_k^1}\Vert }_{L^{\infty }\left({\Omega }_{\infty }\right)}\right].\end{array}$

Thus,

$\begin{array}{l}{\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\right)\\ \le \left[C\left(q\left(\right)\right)\times {\lambda }^{-q_{-}}{\Vert f_1-f_1^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\right)}^{q_{-}}+{\lambda }^{-1}{\Vert f_1-f_1^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\cap {\Omega }_{\infty }\right)}\right]\\ +\left[C\left(q\left(\right)\right)\times {\lambda }^{-q_{+}}{\Vert f_2-f_2^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\right)}^{q_{+}}+{\lambda }^{-1}{\Vert f_2-f_2^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\cap {\Omega }_{\infty }\right)}\right]\end{array}$

this implies that:

$\begin{array}{l}{\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\right)\\ \le \left[C\left(q\left(\right)\right)\times {\lambda }^{-q_{-}}{\Vert f_1-f_1^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\right)}^{q_{-}}+{\lambda }^{-1}{\Vert f_1-f_1^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\right)}\right]\\ +\left[C\left(q\left(\right)\right)\times {\lambda }^{-q_{+}}{\Vert f_2-f_2^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\right)}^{q_{+}}+{\lambda }^{-1}{\Vert f_2-f_2^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\right)}\right]\end{array}$

$f^{\left[n\right]}$ increases to a function f pointwise, a.e., then $f^{\left[n\right]}\le f$ for any n, since $0\le \left(f-f^{\left[n\right]}\right)=\left(f_1-f_1^{\left[n\right]}\right)+\left(f_2-f_2^{\left[n\right]}\right)$ , we have:

$\begin{array}{c}0\le \left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}\left(y\right)=\left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}\left(y\right)+\left(f_2-f_2^{\left[n\right]}\right){\chi }_{I_k^1}\left(y\right)\\ \le \left|f{\chi }_{I_k^1}-f^{\left[n\right]}{\chi }_{I_k^1}\right|(\; y\; )\end{array}$

that is:

$0\le \left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}\left(y\right)+\left(f_2-f_2^{\left[n\right]}\right){\chi }_{I_k^1}\left(y\right)\le \left|\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}\right|\left(y\right)$ (35)

$f^{\left[n\right]}$ converges pointwise to f always everywhere i.e.

$\forall y\in \Omega \mathrm{<mo>\; :\; </mo>}\underset{n\to \infty }{\lim }\left(f\left(y\right)-f^{\left[n\right]}\left(y\right)\right)=0$ , therefore

for any $y\in \Omega$ , if n is sufficiently large, (35) gives:

$0\le \left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}\left(y\right)+\left(f_2-f_2^{\left[n\right]}\right){\chi }_{I_k^1}\left(y\right)\le \left|\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}\right|\left(y\right)\to 0$

which implies for n sufficiently large:

$\{\begin{array}{l}\left(f_1-f_1^{\left[n\right]}\right){\chi }_{I_k^1}\left(y\right)\to 0\\ \left(f_2-f_2^{\left[n\right]}\right){\chi }_{I_k^1}\left(y\right)\to 0\end{array}$ , then $\{\begin{array}{l}{\Vert f_1-f_1^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\right)}\to {\Vert 0\Vert }_{L^{\infty }\left(I_k^1\right)}=0\\ {\Vert f_2-f_2^{\left[n\right]}\Vert }_{L^{\infty }\left(I_k^1\right)}\to {\Vert 0\Vert }_{L^{\infty }\left(I_k^1\right)}=0\end{array}$

therefore:

${\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\right)\to 0$ as $n\to 0$

If we replace ${\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\right)$ by its value in (32), we get:

${\Vert \left(f-f^{\left[n\right]}\right){\chi }_{I_k^r}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\to 0$ as $n\to \infty$ .

Replacing ${\Vert \left(f-f^{\left[n\right]}\right){\chi }_{I_k^r}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}$ by its value in (31), we will find:

${}_1\Vert f-f_n\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\to 0$ as $n\to \infty$ ,

therefore

$0\le \left|{}_1\Vert f^{\left[n\right]}\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }-{}_1\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\right|\le {}_1\Vert f-f^{\left[n\right]}\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\to 0$ as $n\to 0$

¨

Remark 11.

