On the basic sequence structure of variable exponent Lebesgue spaces

José L. Ansorena; Glenier Bello

doi:10.4153/S0008414X26102272

On the basic sequence structure of variable exponent Lebesgue spaces

Part of: Topological linear spaces and related structures Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press: 21 May 2026

José L. Ansorena

and

Glenier Bello

Show author details

José L. Ansorena: Affiliation:
Universidad de La Rioja , Spain e-mail: joseluis.ansorena@unirioja.es
Glenier Bello*: Affiliation:
Universidad de Zaragoza , Spain
*: e-mail: gbello@unizar.es

Article contents

Abstract
Introduction
Musielak–Orlicz function spaces
Orlicz functions generated near zero by power functions
Subsymmetric sequences in variable exponent Lebesgue spaces
Complemented sequences in variable exponent Lebesgue spaces
Data availability statement
Competing interests
Footnotes
References

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Abstract

We study the subsymmetric basic sequence structure of variable exponent Lebesgue spaces $L_{\boldsymbol {P}}$ built from index functions $\boldsymbol {P}\colon \Omega \to (0,\infty ]$ on $\sigma $-finite measure spaces $(\Omega ,\Sigma ,\mu )$. Specifically, we prove that if $\boldsymbol {P}$ is bounded away from infinity, then any complemented subsymmetric basic sequence of $L_{\boldsymbol {P}}$ is equivalent to the canonical basis of $\ell _r$ for some $r\ge 1$ in the essential range of $\boldsymbol {P}$.

Keywords

Variable exponent Lebesgue spaces basic sequences Orlicz sequence spaces complemented spaces

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces

Secondary: 46A45: Sequence spaces (including Köthe sequence spaces) 46B15: Summability and bases 46B25: Classical Banach spaces in the general theory 46B42: Banach lattices

Information

Type: Article
Information: Canadian Journal of Mathematics , First View , pp. 1 - 27

DOI: https://doi.org/10.4153/S0008414X26102272 [Opens in a new window]
Creative Commons: This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial licence (https://creativecommons.org/licenses/by-nc/4.0), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original article is properly cited. The written permission of Cambridge University Press or the rights holder(s) must be obtained prior to any commercial use.
Copyright: © The Author(s), 2026. Published by Cambridge University Press on behalf of Canadian Mathematical Society

1 Introduction

The theory of variable exponent Lebesgue spaces has matured from a topic of abstract interest into an indispensable tool in partial differential equations, fluid dynamics, and harmonic analysis. Unlike their classical counterparts, these spaces allow the exponent to vary with the spatial variable, providing a natural and powerful framework for modeling phenomena with localized behavior, such as electrorheological fluids or materials with inhomogeneous properties. The rich structure of these spaces is deeply intertwined with the properties of the exponent function. In this article, we regard these spaces from a functional analysis perspective.

Within the study of the geometry of Banach spaces, the structure of basic sequences is a fundamental topic already present in Banach’s book [Reference Banach5]. In these early years, specialists were confident of achieving structural results valid for all Banach spaces. In particular, they sought to prove that every Banach space contains an isomorphic copy of $c_0$ or $\ell _p$ for some $1\le p<\infty $ . Since the canonical bases of these spaces are symmetric, the study naturally evolved to determining the structure of symmetric basic sequences of Banach spaces. Any symmetric basis is subsymmetric, that is, unconditional and equivalent to all its subsequences. As a matter of fact, the only feature of symmetric sequences that one needs in many situations is their subsymmetry. Partly because of this, and partly because spreading models of null sequences are subsymmetric, the researchers went on to seek subsymmetric basic sequences.

The study of the basic sequence structure of Banach spaces is part of the study of their subspace structure or, following Banach [Reference Banach5], that of the “linear dimension” of Banach spaces. We point out that the mere existence of an isomorphic copy of a Banach space $\mathbb {X}$ within another Banach space $\mathbb {Y}$ does not entitle us to claim that $\mathbb {X}$ is in a block from which $\mathbb {Y}$ is constructed. To fairly assert this, we must assume that $\mathbb {X}$ can be placed into $\mathbb {Y}$ in a complementary manner, so that $\mathbb {Y}$ is isomorphic to the direct sum of $\mathbb {X}$ and a third Banach space. So, the study of the complemented basic sequence structure of Banach spaces is of special interest.

We refer the reader to [Reference Ansorena and Bello2] for a summary of advances in understanding the structure of subsymmetric basic sequences of Banach spaces. Regarding Lebesgue spaces $L_p$ , the first milestone within this topic is [Reference Paley23], where Paley solved some problems posed in [Reference Banach5]. Namely, he proved that given p, $q\in [1,\infty )$ , $\ell _q$ does not embed in $L_p$ unless $q=2$ or $p\le q\le 2$ . The fact that $\ell _2$ embeds into $L_p$ is a ready consequence of Khintchine’s inequalities [Reference Khintchine16]. Kadets [Reference Kadets12] completed the study by showing that if $p< q< 2,$ then $\ell _q$ do embeds into $L_p$ .

By duality, $\ell _q$ is not a complemented subspace of $L_p$ unless $p=1$ or $q\in \{2, p\}$ . Clearly, $\ell _p$ is a complemented subspace of $L_p$ for any $1\le p<\infty $ . Pełczyński [Reference Pełczyński25] proved that $\ell _2$ is a complemented subspace of $L_p$ for any $1<p<\infty $ . Lindenstrauss and Pełczyński [Reference Lindenstrauss and Pełczyński17] completed the picture by proving that any unconditional basic sequence complemented in $L_1$ is equivalent to the $\ell _1$ -basis.

Kadets and Pełczyński [Reference Kadets and Pełczyński13] carried out a systematic study of basic sequences in Lebesgue spaces. Among other results, they proved that, if $2<p<\infty $ , the canonical bases of $\ell _2$ and $\ell _p$ are the only subsymmetric basic sequences of $L_p$ . In contrast, if $1\le p < 2$ , there exist symmetric basic sequences in $L_p$ other than the canonical $\ell _q$ -basis for $p\le q\le 2$ . Indeed, Bretagnolle and Dacunha-Castelle [Reference Bretagnolle and Dacunha-Castelle8] proved the following result (also see [Reference Lindenstrauss and Tzafriri18, Reference Schütt27]).

Theorem 1.1 (See [Reference Bretagnolle and Dacunha-Castelle8])

Let $1\le p\le 2$ . A Banach space $\mathbb {S}$ with a symmetric basis isomorphically embeds into $L_p$ if and only if $\mathbb {S}=\ell _F$ for some Orlicz function F with

(1.1) $$ \begin{align} p \le \frac{tF'(t)}{F(t)} \le 2, \quad 0 <t \le 1. \end{align} $$ StartLayout 1st Row p less than or equals StartFraction t upper F prime left parenthesis t right parenthesis Over upper F left parenthesis t right parenthesis EndFraction less than or equals 2 comma 0 less than t less than or equals 1 period EndLayout

The article that initiated the study of variable exponent Lebesgue spaces by basic sequence techniques was [Reference Hernández and Ruiz11] (see also [Reference Flores, Hernández, Ruiz and Sanchiz10]). In it, the authors characterized in terms of the essential range of the variable exponent $\boldsymbol {P}$ the indices $q\in [1,\infty )$ such that $\ell _q$ isomorphically embeds into a variable exponent Lebesgue $L_{\boldsymbol {P}}$ over the unit interval. Hernández and Ruiz also proved that $\ell _q$ complementably embeds into $L_{\boldsymbol {P}}$ provided that q belongs to the essential range of $\boldsymbol {P}$ .

In this article, we complement the study carried out in [Reference Hernández and Ruiz11] by proving that, if $q\not =2$ , being q in the essential range of $\boldsymbol {P}$ is a necessary condition for $\ell _q$ to be a complemented subspace of $L_{\boldsymbol {P}}$ . Besides, we widen the scope of the study by considering not only $\ell _q$ -spaces but also subsymmetric basic sequences. Specifically, we characterize both the subsymmetric basic sequence structure and the complemented subsymmetric basic sequence structure of Lebesgue spaces with variable exponent essentially bounded. Oddly enough, we find out that, while symmetric basic sequences other than $\ell _q$ -bases are contained in these spaces, they fail to be complemented unless they are $\ell _q$ -bases. Also, we go forward with the study of variable exponent atomic spaces, that is, Nakano spaces, initiated in [Reference Peirats and Ruiz24].

To conclude this introductory section, we outline the structure of the article. In Section 2, we establish the notation about variable Lebesgue spaces that will be used throughout the article. We see these spaces from the perspective of Musielak–Orlicz spaces, including nonconvex spaces. Without intending to tackle an exhaustive study of these spaces, we sketch the more relevant ideas to our purposes. We revisit some well-known results from our perspective and also prove new ones. In Section 3, we study certain Orlicz functions (those that arise as convex combinations of power functions) that naturally appear when studying the geometry of variable Lebesgue spaces. The study of subsymmetric basic sequences in these spaces is addressed in Sections 4 and 5, which contain the main results of the article. While in Section 5, we study complemented basic sequences, in Section 4, we give the results achieved without imposing complementability.