In the calculation of ${\rho }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\left(\frac{\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\right)$ (in the above proof), we allow the possibility ${\rho }_{L^{q()}\left(\Omega \right)}\left(\frac{\left(f-f^{\left[n\right]}\right){\chi }_{I_k^1}}{\lambda }\right)=\infty$ , it is the case when $f\notin \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ or $q_{+}=\infty$ ( $C\left(q\left(\right)\right)=2^{q_{+}-1}=\infty$ ) then ${}_1\Vert f_n\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\to \infty ={}_1\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }$ .

Remark 12.

If $f\notin \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ , we have defined ${}_1\Vert f\Vert {}_{q(),p(),\Omega }=\infty$ , so in every case, we may write ${}_1\Vert f_n\Vert {}_{q(),p(),\Omega }\to {}_1\Vert f\Vert {}_{q(),p(),\Omega }$ .

Lemma 13. (Lemma of Fatou)

Let Ω be a set such that ${\mathbb{Z}}^d\subset \Omega$ , given $q\left(\right)\in \mathcal{P}\left(\Omega \right)$ , $p\left(\right)\in \mathcal{P}\left({\mathbb{Z}}^d\right)$ and $f\in L_{loc}^{q()}\left(\Omega \right)$ .

If ${\left(f_n\right)}_n\subset \left(\left(L^{q()},l^{p()}\right)\left(\Omega \right),{}_1\Vert \cdot \Vert {}_{q(),p(),\Omega }\right)$ is a sequence of non negative functions such that $f_n\to f$ pointwise a.e.

If $\underset{n\to \infty }{\lim \inf }{}_1\Vert f_n\Vert {}_{q(),p(),\Omega }<\infty$ .

Then, $f\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ and ${}_1\Vert f\Vert {}_{q(),p(),\Omega }\le \underset{n\to +\infty }{\lim \inf }{}_1\Vert f_n\Vert {}_{q(),p(),\Omega }$ .

Proof.

Define a sequence $g_n\left(x\right)=\underset{m\ge n}{\inf }\left|f_m\left(x\right)\right|$ , $x\in \Omega$ .

Then, for all $m\ge n$ , $g_n\left(x\right)\le \left|f_m\left(x\right)\right|$ and so $g_n\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ . By definition, ${\left(g_n\right)}_n$ is an increasing sequence and $\underset{n\to \infty }{\lim }{g}_n\left(x\right)=\underset{n\to \infty }{\lim }\underset{m\ge n}{\inf }\left|f_m\left(x\right)\right|=\underset{m\to \infty }{\lim \inf }\left|f_m\left(x\right)\right|=\left|f\left(x\right)\right|$ a.e $x\in \Omega$ .

Therefore, by monotone convergence lemma: ${}_1\Vert f\Vert {}_{q(),p(),\Omega }=\underset{n\to +\infty }{\lim }{}_1\Vert g_n\Vert {}_{q(),p(),\Omega }\le \underset{n\to +\infty }{\lim }\left(\underset{m\ge n}{\inf }{}_1\Vert f_m\Vert {}_{q(),p(),\Omega }\right)=\underset{n\to \infty }{\lim \inf }{}_1\Vert f_n\Vert {}_{q(),p(),\Omega }<\infty$ , therefore $f\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ .

¨

Lemma 14. (Lemma of Riesz-Fischer)

Let Ω be a set such that ${\mathbb{Z}}^d\subset \Omega$ , given $q\left(\right)\in \mathcal{P}\left(\Omega \right)$ , $p\left(\right)\in \mathcal{P}\left({\mathbb{Z}}^d\right)$ and $f\in L_{loc}^{q()}\left(\Omega \right)$ .

If ${\left(f_n\right)}_n\subset \left(\left(L^{q()},l^{p()}\right)\left(\Omega \right),{}_1\Vert \cdot \Vert {}_{q(),p(),\Omega }\right)$ is a sequence such that:

${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{}_1\Vert f_n\Vert {}_{q(),p(),\Omega }<\infty$ .