2 Musielak–Orlicz function spaces

Given a measure space $(\Omega ,\Sigma ,\mu )$ , we denote by $\Sigma (\mu )$ the set of all measurable sets with finite measure. Let $L(\mu )$ be the vector space of all measurable functions with values in the real or complex field $\mathbb {F}$ . As it is customary, we identify functions and sets that differ on a null set. We say that $\mu $ is separable if the metric space $(\Sigma (\mu ),d_\mu )$ is separable, where $d_\mu $ is the distance given by

$$\begin{align*}d_\mu(A,B)=\mu(A \triangle B), \quad A,\, B\in\Sigma(\mu). \end{align*}$$

We endow $L(\mu )$ with the topology associated with the convergence in measure on finite-measure sets, so $L(\mu )$ becomes a complete topological vector space. We denote by $\mathcal {S}(\mu )$ the vector space of all simple integrable functions, that is, the linear span of

$$\begin{align*}\left\lbrace\chi_E \colon E\in\Sigma(\mu)\right\rbrace. \end{align*}$$

A function space over $(\Omega ,\Sigma ,\mu )$ will be a quasi-Banach lattice $(\mathbb {X},\left \lVert \cdot \right \rVert _{\mathbb {X}})$ such that

• $\mathbb {X}\subseteq L(\mu )$ continuously,
• $\mathcal {S}(\mu )\subseteq \mathbb {X}$ , and
• there is a constant $\boldsymbol {C}\in [0,\infty )$ such that if a sequence $(f_n)_{n=1}^\infty $ in $\mathbb {X}$ converges to $f\, \mu $ -a.e. and $S:=\limsup _n \left \lVert f_n\right \rVert _{\mathbb {X}}<\infty $ , then $f\in \mathbb {X}$ and $\left \lVert f\right \rVert _{\mathbb {X}}\le \boldsymbol {C} S$ .

Let $L^+(\mu )$ be the cone of all measurable functions with values in $[0,\infty ]$ . Following [Reference Ansorena and Bello1], a function quasi-norm will be a homogeneous map

$$\begin{align*}\rho\colon L^+(\mu) \to [0,\infty] \end{align*}$$

such that

(F.1) If $f\le g\, \mu $ -a.e., then $\rho (f)\le \rho (g)$ .
(F.2) There is $\boldsymbol {k}\in [1,\infty )$ , such that, for all f, $g\in L^+(\mu )$ ,
$$\begin{align*}\rho(f+g)\le\boldsymbol{k} \left( \rho( f)+\rho( g)\right). \end{align*}$$
(F.3) $\rho ( \chi _E)<\infty $ for all $E\in \Sigma (\mu )$ .
(F.4) For every $E\in \Sigma (\mu )$ and $\varepsilon>0$ there is $\delta>0$ such that $\mu (A)\le \varepsilon $ whenever $A\in \Sigma $ satisfies $A\subseteq E$ and $\rho (\chi _A)\le \delta $ .
(F.5) There is $\boldsymbol {C} \in [1,\infty )$ such that $\rho (\lim _n f_n)\le \boldsymbol {C} \lim _n \rho (f_n)$ for all non-decreasing sequences $(f_n)_{n=1}^\infty $ in $L^+(\mu )$ .

Let $0<r\le 1$ . If the function quasi-norm $\rho $ satisfies

(F.6) $\rho ^r (f+g)\le \rho ^r( f)+\rho ^r( g)$ for all f, $g\in L^+(\mu )$

we call it a function r-nom (function norm if $r=1$ ). Note that (F.6) implies (F.2) with $\boldsymbol {k}=2^{1/r-1}$ .

Given a function quasi-norm $\rho $ the set

$$\begin{align*}\boldsymbol{L}_{\rho}:=\left\lbrace f\in L(\mu) \colon \left\lVert f\right\rVert_\rho := \rho\left(\left\lvert f\right\rvert\right)<\infty\right\rbrace \end{align*}$$

endowed with the quasi-norm $\left \lVert \cdot \right \rVert _\rho $ is a function space. The other way around, if $(\mathbb {X}, \left \lVert \cdot \right \rVert _{\mathbb {X}})$ is a function space, then $\mathbb {X}=\boldsymbol {L}_{\rho }$ , where $\rho $ is defined by $\rho (f)=\left \lVert f\right \rVert _{\mathbb {X}}$ if $f\in \mathbb {X}$ and $\rho (f)=\infty $ otherwise.

Given $0<r<\infty $ , a function quasi-norm $\rho ,$ and a family $\mathcal { F}=(f_j)_{j\in J}$ in $L^+(\mu ),$ we set

$$\begin{align*}N_{\rho}(\mathcal{ F};r)=\rho\left(\left(\sum_{j\in J} f_j^r\right)^{1/r}\right), \quad M_{\rho}(\mathcal{ F};r)=\left(\sum_{j\in J} \rho^r(f_j)\right)^{1/r}. \end{align*}$$

StartLayout 1st Row upper N Subscript rho Baseline left parenthesis script upper F semicolon r right parenthesis equals rho left parenthesis left parenthesis sigma summation Underscript j element of upper J Endscripts f Subscript j Superscript r Baseline right parenthesis Superscript 1 divided by r Baseline right parenthesis comma upper M Subscript rho Baseline left parenthesis script upper F semicolon r right parenthesis equals left parenthesis sigma summation Underscript j element of upper J Endscripts rho Superscript r Baseline left parenthesis f Subscript j Baseline right parenthesis right parenthesis Superscript 1 divided by r Baseline period EndLayout

We say that $\mathbb {X}=\boldsymbol {L}_\rho $ is lattice r-convex (resp. r-concave) if there is a constant C such that $N_{\rho }(\mathcal { F};r)\le C M_{\rho }(\mathcal { F};r)$ (resp. $M_{\rho }(\mathcal { F};r) \le C N_{\rho }(\mathcal { F};r)$ ) for all families $\mathcal { F}$ in $L^+(\mu )$ . We say that $\mathbb {X}$ is L-convex if there is $\varepsilon>0$ such that $\sup _j \rho (f_j) \ge \varepsilon $ provided that $(f_j)_{j\in J}$ is a finite family in $L^+(\mu )$ for which there is $f\in L^+(\mu )$ such that $\rho (f)=1$ , $f_j\le f$ for all $j\in J$ , and

$$\begin{align*}\frac{1}{\left\lvert J\right\rvert} \sum_{j\in J} f_j\ge (1-\varepsilon)f. \end{align*}$$ StartFraction 1 Over StartAbsoluteValue upper J EndAbsoluteValue EndFraction sigma summation Underscript j element of upper J Endscripts f Subscript j Baseline greater than or equals left parenthesis 1 minus epsilon right parenthesis f period

Since any quasi-Banach is locally p-convex for some $p\in (0,1]$ by the Aoki–Rolewicz theorem (see [Reference Aoki4, Reference Rolewicz26]), the function space $\mathbb {X}$ is L-convex if and only if it is r-convex for some $r\in (0,\infty )$ (see [Reference Kalton14, Theorem 2.2]). If $\mathbb {X}$ is r-concave for some $r\in (0,\infty )$ , we say that $\mathbb {X}$ is L-concave. We say that $\mathbb {X}$ is absolutely continuous if

(F.7) $\lim _n \rho (f_n)=0$ for all non-increasing sequences $(f_n)_{n=1}^\infty $ in $L^+(\mu )$ with $\rho (f_1)<\infty $ and $\lim _n f_n=0$ .

Let $\mathbb {X}_0$ be the closure of $\mathcal {S}(\mu )$ in $\mathbb {X}$ . We say that $\mathbb {X}$ is minimal if $\mathbb {X}_0=\mathbb {X}$ . If $\mathbb {X}$ is L-concave, then $\mathbb {X}$ is absolutely continuous. In turn, if $\mathbb {X}$ is absolutely continuous, then $\mathbb {X}$ is minimal (see, e.g., [Reference Ansorena and Bello2]).

Given a function quasi-norm $\rho $ on $L^+(\mu ),$ we consider the gauge $\rho ^*$ defined by

$$\begin{align*}\rho^*(g)=\sup\left\lbrace \int_\Omega fg \, d\mu \colon f\in L^+(\mu), \, \rho(f)\le 1\right\rbrace, \quad g\in L^+(\mu). \end{align*}$$

(F.8) for every $E\in \Sigma (\mu ),$ there is $C_E\in (0,\infty )$ such that
$$\begin{align*}\int_E f\, d\mu\le C_E \rho(f), \quad f\in L^+(\mu), \end{align*}$$

then $\rho ^*$ is a function norm. Note that (F.8) implies (F.4) and that $\rho ^*$ always satisfies (F.8) as well. We say that $\rho ^*$ is the dual function norm of $\rho $ . The function space built from $\rho ^*$ is called the conjugate space of $\mathbb {X}=\boldsymbol {L}_\rho $ and denoted by $\mathbb {X}'$ . If $\mathbb {X}$ is locally convex, then $\mathbb {X}"=\mathbb {X}$ (see [Reference Bennett and Sharpley6]). We can regard $\mathbb {X}'$ as a closed subspace of the dual space $\mathbb {X}^*$ . If $\mathbb {X}$ is absolutely continuous, then $\mathbb {X}'=\mathbb {X}^*$ (see [Reference Bennett and Sharpley6]).

For $j=1$ , $2$ , let $\mathbb {X}_j$ be a function space over a $\sigma $ -finite measure space $(\Omega _j,\Sigma _j,\mu _j)$ . Let

$$\begin{align*}K\colon \Omega_1\times \Omega_2 \to [0,\infty) \end{align*}$$

be a measurable function. Suppose there is a bounded linear operator $T\colon \mathbb {X}_1 \to \mathbb {X}_2$ given by

$$\begin{align*}T(f)(\omega_2)=\int_{\Omega_1} K(\omega_1,\omega_2) f (\omega_1) \, d \mu_1(\omega_1), \quad f\in \mathbb{X}_1, \, \omega_2\in \Omega_2. \end{align*}$$

Then, there is a bounded linear operator $T'\colon \mathbb {X}_2' \to \mathbb {X}_1'$ given by

$$\begin{align*}T'(g)(\omega_1)=\int_{\Omega_2} K(\omega_1,\omega_2) g (\omega_2) \, d \mu_2(\omega_2), \quad g\in \mathbb{X}_2, \, \omega_1\in \Omega_1. \end{align*}$$

We say that T is a kernel operator and that $T'$ is the adjoint operator of T.

Given a measurable function $f\colon \Omega \to G$ , where G is a topological commutative monoid (for instance, $G=[0,\infty ]$ or G a vector space), we set

$$\begin{align*}\operatorname{\mathrm{supp}}(f)= \Omega\setminus f^{-1}(0). \end{align*}$$

Given a function space $\mathbb {X}$ over a $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ , and a pairwise disjointly supported sequence $\mathcal {X}=(x_n)_{n=1}^\infty $ in $\mathbb {X}\setminus \{0\}$ , we set

$$\begin{align*}\mathbb{S}[\mathbb{X},\mathcal{X}]=\left\lbrace(a_n)_{n=1}^\infty \colon \sum_{n=1}^\infty a_n \, x_n \in \mathbb{X}\right\rbrace, \end{align*}$$ double struck upper S left bracket double struck upper X comma script upper X right bracket equals StartSet left parenthesis a Subscript n Baseline right parenthesis Subscript n equals 1 Superscript infinity Baseline colon sigma summation Underscript n equals 1 Overscript infinity Endscripts a Subscript n Baseline x Subscript n Baseline element of double struck upper X EndSet comma

and we endow $\mathbb {S}[\mathbb {X},\mathcal {X}]$ with the topology it inherits from $\mathbb {X}$ .

An Orlicz function will be a non-decreasing left-continuous function

$$\begin{align*}F\colon[0,\infty)\to[0,\infty] \end{align*}$$

such that $\lim _{t\to 0^+} F(t)=0$ and $F(\infty ):=\lim _{t\to \infty } F(t)>0$ . Note that we allow Orlicz functions to take the value $\infty $ . If it is the case for the Orlicz function F, then there is $0<c<\infty $ such that $F^{-1}(\infty )=(c,\infty )$ .

Given a $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ , a function

$$\begin{align*}M\colon\Omega\times [0,\infty)\to [0,\infty], \end{align*}$$

and $t\in [0,\infty )$ , we denote by $\nu _M(\cdot ,t)$ the measure on $(\Omega ,\Sigma )$ given by

$$\begin{align*}\nu_M(A,t)=\int_A M(\omega,t)\, d\mu(\omega), \quad A\in\Sigma. \end{align*}$$

We say that M is a Musielak–Orlicz function (cf. [Reference Musielak21, Chapter 7]) if

• $M(\omega ,\cdot )$ is an Orlicz function for all $\omega \in \Omega $ , and
• for every $E\in \Sigma (\mu ),$ there exists $t_E\in (0,\infty )$ such that $\nu _M(E,t_E)<\infty $ and $M(\omega ,t_E)>0$ for all $\omega \in E$ .

If $f\in L^+(\mu )$ and M is a Musielak–Orlicz function, then $M(\cdot ,f(\cdot ))$ is a measurable function. Hence, associated with $M,$ there is a gauge, usually called a modular,

$$\begin{align*}\rho_M\colon L^+(\mu)\to[0,\infty], \quad f\mapsto \int_\Omega M(\omega,f(\omega))\, d\mu(\omega). \end{align*}$$

Consider the sets

$$ \begin{align*} L^+_M&=\left\lbrace f\in L^+(\mu) \colon \rho_M(tf)<\infty \text{ for some } t\in(0,\infty)\right\rbrace,\\ L_M&=\left\lbrace f\in L(\mu) \colon \left\lvert f\right\rvert \in L_M^+\right\rbrace,\\ H^+_M&=\left\lbrace f\in L^+(\mu) \colon \rho_M(tf)<\infty \text{ for all } t\in(0,\infty)\right\rbrace, \\ H_M&=\left\lbrace f\in L(\mu) \colon \left\lvert f\right\rvert \in H_M^+\right\rbrace, \end{align*} $$

StartLayout 1st Row 1st Column upper L Subscript upper M Superscript plus 2nd Column equals StartSet f element of upper L Superscript plus Baseline left parenthesis mu right parenthesis colon rho Subscript upper M Baseline left parenthesis t f right parenthesis less than infinity for some t element of left parenthesis 0 comma infinity right parenthesis EndSet comma 2nd Row 1st Column upper L Subscript upper M 2nd Column equals StartSet f element of upper L left parenthesis mu right parenthesis colon StartAbsoluteValue f EndAbsoluteValue element of upper L Subscript upper M Superscript plus Baseline EndSet comma 3rd Row 1st Column upper H Subscript upper M Superscript plus 2nd Column equals StartSet f element of upper L Superscript plus Baseline left parenthesis mu right parenthesis colon rho Subscript upper M Baseline left parenthesis t f right parenthesis less than infinity for all t element of left parenthesis 0 comma infinity right parenthesis EndSet comma 4th Row 1st Column upper H Subscript upper M 2nd Column equals StartSet f element of upper L left parenthesis mu right parenthesis colon StartAbsoluteValue f EndAbsoluteValue element of upper H Subscript upper M Superscript plus Baseline EndSet comma EndLayout

and, given $\varepsilon>0$ ,

$$\begin{align*}B_M(\varepsilon)=\left\lbrace f\in L(\mu) \colon \rho_M(\left\lvert f\right\rvert)<\varepsilon\right\rbrace. \end{align*}$$

The modular $\rho =\rho _M$ satisfies properties (F.1), (F.4), (F.5), and (F.7), property (F.9) below instead of (F.2), and property (F.10) below instead of (F.3).

(F.9) There are $\boldsymbol {k}$ and $\boldsymbol {r}$ in $(0,\infty )$ such that, for all f, $g\in L^+(\mu )$ ,
$$\begin{align*}\rho(f+g)\le\boldsymbol{k} \left( \rho( \boldsymbol{r} f)+\rho(\boldsymbol{r} g)\right). \end{align*}$$
(F.10) For every $E\in \Sigma (\mu ),$ there is $u_E\in (0,\infty )$ such that $\rho (u_E \chi _E)<\infty $ .

In fact, we can choose $\boldsymbol {C}=1$ in (F.5), $\boldsymbol {k}=1$ and $\boldsymbol {r}=2$ in (F.9), and $u_E=t_E$ in (F.10). Consequently,

$$\begin{align*}u B_M(\varepsilon), \quad u,\, \varepsilon\in(0,\infty), \end{align*}$$

is a neighborhood basis at the origin for a complete vector topology on $L_M$ . Besides, the F-space $L_M$ continuously embeds into $L(\mu )$ . We say that $L_M$ is the Musielak–Orlicz space over $(\Omega ,\Sigma ,\mu )$ built from M. We denote by $L_M^0$ the closed subspace of $L_M$ generated by simple integrable functions.