Then, there exists $f\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ such that: $F_i={\displaystyle \underset{n=1}{\overset{i}{\sum }}}{f}_n\to f$ in norm as $i\to \infty$ and ${}_1\Vert f\Vert {}_{q(),p(),\Omega }\le {\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{}_1\Vert f_n\Vert {}_{q(),p(),\Omega }$ .

Proof.

Define the function F on Ω by:

$F\left(x\right)={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}\left|f_n\left(x\right)\right|$ and define the sequence ${\left(F_i\right)}_i$ by: $F_i\left(x\right)={\displaystyle \underset{n=1}{\overset{i}{\sum }}}\left|f_n\left(x\right)\right|$ .

The sequence ${\left(F_i\right)}_i$ is non-negative and increases pointwise almost everywhere to F. Further, for each i, $F_i\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ and its norm is uniformly bounded, since ${}_1\Vert F_i\Vert {}_{q(),p(),\Omega }\le {\displaystyle \underset{n=1}{\overset{i}{\sum }}}{}_1\Vert f_n\Vert {}_{q(),p(),\Omega }\le {\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{}_1\Vert f_n\Vert {}_{q(),p(),\Omega }<\infty$ by hypothesis

By the monotone convergence theorem $F\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ . In particular from Proposition 8-1) F is finite a.e.

Hence, if we define the sequence ${\left(G_i\right)}_i$ by $G_i\left(x\right)={\displaystyle \underset{n=1}{\overset{i}{\sum }}}{f}_n\left(x\right)$ .

Then, this sequence also converges pointwise almost everywhere since absolute convergence implies convergence. Denote its sum by f ( $G_i={\displaystyle \underset{k=1}{\overset{i}{\sum }}}{f}_k\to f$ as $i\to \infty$ ).

Let $G_0=0$ , then for any $j\ge 0$ , $G_i-G_j\to f-G_j$ pointwise almost everywhere.

Furthermore, $\underset{i\to \infty }{\lim \inf }{}_1\Vert G_i-G_j\Vert {}_{q(),p(),\Omega }\le \underset{i\to \infty }{\lim \inf }{\displaystyle \underset{n=j+1}{\overset{i}{\sum }}}{}_1\Vert f_n\Vert {}_{q(),p(),\Omega }={\displaystyle \underset{n=j+1}{\overset{\infty }{\sum }}}{}_1\Vert f_n\Vert {}_{q(),p(),\Omega }<\infty$ . By Fatou’s lemma, if we take $j=0$ then:

${}_1\Vert f\Vert {}_{q(),p(),\Omega }\le \underset{i\to +\infty }{\lim \inf }{}_1\Vert G_i\Vert {}_{q(),p(),\Omega }\le {\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}{}_1\Vert f_n\Vert {}_{q(),p(),\Omega }<\infty$

More generally, for each j, the same argument shows that:

${}_1\Vert f-G_j\Vert {}_{q(),p(),\Omega }\le \underset{i\to \infty }{\lim \inf }{}_1\Vert G_i-G_j\Vert {}_{q(),p(),\Omega }\le {\displaystyle \underset{n=j+1}{\overset{\infty }{\sum }}}{}_1\Vert f_n\Vert {}_{q(),p(),\Omega }$ .

Since the sum in the right-hand side tends to zero, we see that $G_j\to f$ in norm, which completes the norm.¨

Proposition 15.

Let Ω be a set such that ${\mathbb{Z}}^d\subset \Omega$ , given $q\left(\right)\in \mathcal{P}\left(\Omega \right)$ , $p\left(\right)\in \mathcal{P}\left({\mathbb{Z}}^d\right)$ .

$\left(\left(L^{q()},l^{p()}\right)\left(\Omega \right),{}_1\Vert \cdot \Vert {}_{q(),p()}\right)$ is a Banach space

Proof.

It is sufficient to show that every Cauchy sequence in $\left(\left(L^{q()},l^{p()}\right)\left(\Omega \right),{}_1\Vert \cdot \Vert {}_{q(),p(),\Omega }\right)$ converges in norm.

Let $\left(f_n\right)\subset \left(\left(L^{q()},l^{p()}\right)\left(\Omega \right),{}_1\Vert \cdot \Vert {}_{q(),p(),\Omega }\right)$ be a Cauchy sequence.

Choose $n_1$ such that: ${}_1\Vert f_i-f_j\Vert {}_{q(),p(),\Omega }<2^{-1}$ for $i\mathrm{,}j\ge n_1$

Choose $n_2>n_1$ such that: ${}_1\Vert f_i-f_j\Vert {}_{q(),p(),\Omega }<2^{-2}$ for $i\mathrm{,}j\ge n_2$

and so on...

This construction yields a subsequence ${\left(f_{n_j}\right)}_j\mathrm{,}{n}_{j+1}\ge n_j$ such that:

${}_1\Vert f_{n_{j+1}}-f_{n_j}\Vert {}_{q(),p(),\Omega }<2^{-j}$

Define a new sequence ${\left(g_j\right)}_j$ by:

$\{\begin{array}{l}{g}_1=f_{n_1}\\ g_j=f_{n_j}-f_{n_{j-1}}\text{if}j>1\end{array}$

Then, for all j, we get the sum:

${\displaystyle \underset{i=1}{\overset{j}{\sum }}}{g}_i=f_{n_j}.$

Further, we have that:

${\displaystyle \underset{j=1}{\overset{\infty }{\sum }}}{}_1\Vert g_j\Vert {}_{q(),p(),\Omega }\le {}_1\Vert f_{n_1}\Vert {}_{q(),p(),\Omega }+{\displaystyle \underset{j=1}{\overset{\infty }{\sum }}}{2}^{-j}<\infty \mathrm{.}$

Therefore, by the Riesz-Fischer lemma, there exists $f\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ such that:

$f_{n_j}\to f$ in norm.

Finally, by the triangle inequality, we have that:

${}_1\Vert f-f_n\Vert {}_{q(),p(),\Omega }\le {}_1\Vert f-f_{n_j}\Vert {}_{q(),p(),\Omega }+{}_1\Vert f_{n_j}-f_n\Vert {}_{q(),p(),\Omega }$

Since ${\left(f_n\right)}_n$ is a Cauchy sequence, for n sufficiently large we can choose $n_j$ to make the right-hand side as small as desired.

Hence, $f_n\to f$ in norm.

¨

We will need the following lemma.

Lemma 16. [27]

Given $q\left(\right)\mathrm{,}{q}_1\left(\right)\mathrm{,}{q}_2\left(\right)\in \mathcal{P}\left(\Omega \right)$ .

1) ${\Vert f\Vert }_{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\le K{\Vert f\Vert }_{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\iff \{\begin{array}{l}{q}_1\left(x\right)\le q_2\left(x\right)a\mathrm{.}e\mathrm{.}x\in \Omega \\ {\displaystyle {\int }_{\left\{x\in \Omega :q_1\left(x\right)<q_2\left(x\right)\right\}}}{\lambda }^{-\frac{q_2\left(x\right)\times q_1\left(x\right)}{q_2\left(x\right)-q_1\left(x\right)}}\text{d}x<\infty \text{for}\text{some}\lambda >1\end{array}$

In particular, if Ω is bounded set:

$\begin{array}{l}{q}_1\left(x\right)\le q_2\left(x\right)a.e.x\in \Omega \\ \left|\Omega \backslash {\Omega }^{q_1()}\right|<\infty \end{array}\}\Rightarrow {\Vert f\Vert }_{q_1()}\le \left(1+\left|\Omega \backslash {\Omega }^{q_1()}\right|\right){\Vert f\Vert }_{q_2(\; \hspace{0.17em}\; )}$

2) ${\Vert f\Vert }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\le {\Vert f\Vert }_{\infty }\iff L^{\infty }\left(\Omega \right)\subset L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\iff 1\in L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\iff {\displaystyle {\int }_{\Omega \backslash {\Omega }_{\infty }^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}}}{\lambda }^{-q\left(x\right)}\text{d}x<\infty$ for some $\lambda >0$ .

In particular, the embedding holds if $\left|\Omega \right|<\infty$ .

Proposition 17.