By property (F.7), the dominated convergence theorem holds in $L_M$ for sequences dominated by a function in $H_M^+$ . We infer that $H_M$ is the closed linear span of

$$\begin{align*}\left\lbrace\chi_E \colon E \in\Sigma(\mu), \chi_E\in H^+_{M}\right\rbrace. \end{align*}$$

Hence, $H_M\subseteq L_M^0\subseteq L_M$ , and $L_M^0=H_{M}$ if and only if $\chi _E\in H^+_{M}$ for all $E\in \Sigma (\mu )$ . Consequently, $H_{M}=L_M^0$ provided that

(F.11) there are C, $D\in (1,\infty )$ such that
$$ \begin{align*} M(\omega,Dt)\le C M(\omega,t), \quad (\omega,t)\in\Omega\times[0,\infty). \end{align*} $$

(F.12) there are c, $d\in (0,1)$ such that
$$ \begin{align*} M(\omega,dt)\le c M(\omega,t), \quad (\omega,t)\in\Omega\times[0,\infty), \end{align*} $$

then $L_M$ is a function space. Indeed, the Luxemburg map

$$\begin{align*}f\mapsto \rho^L_M(f):= \inf\left\lbrace t\in(0,\infty) \colon \rho_M\left(f/t\right) \le 1\right\rbrace, \quad f\in L^+(\mu), \end{align*}$$

besides being homogeneous, inherits from $\rho _M$ the properties (F.1), (F.4), (F.5), (F.9), and (F.10). Hence, $\rho _M^L$ is a function quasi-norm such that $L_M=\boldsymbol {L}_{\rho ^L_M}$ .

Let $0<r<\infty $ and M be a Musielak–Orlicz function. If M is r-convex, that is, the mapping

$$\begin{align*}t \mapsto M\left(\omega, t^{1/r}\right), \quad t\in[0,\infty), \end{align*}$$

defines a convex function for all $\omega \in \Omega $ , then (F.12) holds, and the function space $L_M$ is lattice r-convex. Let $0<s<\infty $ . If M is s-concave, that is, the mapping

$$\begin{align*}t\mapsto M\left(\omega, t^{1/s}\right), \quad t\in[0,\infty), \end{align*}$$

defines a concave function for all $\omega \in \Omega $ , then it satisfies (F.11). Therefore, if M is s-concave and satisfies (F.12), then $L_M$ is a lattice s-concave absolutely continuous function space, and $H_M=L_M$ .

We will use the following two results, whose straightforward proofs we omit.

Lemma 2.1 Let M be a Musielak–Orlicz function satisfying (F.12). Let $f\in L_M\setminus \{0\}$ and $a=1/\left \lVert f\right \rVert _M$ . Then, either

(a) $\rho _M(a\left \lvert f\right \rvert )=1$ or
(b) $\rho _M(a\left \lvert f\right \rvert )<1$ and $\rho _M (t\left \lvert f\right \rvert )=\infty $ for all $t\in (a,\infty )$ .

In particular, (a) holds for all $f\in H_M$ .

Theorem 2.2 (See [Reference Ansorena and Marcos3])

Let M and N be Musielak–Orlicz functions on a $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ . Suppose that there is $c\in (0,\infty )$ such that

• $\max \left \lbrace \nu _M(\Omega ,c),\nu _N(\Omega ,c)\right \rbrace <\infty $ ,
• $\min \left \lbrace M(\omega ,c), N(\omega ,c)\right \rbrace>0$ for all $\omega \in \Omega $ , and
• $M(\omega ,t)\le N(\omega ,t)$ for all $\omega \in \Omega $ and $t\in [c,\infty )$ .

Then $L_N(\mu )\subseteq L_M(\mu )$ continuously.

Next, we detail three types of Musielak–Orlicz spaces we will deal with.

2.1 Musielak–Orlicz sequence spaces

We regard Musielak–Orlicz functions over $\mathbb {N}$ , which we call Musielak–Orlicz sequences, as sequences of Orlicz functions. Given a Musielak–Orlicz sequence $\boldsymbol {F}=(F_n)_{n=1}^\infty $ , we denote by $\ell _{\boldsymbol {F}}$ the Musielak–Orlicz sequence space built from $\boldsymbol {F}$ . These Musielak–Orlicz atomic spaces naturally appear when studying the unconditional basic sequence structure of Musielak–Orlicz function spaces. Indeed, if M is a Musielak–Orlicz function over a $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ , and $\mathcal {X}=(x_n)_{n=1}^\infty $ is a pairwise disjointly supported sequence in $L_M\setminus \{0\}$ , then $\mathbb {S}[L_M,\mathcal {X}]=\ell _{\boldsymbol {F}}$ , where $\boldsymbol {F}=(F_n)_{n=1}^\infty $ is given by

$$\begin{align*}F_n(t)=\int_{\Omega} M\left(\omega, t \left\lvert x_n(\omega)\right\rvert\right) \, d\mu(\omega), \quad n\in\mathbb{N}. \end{align*}$$

We bring up a result in Musielak’s memoir [Reference Musielak21] that we will use.

Theorem 2.3 [Reference Musielak21, Theorem 8.11]

Consider two Musielak–Orlicz sequences $\boldsymbol {F}=(F_n)_{n=1}^\infty $ and $\boldsymbol {G}=(G_n)_{n=1}^\infty $ . Then, $\ell _{\boldsymbol {F}} \subseteq \ell _{\boldsymbol {G}}$ continuously if and only if there exist $\boldsymbol {a}=(a_n)_{n=1}^\infty \in \ell _1$ and $\delta ,b,$ and C in $(0,\infty )$ such that

$$\begin{align*}G_n(t) \le a_n+ C F_n(bt) \end{align*}$$

for all $n\in \mathbb {N}$ and all $t\in (0,\infty )$ with $F_n(t)< \delta $ .

2.2 Orlicz sequence spaces

Given an Orlicz function F and a measure space $(\Omega ,\Sigma ,\mu )$ , the Orlicz space $L_F(\mu )$ is the Musielak–Orlicz space built from the Musielak–Orlicz function M over $(\Omega ,\Sigma ,\mu )$ given by $M(\omega ,t)=F(t)$ for all $\omega \in \Omega $ and $t\in [0,\infty )$ . If $\mu $ is the counting measure on $\mathbb {N}$ , we set $\ell _F=L_F(\mu )$ , and we call $\ell _F$ the Orlicz sequence space built from F. Orlicz sequence spaces are Musielak–Orlicz sequence spaces built from constant Musielak–Orlicz sequences. Applying Theorem 2.3 in this particular situation yields the following well-known result.

Corollary 2.4 Given two Orlicz functions F and G, then $\ell _F\subseteq \ell _G$ if and only if there are b, $c,$ and C in $(0,\infty )$ such that

$$\begin{align*}G(t)\le C F(bt), \quad 0\le t \le c. \end{align*}$$

Given a convex Orlicz function F, we define its conjugate function $F^*\colon [0,\infty )\to [0,\infty ]$ as the optimal function G such that

$$\begin{align*}uv\le F(u)+G(v), \quad u,v\in[0,\infty]. \end{align*}$$

$F^*$ is a convex Orlicz function, and the conjugate space of $L_F(\mu )$ is $L_{F^*}(\mu )$ . We record some properties of the mapping $F\mapsto F^*$ that we will need.

Given $p\in (0,\infty ],$ we consider the power Orlicz functions

$$\begin{align*}\boldsymbol{\Phi}_p,\, \boldsymbol{\Psi}_p\colon[0,\infty)\to [0,\infty), \quad \boldsymbol{\Phi}_p(t)= t^p, \, \boldsymbol{\Psi}_p(t)= \frac{ t^p}{p}, \end{align*}$$

with the convention that $1^\infty =0$ . If $p\ge 1$ , so that $\boldsymbol {\Phi }_p$ and $\boldsymbol {\Psi }_p$ are convex, we denote by $p'$ its conjugate exponent defined by $1/p+1/p'=1$ . Note that $\boldsymbol {\Phi }_\infty =\boldsymbol {\Psi }_\infty =\infty \chi _{(1,\infty )} $ .

We record a couple of lemmas about Orlicz functions. The first of them is clear and well-known.

Lemma 2.5 $(\boldsymbol {\Psi }_p)^*=\boldsymbol {\Psi }_{p'}$ for all $p\in [1,\infty ]$ .