Let Ω be a set such that ${\mathbb{Z}}^d\subset \Omega$ , given $q\left(\right)\mathrm{,}{q}_1\left(\right)\mathrm{,}{q}_2\left(\right)\in \mathcal{P}\left(\Omega \right)$ , $p\left(\right),p_1\left(\right)\mathrm{,}{p}_2\left(\right)\in \mathcal{P}\left({\mathbb{Z}}^d\right)$ and $f\in L_{loc}^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)$ .

1)

If $\max \left\{\frac{1}{q_{-}},\frac{1}{p_{-}}\right\}\le s<\infty$ , $\left|{\Omega }_{\infty }^{q()}\right|=0$ and $f\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ , then ${}_1\Vert {\left|f\right|}^s\Vert {}_{q(),p(),\Omega }={}_1\Vert f\Vert {}_{sq\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}sp\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }^s$

2)

•) Let $f\in \left(L^{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)$ .

If $q_1\left(\right)\le q_2\left(\right)$ on Ω, then $f\in \left(L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)$ and ${\Vert f\Vert }_{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\le K_{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}{q}_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\times {\Vert f\Vert }_{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }$ or

$\left(L^{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)\subset \left(L^{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)$ .

••) In particular, when $\left|\Omega \backslash {\Omega }_{\infty }^{q_1()}\right|<\infty$ , we have:

${\Vert f\Vert }_{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\le \left(1+\left|\Omega \backslash {\Omega }_{\infty }^{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right|\right){\Vert f\Vert }_{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\le \left(1+\left|\Omega \right|\right){\Vert f\Vert }_{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }$

3)

•) Let $f\in \left(L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)$ and $p_1\left(\right)\le p_2\left(\right)$ on ${\mathbb{Z}}^d$ .

Then, $f\in \left(L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)$ and ${}_1\Vert f\Vert {}_{q(),p_2(),\Omega }\le C\times {}_1\Vert f\Vert {}_{q(),p_1(),\Omega }$ or

$\left(L^{q()},l^{p_1()}\right)\left(\Omega \right)\subset \left(L^{q()},l^{p_2()}\right)\left(\Omega \right)$ .

••) In particular when $\left|\Omega \right|<\infty$ , we have:

${\Vert f\Vert }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\le \left(1+\left|\Omega \right|\right){\Vert f\Vert }_{q_{+}\mathrm{,}{p}_{-}\mathrm{,}\Omega }$

4)

If $\{\begin{array}{l}q\left(\right)=q=\text{constant}\text{real}\\ p\left(\right)=p=\text{constant}\text{real}\end{array}$

both in $\left[\mathrm{1,}\infty \right]$ then $\left(L^{q()},l^{p()}\right)\left(\Omega \right)=\left(L^q\mathrm{,}{l}^p\right)\left(\Omega \right)$ with constant exponents.

$\left(L^q\mathrm{,}{l}^p\right)\left(\Omega \right)$ with constant exponents have been widely studied by many researchers (see [1] [15] [16] [17] [18] ).

5)

If $\left|\Omega \right|<\infty$ . Then, there exist positive constant reals $c\mathrm{,}C$ such that:

$c\times {}_1\Vert f\Vert {}_{q_{-}\mathrm{,}{p}_{+}\mathrm{,}\Omega }\le {}_1\Vert f\Vert {}_{q(),p(),\Omega }\le C\times {}_1\Vert f\Vert {}_{q_{+}\mathrm{,}{p}_{-}\mathrm{,}\Omega }$

otherwise,

$\left(L^{q_{+}}\mathrm{,}{l}^{p_{-}}\right)\left(\Omega \right)\subset \left(L^{q()},l^{p()}\right)\left(\Omega \right)\subset \left(L^{q_{-}}\mathrm{,}{l}^{p_{+}}\right)\left(\Omega \right)$ .

6)

If

$\begin{array}{l}\frac{1}{q_1\left(\right)}+\frac{1}{q_2\left(\right)}=\frac{1}{q\left(\right)}\le 1\\ \frac{1}{p_1\left(\right)}+\frac{1}{p_2\left(\right)}=\frac{1}{p\left(\right)}\le 1\\ \left(f,g\right)\in \left(L^{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}},l^{p_1()}\right)\left(\Omega \right)\times \left(L^{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}},l^{p_2()}\right)\left(\Omega \right)\end{array}\}$

Then,

$\{\begin{array}{l}fg\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)\\ {}_1\Vert fg\Vert {}_{q(),p(),\Omega }\le C\times {}_1\Vert f\Vert {}_{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}{p}_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\times {}_1\Vert g\Vert {}_{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}{p}_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\end{array}$

7)

Let $f\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)$ .

Then,

${}_1\Vert f\Vert {}_{q_{+}+q_{-}\mathrm{,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>\; <mo\; stretchy="false">\; )\; </mo>,}\Omega }\le C\left(p\left(\right)\right)\left[{}_1\Vert f_1\Vert {}_{q_{+}\mathrm{,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }+{}_1\Vert f_2\Vert {}_{q_{-}\mathrm{,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\right]\le 2\times {}_1\Vert f\Vert {}_{q(),p(),\Omega }$

with $f_1\in \left(L^{q_{+}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)$ and $f_2\in \left(L^{q_{-}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)$ ,

otherwise,

$\left(L^{q()},l^{p()}\right)\left(\Omega \right)\subset \left(L^{q_{+}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)+\left(L^{q_{-}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)\subset \left(L^{q_{+}+q_{-}}\mathrm{,}{l}^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)$ .

8)

For any $r>0$ , the norms ${}_1\Vert \cdot \Vert {}_{q(),p()}$ and ${}_r\Vert \cdot \Vert {}_{q(),p()}$ are equivalent.

Proof.

1) Under the hypotheses of 1), we know that ${\Vert {\left|f\right|}^s\Vert }_{L^{q()}\left(\Omega \right)}={\Vert f\Vert }_{L^{sq()}\left(\Omega \right)}^s$ :

$f\in \left(L^{q()},l^{p()}\right)\left(\Omega \right)\Rightarrow {\Vert f{\chi }_{I_k^1}\Vert }_{L^{q()}\left(\Omega \right)}<\infty ,k\in {\mathbb{Z}}^d$

Combining these two results, we will get:

${\Vert {\left|f{\chi }_{I_k^1}\right|}^s\Vert }_{L^{q()}\left(\Omega \right)}={\Vert f{\chi }_{I_k^1}\Vert }_{L^{sq()}\left(\Omega \right)}^s$ $\Rightarrow {\Vert {\left\{{\Vert {\left|f{\chi }_{I_k^1}\right|}^s\Vert }_{L^{q()}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}={\Vert {\left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{sq()}\left(\Omega \right)}^s\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}$ , using properties 4-2), we get:

${\Vert {\left\{{\Vert {\left|f{\chi }_{I_k^1}\right|}^s\Vert }_{L^{q()}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}={\Vert \left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{sq()}\left(\Omega \right)}\right\}\Vert }_{l^{sp\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}^s$ , therefore

${\Vert {\left\{{\Vert {\left|f{\chi }_{I_k^1}\right|}^s\Vert }_{L^{q()}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}={\left({\Vert {\left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{sq()}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{sp\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}\right)}^s$

that is

${}_1\Vert {\left|f\right|}^s\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }={}_1\Vert f\Vert {}_{sq\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}sp\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }^s$

2)

•) $q_1\left(\right)\le q_2\left(\right)$ on Ω $\Rightarrow$ $\frac{1}{q_1\left(\right)}\ge \frac{1}{q_2\left(\right)}$ on Ω, therefore, there exists $q_3\left(\right)>0$ on Ω such that $\frac{1}{q_1\left(\right)}=\frac{1}{q_2\left(\right)}+\frac{1}{q_3\left(\right)}$ on Ω, from Holder’s inequality