Lemma 2.6 Let F be a convex Orlicz function. Let c, C, $A,$ and v in $(0,\infty )$ be such that

$$\begin{align*}uv \le C F(u) + A, \quad 0\le u \le c. \end{align*}$$

Then, $F^*(w)\le A/C$ , where

$$\begin{align*}w=\min{\left\lbrace\frac{v}{C}, \frac{F(c)}{c}\right\rbrace}. \end{align*}$$ w equals min left brace StartFraction v Over upper C EndFraction comma StartFraction upper F left parenthesis c right parenthesis Over c EndFraction right brace period

Proof On the one hand, if $u\in [0,c]$ ,

$$\begin{align*}uw\le \frac{uv}{C}\le F(u)+\frac{A}{C}. \end{align*}$$

On the other hand, if $u\in [c,\infty )$ , by convexity,

$$\begin{align*}u w \le u \frac{F(c)}{c} \le F(u) \le F(u)+ \frac{A}{C}. \\[-42pt]\end{align*}$$

2.3 Variable exponent Lebesgue spaces

A variable exponent on a measure space $(\Omega ,\Sigma ,\mu )$ will be a measurable function

$$\begin{align*}\boldsymbol{P}\colon \Omega\to (0,\infty]. \end{align*}$$

The variable exponent Lebesgue space $L_{\boldsymbol {P}}$ associated with $\boldsymbol {P}$ will be the F-space over $\Omega $ built from the Musielak–Orlicz function

$$\begin{align*}(\omega,t)\mapsto M_{\boldsymbol{P}}(\omega,t)= \boldsymbol{\Phi}_{\boldsymbol{P}(\omega)}(t), \quad (\omega,t)\in\Omega\times[0,\infty), \end{align*}$$

that is, $L_{\boldsymbol {P}}=L_{M_{\boldsymbol {P}}}$ . We denote by $\rho _{\boldsymbol {P}}:=\rho _{M_{\boldsymbol {P}}}$ the modular constructed from $M_{\boldsymbol {P}}$ . If the implicit measure $\mu $ over $(\Omega ,\Sigma )$ we are considering could be in doubt, we put $L_{\boldsymbol {P}}=L_{\boldsymbol {P}}(\mu )$ . Set also $H_{\boldsymbol {P}}=H_{M_{\boldsymbol {P}}}$ .

Recall that the support of a Borel measure $\nu $ on a Lindelöf space X, denoted by $\operatorname {\mathrm {supp}}(\nu )$ , is the smallest closed set K such that $\nu (X\setminus K)=0$ . Let $\mu \circ \boldsymbol {P}^{-1}$ be the pushforward of $\mu $ by $\boldsymbol {P}$ . Let

$$\begin{align*}R(\boldsymbol{P}):=\operatorname{\mathrm{supp}}\left(\mu\circ \boldsymbol{P}^{-1}\right)\subseteq[0,\infty] \end{align*}$$

be the essential range of $\boldsymbol {P}$ , that is, the set of all $p\in [0,\infty ]$ such that

$$\begin{align*}\mu\left(\boldsymbol{P}^{-1}(V)\right)>0 \end{align*}$$

for all measurable neighborhoods V of p. Set

$$\begin{align*}\boldsymbol{P}^-:=\min\left(R(\boldsymbol{P})\right), \quad \boldsymbol{P}^+=\max\left(R(\boldsymbol{P})\right). \end{align*}$$

Suppose that $\boldsymbol {P}^->0$ . Then, $M_{\boldsymbol {P}}$ is $\boldsymbol {P}^-$ -convex, whence $L_{\boldsymbol {P}}$ is a lattice $\boldsymbol {P}^-$ -convex function space over $(\Omega ,\Sigma ,\mu )$ . We denote by $\rho ^L_{\boldsymbol {P}}$ the function quasi-norm constructed from $\rho _{\boldsymbol {P}}$ .

We will also consider the Musielak–Orlicz function over $(\Omega ,\Sigma ,\mu )$ given by

$$\begin{align*}(\omega,t) \mapsto M^a_{\boldsymbol{P}}(\omega,t):= \boldsymbol{\Psi}_{\boldsymbol{P}(\omega)}(t). \end{align*}$$

Let $\rho ^a_{\boldsymbol {P}}$ be the modular associated with $M^a_{\boldsymbol {P}}$ . In turn, let $\rho _{\boldsymbol {P}}^{L,a}$ be the function quasi-norm constructed from $\rho ^a_{\boldsymbol {P}}$ . Since $p^{1/p}$ is bounded away from zero at infinity as long as p is bounded away from zero, the function quasi-norms $\rho ^L_{\boldsymbol {P}}$ and $\rho _{\boldsymbol {P}}^{L,a}$ are equivalent. In fact,

$$\begin{align*}\boldsymbol{P}^{-} \rho^{L,a}_{\boldsymbol{P}} \le \rho^L_{\boldsymbol{P}} \le \sup\left\lbrace p^{1/p} \colon \boldsymbol{P}^{-}\le p <\infty\right\rbrace \rho^{L,a}_{\boldsymbol{P}}. \end{align*}$$

Thus, there are two natural quasi-norms $L_{\boldsymbol {P}}$ can be equipped with, namely, those constructed from $\rho ^L_{\boldsymbol {P}}$ and $\rho _{\boldsymbol {P}}^{L,a}$ . We denote by $\left \lVert \cdot \right \rVert _{\boldsymbol {P}}$ the one constructed from the former function quasi-norm, that is, $\left \lVert f\right \rVert _{\boldsymbol {P}}=\rho _{\boldsymbol {P}}^L\left (\left \lvert f\right \rvert \right )$ for all $f\in L_{\boldsymbol {P}}$ .

Both $\rho ^L_{\boldsymbol {P}}$ and $\rho ^{L,a}_{\boldsymbol {P}}$ satisfy (F.6) with

$$\begin{align*}r=\boldsymbol{P}^c:=\min\{1,\boldsymbol{P}^{-}\}. \end{align*}$$

In other words, $\rho ^L_{\boldsymbol {P}}$ and $\rho ^{L,a}_{\boldsymbol {P}}$ are function $\boldsymbol {P}^c$ -norms.

When dealing with variable exponent Lebesgue spaces, we must be aware that the identity

$$\begin{align*}\boldsymbol{\Psi}_p(st) = \boldsymbol{\Phi}_p(s) \boldsymbol{\Psi}_p(t) \end{align*}$$

breaks down when $p=\infty $ and $0\le s <1 <t <\infty $ . So, we must pay attention to the set

$$\begin{align*}\Omega^{\boldsymbol{P}}_\infty:=\boldsymbol{P}^{-1}(\infty). \end{align*}$$

In fact, for any $f\in L^+(\mu )$ , if $\left \lVert \cdot \right \rVert _{\infty }$ denotes the norm of $L_\infty ({\left . \mu \right |{}_{{\Omega ^{\boldsymbol {P}}_\infty }}})$ ,

$$\begin{align*}\rho^L_{\boldsymbol{P}}(f)=\max\left\lbrace \left\lVert{\left. f \right|{}_{{\Omega^{\boldsymbol{P}}_\infty}}}\right\rVert_{\infty} , \inf \left\lbrace t>0 \colon \int_{\Omega\setminus\Omega^{\boldsymbol{P}}_\infty} \frac{{f(\omega)}^{\boldsymbol{P}(\omega)}}{t^{\boldsymbol{P}(\omega)}} d\mu(\omega) \le 1\right\rbrace\right\rbrace. \end{align*}$$

If $\boldsymbol {P}^+<\infty $ , then $L_{\boldsymbol {P}}$ is $\boldsymbol {P}^+$ -concave. Consequently, if $\boldsymbol {P}^->0$ and $\boldsymbol {P}^+<\infty $ , then $L_{\boldsymbol {P}}$ is lattice $\boldsymbol {P}^+$ -concave, whence $H_{\boldsymbol {P}}=L_{\boldsymbol {P}}$ and $L_{\boldsymbol {P}}$ is absolutely continuous.

If $\mu $ is a finite measure and $\boldsymbol {P}$ and $\boldsymbol {Q}$ are variable exponents over $(\Omega ,\Sigma ,\mu )$ with $\boldsymbol {P}\le \boldsymbol {Q}$ , then $L_{\boldsymbol {Q}} \subseteq L_{\boldsymbol {P}}$ continuously by Theorem 2.2.

Given a measurable function $h\colon \Omega \to (0,\infty )$ , the pointwise multiplier

$$\begin{align*}f\mapsto f h \end{align*}$$

defines an isomorphism from $L_{\boldsymbol {P}}(\nu )$ onto $L_{\boldsymbol {P}}(\mu )$ , where

$$\begin{align*}d \nu = h_{\boldsymbol{P}}\, d\mu, \quad h_{\boldsymbol{P}}=h^{\boldsymbol{P}} \chi_{\Omega\setminus\Omega^{\boldsymbol{P}}_\infty} + h \chi_{\Omega^{\boldsymbol{P}}_\infty}. \end{align*}$$

Since we can choose h so that $\int _\Omega h_{\boldsymbol {P}} \, d\mu =1$ , any variable exponent Lebesgue space is isomorphic to a variable exponent Lebesgue space over a probability space. If this probability space is nonatomic and separable, then, by [Reference Carathéodory9], the corresponding variable exponent Lebesgue space is isomorphic to a variable exponent Lebesgue space over $[0,1]$ .