${\Vert f{\chi }_{I_k^1}\Vert }_{L^{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}={\Vert f{\chi }_{I_k^1}\times {\chi }_{I_k^1}\Vert }_{L^{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\le K\times {\Vert f{\chi }_{I_k^1}\Vert }_{L^{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\times {\Vert {\chi }_{I_k^1}\Vert }_{L^{q_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}$ , but $\left|I_k^1\right|=1<\infty$ , therefore ${\Vert {\chi }_{I_k^1}\Vert }_{L^{q_3\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\le \left|I_k^1\right|+1=2$ (Lemma 2.39 of [27] ), then we get: ${\Vert f{\chi }_{I_k^1}\Vert }_{L^{q_1()}\left(\Omega \right)}\le 2K\times {\Vert f{\chi }_{I_k^1}\Vert }_{L^{q_2()}\left(\Omega \right)}$ $\Rightarrow {\Vert {\left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{q_1()}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}\le 2K\times {\Vert {\left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{q_2()}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}$ ,

that is

${}_1\Vert f\Vert {}_{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\le C\times {}_1\Vert f\Vert {}_{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }$ or $\left(L^{q_2()},l^{p()}\right)\left(\Omega \right)\subset \left(L^{q_1()},l^{p()}\right)\left(\Omega \right)$ .

this result generalizes (3).

••) In the particular case, when $\left|\Omega \backslash {\Omega }_{\infty }^{q_1()}\right|<\infty$ , we have:

${\Vert f{\chi }_{I_k^1}\Vert }_{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\le \left(1+\left|\Omega \backslash {\Omega }_{\infty }^{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right|\right){\Vert f{\chi }_{I_k^1}\Vert }_{q_2\mathrm{<mo\; stretchy="false">\; (\; \hspace{0.17em}\; )\; </mo>}}$

then follows the inequality:

${\Vert f\Vert }_{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\le \left(1+\left|\Omega \backslash {\Omega }_{\infty }^{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right|\right)\times {\Vert f\Vert }_{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\stackrel{\Omega \backslash {\Omega }_{\infty }^{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\subset \Omega }{\le }\left(1+\left|\Omega \right|\right){\Vert f\Vert }_{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }$

3)

•) $f\in \left(L^{q()},l^{p_2()}\right)\left(\Omega \right)\Rightarrow {\Vert f{\chi }_{I_k^1}\Vert }_{L^{q()}\left(\Omega \right)}<\infty$ . Under the hypotheses of 3), since $p_1\left(\right)\le p_2\left(\right)$ on ${\mathbb{Z}}^d$ , we have from (17):

${\Vert {\left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{q()}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}\le {\Vert {\left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{q()}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\mathrm{,}$

that is

${}_1\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}{p}_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>},\Omega }\le {}_1\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}{p}_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }$ or $\left(L^{q()},l^{p_1()}\right)\left(\Omega \right)\subset \left(L^{q()},l^{p_2()}\right)\left(\Omega \right)$ .

This result generalizes (2)

••)

$\{\begin{array}{l}q\left(\right)\le q_{+}\\ p\left(\right)\ge p_{-}\\ {\Omega }_{\infty }\subset \Omega \Rightarrow \left|\Omega \backslash {\Omega }_{\infty }\right|\le \left|\Omega \right|<\infty \end{array}$ ,

if we apply 2) ••) and3) •), we get:

${\Vert f\Vert }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\le \left(1+\left|\Omega \backslash {\Omega }_{\infty }\right|\right)\times {\Vert f\Vert }_{q_{+}\mathrm{,}{p}_{-}\mathrm{,}\Omega }\le \left(1+\left|\Omega \right|\right)\times {\Vert f\Vert }_{q_{+}\mathrm{,}{p}_{-}\mathrm{,}\Omega }$

4)

We know that:

${}_r\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }={\Vert {\left\{{\Vert f{\chi }_{I_k^r}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}$ .

If $p\left(\right)=p=\text{constant}\in \left[\mathrm{1,}\infty \right]$ , then from (5) this last equality becomes:

${}_r\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }=\{\begin{array}{ll}{\left[{\displaystyle \underset{k\in {\mathbb{Z}}^d}{\sum }}{\Vert f{\chi }_{I_k^r}\Vert }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}^p\right]}^{\frac{1}{p}}\hfill & \text{if}p<\infty \hfill \\ \underset{k\in {\mathbb{Z}}^d}{\sup }{\Vert f{\chi }_{I_k^r}\Vert }_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\hfill & \text{if}p=\infty \hfill \end{array}$ .