To understand the structure of a variable exponent Lebesgue space we must take into account that if $(\Omega _j)_{j=1}^2$ is a partition of the measure space $(\Omega ,\Sigma ,\mu )$ then, for any variable exponent $\boldsymbol {P}\colon \Omega \to (0,\infty ]$ , setting $\boldsymbol {P}_j={\left . \boldsymbol {P} \right |{}_{{\Omega _j}}}$ and $\mu _j={\left . \mu \right |{}_{{\Omega _j}}}$ for $j=1$ , $2$ ,

(2.1) $$ \begin{align} L_{\boldsymbol{P}} \simeq L_{\boldsymbol{P}_1} \oplus L_{\boldsymbol{P}_2}. \end{align} $$

If J is a countable set and $\boldsymbol {Q}\colon J\to (0,\infty ]$ is a function, we denote by $\ell _{\boldsymbol {Q}}$ the variable exponent Lebesgue space over the counting measure on J associated with the variable exponent $\boldsymbol {Q}$ . These variable exponent Lebesgue atomic spaces were introduced by Nakano [Reference Nakano22] in the case when $\boldsymbol {Q}^{-}\ge 1$ , and by Bourgin [Reference Bourgin7] in the case when $\boldsymbol {Q}^{+}\le 1$ . In general, as is customary, we will call them Nakano sequence spaces.

Splitting $\sigma $ -finite measures into their purely atomic part and their nonatomic part, we infer that any variable exponent Lebesgue space is isomorphic to

$$\begin{align*}L_{\boldsymbol{P}}\oplus \ell_{\boldsymbol{Q}} \end{align*}$$

for some nonatomic probability measure space $(\Omega ,\Sigma ,\mu )$ , some countable set J and some variable exponents $\boldsymbol {P}\colon \Omega \to (0,\infty ]$ and $\boldsymbol {Q}\colon J \to (0,\infty ]$ . Summing up, any variable exponent Lebesgue space over any separable $\sigma $ -finite measure space is isomorphic to $L_{\boldsymbol {P}}\oplus \ell _{\boldsymbol {Q}}$ for some countable set J and some variable exponents $\boldsymbol {P}\colon [0,1]\to (0,\infty ]$ and $\boldsymbol {Q}\colon J\to (0,\infty ]$ .

The dual space of $L_{\boldsymbol {P}}$ only depends on the behavior of $\boldsymbol {P}$ on

$$\begin{align*}\Omega^{\boldsymbol{P}}_c=\boldsymbol{P}^{-1}([1,\infty]). \end{align*}$$

Indeed, we have the following.

Proposition 2.7 Let $\boldsymbol {P}\colon \Omega \to (0,1)$ be a variable exponent on a nonatomic $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ . Then, the dual space of $L_{\boldsymbol {P}}$ is null.

Proof Set $\Omega _{p,A}=\boldsymbol {P}^{-1}((0,p]) \cap A$ for each $p\in (0,1)$ and $A\in \Sigma (\mu )$ . Since

$$\begin{align*}\bigcup_{\substack{0<p<1\\ A\in \Sigma(\mu)}} \left\lbrace f\in L_{\boldsymbol{P}} \colon \operatorname{\mathrm{supp}}(f)\subseteq \Omega_{p,A}\right\rbrace \end{align*}$$ union Underscript StartLayout 1st Row 0 less than p less than 1 2nd Row upper A element of normal upper Sigma left parenthesis mu right parenthesis EndLayout Endscripts StartSet f element of upper L Subscript bold italic upper P Baseline colon supp left parenthesis f right parenthesis subset of or equal to normal upper Omega Subscript p comma upper A Baseline EndSet

is dense in $L_{\boldsymbol {P}}$ , it suffices to prove the result in the case when $\mu $ is finite and $\boldsymbol {P}(\Omega )\subseteq (0,p]$ for some $0<p<1$ . In this case, since $L_p(\mu )\subseteq L_{\boldsymbol {P}}$ continuously and $\mathcal {S}(\mu )$ is dense in $L_{\boldsymbol {P}}$ , the result follows from the fact that $(L_p(\mu ))^*=\{0\}$ .

We say that a variable exponent $\boldsymbol {P}$ over a $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ is convex if $\boldsymbol {P}(\Omega )\subseteq [1,\infty ]$ . The conjugate variable exponent of $\boldsymbol {P}$ is the convex variable exponent $\boldsymbol {Q}$ over $(\Omega ,\Sigma ,\mu )$ defined by

$$\begin{align*}\boldsymbol{Q}(\omega)=\left(\boldsymbol{P}(\omega)\right)'= \frac{\boldsymbol{P}(\omega)}{\boldsymbol{P}(\omega)-1}. \end{align*}$$ StartLayout 1st Row bold italic upper Q left parenthesis omega right parenthesis equals left parenthesis bold italic upper P left parenthesis omega right parenthesis right parenthesis prime equals StartFraction bold italic upper P left parenthesis omega right parenthesis Over bold italic upper P left parenthesis omega right parenthesis minus 1 EndFraction period EndLayout

The well-known duality relation between $L_{\boldsymbol {P}}$ and $L_{\boldsymbol {Q}}$ relies on the following lemma.

Lemma 2.8 Let $\boldsymbol {Q}$ be the conjugate variable exponent of a variable exponent $\boldsymbol {P}$ over a $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ . Let $g\in L^+(\mu )$ with $\rho ^L_{\boldsymbol {Q}}(g)=1$ and $\operatorname {\mathrm {supp}}(g)\subseteq \boldsymbol {P}^{-1}([p,\infty ])$ for some $p>1$ . Set $f={g^{\boldsymbol {Q}-1}}$ , with the convention that $0^0=0$ . Then,

$$\begin{align*}\int_\Omega f g \, d\mu= \rho_{\boldsymbol{P}} (f) = \rho_{\boldsymbol{Q}}(g)=1. \end{align*}$$

Proof Our conditions on g yields $g\in H^{+}_{\boldsymbol {Q}}$ , whence $\rho _{\boldsymbol {Q}}(g)=1$ by Lemma 2.1. Since $fg=f^{\boldsymbol {P}}=g^{\boldsymbol {Q}}$ , we are done.

With the information we have gathered, proving that the conjugate space of $L_{\boldsymbol {P}}$ is $L_{\boldsymbol {Q}}$ takes little effort. On the one hand, by Lemma 2.5,

$$\begin{align*}\int_\Omega fg\, d\mu \le 2, \quad f,g\in L^+(\mu), \, \max\left\lbrace\rho^{L,a}_{\boldsymbol{P}}(f) , \rho^{L,a}_{\boldsymbol{Q}} (g) \right\rbrace\le 1. \end{align*}$$

Therefore, the dual function norm $\rho ^*$ of $\rho _{\boldsymbol {Q}}^L$ satisfies $\rho ^*\le 2 \rho _{\boldsymbol {Q}}^{L,a}$ . On the other hand, by Lemma 2.8 and homogeneity, $\rho ^*(g)\ge \rho ^L_{\boldsymbol {Q}}(g)$ for all $g\in L^+(\mu )$ with $\operatorname {\mathrm {supp}}(g)\subseteq \boldsymbol {P}^{-1}([p,\infty ])$ for some $p>1$ . By the Fatou property, this inequality extends to any $g\in L^+(\mu )$ with $\operatorname {\mathrm {supp}}(g)\subseteq \boldsymbol {P}^{-1}((1,\infty ])$ . Since the conjugate space of $L_1(\Omega ^{\boldsymbol {Q}}_\infty )$ is $L_\infty (\Omega ^{\boldsymbol {Q}}_\infty )$ , the inequality extends to any $g\in L^+(\mu )$ .

We close this section with a result that connects variable exponent Lebesgue spaces to classical Lebesgue spaces. Given $0<p\le 2$ and $t\in (0,\infty )$ , we say that a random variable is p-stablewith parameter t if its characteristic function is

$$\begin{align*}v\mapsto e^{-t\left\lvert v\right\rvert^p}, \quad v\in\mathbb{R}. \end{align*}$$

Note that if $s\in (0,\infty )$ and X is a p-stable random variable with parameter t, then $sX$ is a p-stable random variable with parameter $s^p t$ . We say that a continuous-time stochastic process $(X_t)_{t\in [0,1]}$ is p-stable if $X_0=0$ a.e., it has stationary and independent increments, and, for all $0\le s <t\le 1$ , $X_t-X_s$ is a p-stable random variable with parameter $t-s$ .

Proposition 2.9 Let $\boldsymbol {P}$ be a variable exponent on a nonatomic $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ . Suppose that $\boldsymbol {P}^-<2$ . Then, $L_p$ is almost isometrically isomorphic to a subspace of $L_{\boldsymbol {P}}$ for all $p\in (\boldsymbol {P}^-,2]$ .