Suppose that both $q\left(\right)=q$ and $p\left(\right)=p$ are constants belonging to $\left[\mathrm{1,}\infty \right]$ , the last equality gives: ${}_r\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }=\{\begin{array}{ll}{\left[{\displaystyle \underset{k\in {\mathbb{Z}}^d}{\sum }}{\Vert f{\chi }_{I_k^r}\Vert }_q^p\right]}^{\frac{1}{p}}\hfill & \text{if}p<\infty \hfill \\ \underset{k\in {\mathbb{Z}}^d}{\sup }{\Vert f{\chi }_{I_k^r}\Vert }_q\hfill & \text{if}p=\infty \hfill \end{array}={}_r\Vert f\Vert {}_{q\mathrm{,}p\mathrm{,}\Omega }$ , see (1).

We conclude that our space $\left(L^{q()},l^{p()}\right)\left(\Omega \right)$ generalizes both: $\left(L^{q()},l\right)\left(\Omega \right)$ studied in [19] [20] [21] ; and $\left(L^q\mathrm{,}{l}^p\right)\left(\Omega \right)$ in [16] [17] [18] .

5)

We have $\{\begin{array}{l}{q}_{-}\le q_{\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\le q_{+}\\ p_{-}\le p_{\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\le p_{+}\end{array}$ , we will use 2) and 3) to get:

${}_1\Vert f\Vert {}_{q_{-}\mathrm{,}{p}_{+}\mathrm{,}\Omega }\le K\times {}_1\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\le K\times C\times {}_1\Vert f\Vert {}_{q_{+}\mathrm{,}{p}_{-}\mathrm{,}\Omega }$

or $c\times {}_1\Vert f\Vert {}_{q_{-}\mathrm{,}{p}_{+}\mathrm{,}\Omega }\le {}_1\Vert f\Vert {}_{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>,}\Omega }\le C\times {}_1\Vert f\Vert {}_{q_{+}\mathrm{,}{p}_{-}\mathrm{,}\Omega }$

otherwise,

$\left(L^{q_{+}}\mathrm{,}{l}^{p_{-}}\right)\left(\Omega \right)\subset \left(L^{q()},l^{p()}\right)\left(\Omega \right)\subset \left(L^{q_{-}}\mathrm{,}{l}^{p_{+}}\right)(\; \Omega \; )$

6)

$\left(f\mathrm{,}g\right)\in \left(L^{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)\times \left(L^{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\mathrm{,}{l}^{p_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\right)\left(\Omega \right)$ , from Holder’s inequality:

${\Vert \left(fg\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\le C\times {\Vert f{\chi }_{I_k^1}\Vert }_{L^{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\times {\Vert g{\chi }_{I_k^1}\Vert }_{L^{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\mathrm{,}k\in {\mathbb{Z}}^d$ (36)

Case 1: $p_1\left(\right)=p_2\left(\right)=\infty$

This implies that $p\left(\right)=\infty$ .

${\Vert \cdot \Vert }_{l^{p\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left({\mathbb{Z}}^d\right)}\left(1\le p\left(\right)\le \infty \right)$ is order preserving, therefore the last inequality (36) implies that:

$\begin{array}{l}{\Vert {\left\{{\Vert \left(fg\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{\infty }\left({\mathbb{Z}}^d\right)}\\ \le C\times {\Vert {\left\{{\Vert f{\chi }_{I_k^1}\Vert }_{L^{q_1\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\times {\left\{{\Vert g{\chi }_{I_k^1}\Vert }_{L^{q_2\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}\left(\Omega \right)}\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{l^{\infty }(\; \mathbb{Z}\; d\; )}\end{array}$

now we apply (21) with $p\left(\right)=q\left(\right)=r\left(\right)=\infty$ to get:

$\begin{array}{l}{\Vert {\{{\Vert \left(fg\right){\chi }_{I_k^1}\Vert }_{L^{q\mathrm{<mo\; stretchy="false">\; (\; </mo>}\mathrm{<mo\; stretchy="false">\; )\; </mo>}}}}_{}}_{}\end{array}$