Proof For each $0<s<\infty $ , let $C_{p,s}\in (0,\infty ]$ be the s-norm of a p-stable random variable with parameter one. Pick $r:=\boldsymbol {P}^-<q<p$ . By (2.1), we can assume that $P(\omega )\in [\boldsymbol {P}^-,q]$ for all $\omega \in \Omega $ . We can also assume that $\mu $ is a probability measure and that there is a p-stable continuous-time stochastic process on $(\Omega ,\Sigma ,\mu )$ (see, e.g., [Reference Kanter15, Section 4]). Then, there is a linear map $T\colon L_p \to L(\mu )$ such that, for all $f\in L_p$ , $T(f)$ is a p-stable random variable with parameter $\left \lVert f\right \rVert _p^p$ (see, e.g., [Reference Kanter15, Section 5]). Therefore, $\left \lVert T(f)\right \rVert _s=C_{p,s} \left \lVert f\right \rVert _p$ for all $0<s<p$ and $f\in L_p$ . Since $L_q\subseteq L_{\boldsymbol {P}} \subseteq L_r$ and $C_{p,s}<\infty $ for all $0<s<p$ (see, e.g., [Reference Kanter15, Section 5]),

$$\begin{align*}\frac{1}{C_{p,q}} \left\lVert T(f)\right\rVert_{\boldsymbol{P}}\le \left\lVert f\right\rVert_p \le \frac{1}{C_{p,r}} \left\lVert T(f)\right\rVert_{\boldsymbol{P}}, \quad f\in L_p. \end{align*}$$

Consequently, $L_p$ is $(C_{p,q}/C_{p,r})$ -isomorphic to a subspace of $L_{\boldsymbol {P}}$ . Since

$$\begin{align*}\lim_{q\to r^+} C_{p,q}=C_{p,r}, \end{align*}$$

we are done.

3 Orlicz functions generated near zero by power functions

Given a signed measure $\mu $ on $(0,\infty ],$ we define

$$\begin{align*}\boldsymbol{\psi}_\mu\colon [0,1]\to \mathbb{R},\quad t \mapsto \int_{(0,\infty]} \boldsymbol{\Psi}_p(t) \, d\mu(p). \end{align*}$$

Let $\delta _p$ be the Dirac measure on $p\in (0,\infty ]$ . We have

$$\begin{align*}\boldsymbol{\psi}_p:=\boldsymbol{\psi}_{\delta_p}={\left. \boldsymbol{\Psi}_p \right|{}_{{[0,1]}}}. \end{align*}$$

We will use the elementary inequality

(3.1) $$ \begin{align} \frac{ v^s-u^s}{s} \le \frac{v^r-u^r}{r} , \quad 0<r\le s\le\infty, \, 0\le u\le v \le 1, \end{align} $$

several times.

Lemma 3.1 The mapping $p\mapsto H(p):= \boldsymbol {\psi }_p$ defines a nonincreasing continuous function from $(0,\infty ]$ into the Banach lattice $\mathcal {C}([0,1])$ .

Proof Applying (3.1) with $u=0$ gives that H is nonincreasing. This inequality also gives

$$\begin{align*}\left\lVert H(r)-H(s)\right\rVert_\infty =\boldsymbol{\psi}_r(1)-\boldsymbol{\psi}_s(1)=\frac{1}{r}-\frac{1}{s}, \quad 0<r\le s\le\infty.\\[-38pt] \end{align*}$$

Definition 3.2 Given $R\subseteq (0,\infty ]$ nonempty, we denote by $\mathcal {K}(R)$ the the closed convex hull in $\mathcal {C}([0,1])$ of $\{\boldsymbol {\psi }_p \colon p\in R\}$ . We denote by $\mathcal {O}(R)$ the set of all Orlicz functions F for which there are $0<c\le 1$ and $\varphi \in \mathcal {K}(R)$ such that

$$\begin{align*}F(t)\approx \varphi(t), \quad 0\le t \le c. \end{align*}$$

Put

$$\begin{align*}\mathcal{O}((0,\infty])=\bigcup_{p>0} \mathcal{O}([p,\infty]). \end{align*}$$

Given $r\in (0,\infty ],$ we set

$$\begin{align*}\mathcal{O}(R,r^+)=\bigcap_{s>r} \mathcal{O}(R\cap [r,s]). \end{align*}$$

Finally, we put $\mathcal {O}(r)=\mathcal {O}(\{r\})$ and $\mathcal {O}(r^+)=\mathcal {O}([r,\infty ],r^+)$ .

Note that $\mathcal {O}(\infty )=\mathcal {O}(\infty ^+)$ consists of all Orlicz functions that are null in a neighborhood of the origin.

Lemma 3.3 Let $R\subseteq (0,\infty ]$ be closed, nonempty, and bounded away from zero. Then, any function in $\mathcal {K}(R)$ extends to a function in $\mathcal {O}(R)$ .

Proof Just notice that any function $F\in \mathcal {K}(R)$ is nondecreasing, continuous, and satisfies $F(0)=0$ .

Given $f\colon [0,1] \to \mathbb {R}$ and $p\in (0,\infty ],$ we consider the function

$$\begin{align*}\pi_p(f)\colon[0,1] \to \mathbb{R}, \quad \pi_p(f)(t)=f\left(t^p\right). \end{align*}$$

Definition 3.4 Let $ 0<r\le s\le \infty $ . We denote by $\mathcal {K}_{r,s}$ the set consisting of all nondecreasing functions $\varphi \colon [0,1]\to [0,\infty )$ such that $\pi _r(\varphi )$ is convex, $\pi _s(\varphi )$ is concave,

$$\begin{align*}\frac{t^s-u^s}{s} \le \varphi(t)-\varphi(u)\le \frac{t^r-u^r}{r}, \quad 0\le u <t \le 1, \end{align*}$$

and

$$\begin{align*}\varphi(bt) \le b^s \varphi(t), \quad 1<b<\infty, \, 0\le t \le \frac{1}{b}. \end{align*}$$

Lemma 3.5 Let $R\subseteq (0,\infty ]$ be closed, nonempty, and bounded away from zero. Set $r=\min (R)$ and $s=\max (R)$ . Then $\mathcal {K}(R) \subseteq \mathcal {K}_{r,s}$ .

Proof We infer from inequality (3.1) that $\boldsymbol {\psi }_p\in \mathcal {K}_{r,s}$ for all $p\in R$ . Since $\mathcal {K}_{r,s}$ is convex and closed, we are done.

Lemma 3.6 For $j=1$ , $2,$ let $R_j\subseteq (0,\infty ]$ be closed, nonempty, and bounded away from zero. Set $r_1:=\max (R_1)$ and $r_2:=\min (R_2)$ .

(i) If $r_1=r_2$ , then $\mathcal {O}(R_1)\cap \mathcal {O}(R_2) =\mathcal {O}(r_1)$ .
(ii) If $r_1<r_2$ , then $\mathcal {O}(R_1)\cap \mathcal {O}(R_2) =\emptyset $ .

Proof Pick $F\in \mathcal {O}(R_1)\cap \mathcal {O}(R_2)$ and $r_1\le r \le s\le r_2$ . There are $0<c\le 1$ , $\varphi _j\in \mathcal {O}(R_j),$ and $C_j\in (0,\infty )$ , $j=1$ , $2$ , such that

$$\begin{align*}\frac{\varphi_1(t)}{C_1} \le F(t)\le C_2 \varphi_2(t), \quad 0\le t \le c. \end{align*}$$

By Lemma 3.5,

$$\begin{align*}\frac{\boldsymbol{\psi}_r(t)}{C_1} \le F(t)\le C_2 \boldsymbol{\psi}_s(t) , \quad 0\le t \le c. \end{align*}$$ StartLayout 1st Row StartFraction bold italic psi Subscript r Baseline left parenthesis t right parenthesis Over upper C 1 EndFraction less than or equals upper F left parenthesis t right parenthesis less than or equals upper C 2 bold italic psi Subscript s Baseline left parenthesis t right parenthesis comma 0 less than or equals t less than or equals c period EndLayout

Consequently, $r\ge s$ .

Lemma 3.7 Let $0<q<r\le \infty $ . Then, $\mathcal {O}(q^+)\cap \mathcal {O}(r^+)=\emptyset $ .

Proof Choose $q<s_1<r<s_2$ . Since $\mathcal {O}(q^+)\cap \mathcal {O}(r^+)\subseteq \mathcal {O}([q,s_1]) \cap \mathcal {O}([r,s_2])$ , the result follows from Lemma 3.6.

Proposition 3.8 Let $R\subseteq (0,\infty ]$ be closed, nonempty, and bounded away from zero.

(i) $\varphi \in \mathcal {K}(R)$ if and only if there is a probability measure $\mu $ over R such that $\varphi =\boldsymbol {\psi }_\mu $ .
(ii) $\mathcal {K}(R)$ is a compact set.

Proof Let us identify the space $\mathcal {M}(R)$ of all signed measures over R with the dual space of $\mathcal {C}(R)$ . The set $\mathcal {P}(R)$ of all probability measures over R is a convex subspace of $\mathcal {M}(R)$ . Besides, $\mathcal {P}(R)$ is closed relative to the weak* topology, and

$$\begin{align*}\{\delta_p\colon p\in R\} \end{align*}$$

is the set of extreme points of $\mathcal {P}(R)$ .

By (3.1), the map

$$\begin{align*}G\colon [0,1]\to \mathcal{C}(R), \quad G(t)(p)= \boldsymbol{\psi}_p(t) \end{align*}$$

is continuous. Consequently, the map

$$\begin{align*}D\colon [0,1] \times \mathcal{M}(R), \quad (t,\mu)\mapsto \int_R \boldsymbol{\psi}_p(t) \, d\mu(p) \end{align*}$$

is continuous. We infer that the map

$$\begin{align*}E\colon \mathcal{M}(R) \to \mathcal{C}([0,1]) , \quad \mu \mapsto D(\cdot,\mu)=\boldsymbol{\psi}_\mu, \end{align*}$$

besides linear, is continuous. Note that $E(\delta _p)=\boldsymbol {\psi }_p$ for all $p\in R$ and that

$$\begin{align*}E(\mathcal{P}(R))=\mathcal{L}(R):=\left\lbrace\boldsymbol{\psi}_\mu \colon \mu\in\mathcal{P}(R)\right\rbrace. \end{align*}$$

Therefore, $\mathcal {L}(R)$ is compact by the Banach–Alaoglu theorem, and the convex hull of $\{\boldsymbol {\psi }_p\colon p\in R\}$ is dense in $\mathcal {L}(R)$ by the Krein–Milman theorem. Consequently, $\mathcal {L}(R)=\mathcal {K}(R)$ .

Example 3.9 Let $0<r<\infty $ and $a\in \mathbb {R}$ . Define $F_{r,a}\colon [0,\infty )\to [0,\infty )$ by

$$\begin{align*}F_{r,a}(t)=\begin{cases} 0 & \text{ if } t=0, \\ t^r (-\log (t) )^{a} & \text{ if } 0 <t \le 1/e, \\ t^r& \text{ otherwise.} \end{cases} \end{align*}$$

upper F Subscript r comma a Baseline left parenthesis t right parenthesis equals StartLayout Enlarged left brace 1st Row 1st Column 0 2nd Column if t equals 0 comma 2nd Row 1st Column t Superscript r Baseline left parenthesis minus log left parenthesis t right parenthesis right parenthesis Superscript a Baseline 2nd Column if 0 less than t less than or equals 1 divided by e comma 3rd Row 1st Column t Superscript r Baseline 2nd Column otherwise period EndLayout

Suppose that $a>0$ . Aiming to prove that $F_{r,a} \in \mathcal {O}(r^+)$ , we fix $s>r$ and consider the probability measure $\mu _{a,r,s}$ on $(0,\infty )$ given by

$$\begin{align*}d\mu_{a,r,s}(p)= \frac{a \left(p-r\right)^{a-1}} { (s-r)^a } \chi_{[r,s]}(p)\, dp. \end{align*}$$ d mu Subscript a comma r comma s Baseline left parenthesis p right parenthesis equals StartFraction a left parenthesis p minus r right parenthesis Superscript a minus 1 Baseline Over left parenthesis s minus r right parenthesis Superscript a Baseline EndFraction chi Subscript left bracket r comma s right bracket Baseline left parenthesis p right parenthesis d p period

By definition, $\mu _{a,r,s}$ is supported on $[r,s]$ . Set $\varphi _{a,r,s}=\boldsymbol {\psi }_{\mu _{a,r,s}}$ . We have

$$\begin{align*}\varphi_{a,r,s}(t) =\int_r^s \frac{t^p}{p} \frac{a \left(p-r\right)^{a-1}} { (s-r)^a }\, dp = C(t) t^r \left(- \log(t)\right)^a \end{align*}$$

for all $0\le t< 1$ , where

$$\begin{align*}C(t)=\frac{a }{(s-r)^a} \int_0^{(r-s)/\log(t)} \frac{ v^{a-1} e^{-v}}{r-v \log(t)} dv. \end{align*}$$

Fix $0\le c<1$ . Since $C(t)$ is bounded away from zero and infinity when t runs over $[0,c]$ , $\varphi _{a,r,s}\approx F_{r,a}$ on $[0,c]$ . Therefore, $F_{r,a} \in \mathcal {O}([r,s])$ .

Lemma 3.10 Let $R\subseteq (0,\infty ]$ be closed, nonempty, and bounded away from zero. Let $\mu \in \mathcal {P}(R)$ and F be an Orlicz function with $F\approx \boldsymbol {\psi }_\mu $ . If $r=\min (\operatorname {\mathrm {supp}}(\mu ))$ , then $F\in \mathcal {O}(R,r^+)$ .

Proof Fix $s>r$ . We have

$$\begin{align*}\lambda=\mu( [r,s])>0. \end{align*}$$

Let $\nu $ be the probability measure obtained by restricting $\mu /\lambda $ to $[r,s]$ . Let $0<t\le 1$ . By Lemma 3.1,

$$\begin{align*}\boldsymbol{\psi}_\mu(t)\le \boldsymbol{\psi}_\nu(t) \le \frac{1}{\lambda} \boldsymbol{\psi}_\mu(t). \end{align*}$$

Since $\nu \in \mathcal {P} (R\cap [r,s])$ , we are done.

Corollary 3.11 Let $R\subseteq (0,\infty ]$ be closed, nonempty, and bounded away from zero. Let $F\in \mathcal {O}(R)$ and $b\in (0,\infty )$ . Then, there is $c\in (0,\infty )$ such that $F(bt)\approx F(t)$ for $0\le t \le c$ .

Proof If $F\in \mathcal {O}(\infty )$ , the result is clear. Otherwise, there are $0<r<s<\infty $ such that $F\in \mathcal {O}([r,s])$ . Applying Lemma 3.5 puts an end to the proof.

Lemma 3.12 Suppose that a decreasing sequence $(r_n)_{n=1}^\infty $ in $(0,\infty )$ converges to $r>0$ . Set $R=\{r\}\cup \left \lbrace r_n \colon n\in \mathbb {N}\right \rbrace $ . Let $(b_n)_{n=1}^\infty $ be a positive sequence with $\sum _{n=1}^\infty b_n=1$ . Define $F\colon [0,\infty )\to [0,\infty ]$ by

$$\begin{align*}F(t)=\sum_{n=1}^\infty t^{r_n} b_n. \end{align*}$$ upper F left parenthesis t right parenthesis equals sigma summation Underscript n equals 1 Overscript infinity Endscripts t Superscript r Super Subscript n Superscript Baseline b Subscript n Baseline period

Then, $F\in \mathcal {O}(R,r^+)\setminus \mathcal {O}(r)$ .

Proof By Lemma 3.10, $F\in \mathcal {O}(R,r^+)$ . By the dominated convergence theorem,

$$\begin{align*}\lim_{t\to 0^+} \frac{F(t)}{t^r}=0. \end{align*}$$

Hence, $F\notin \mathcal {O}(r)$ .

Recall that $r\in [0,\infty ]$ is an accumulation point of a set $R\subseteq [0,\infty ]$ if $R\cap(V\setminus \{r\})\not =\emptyset $ for every neighborhood V of r. If $R\cap (r,s)\not =\emptyset $ for all $s>r$ , we say that r is a right-sided accumulation point of R.

Proposition 3.13 Let $R\subseteq (0,\infty ]$ be closed, nonempty, and bounded away from zero. Given $r\in R$ , $\mathcal {O}(R,r^+)\setminus \mathcal {O}(r)$ is nonempty if and only if r is a right-sided accumulation point of R.

Proof Let $F\notin \mathcal {O}(r)$ be such that $F\in \mathcal {O}(R\cap [r,s])$ for all $s>r$ . Then, $\{r\}\subsetneq R\cap [r,s]$ for all $s>r$ .

Suppose that r is a right-sided accumulation point of R. Let $(r_n)_{n=1}^\infty $ in R be decreasing to r. Pick $(b_n)_{n=1}^\infty $ with $\sum _{n=1}^\infty b_n=1$ . If F is as in Lemma 3.12, $F\in \mathcal {O}(R\cap [r,s])$ for all $s>r$ .

Corollary 3.14 Let $R\subseteq (0,\infty ]$ be closed, nonempty, and bounded away from zero. The following are equivalent:

• $\mathcal {O}(R)=\cup _{r\in R} \mathcal {O}(r)$ .
• R has no right-sided accumulation point.

Proof Just combine Proposition 3.13 with Lemma 3.7.

Proposition 3.15 Let $R\subseteq (0,\infty ]$ be closed, nonempty, and bounded away from zero. Then,

$$\begin{align*}\mathcal{O}(R,r^+), \quad r\in R, \end{align*}$$

is a partition of $\mathcal {O}(R)$ .

Proof Just combine Lemma 3.7, Proposition 3.8, and Lemma 3.10.

Corollary 3.16 $\mathcal {O}(r^{-}) :=\cap _{s<r} \mathcal {O}([s,r]) =\mathcal {O}(r)$ for all $r\in (0,\infty ]$ .

Proof Let $F\in \mathcal {O}(r^{-})$ . By Lemma 3.15, for each $s<r,$ there is $p(s)\in [s,r]$ such that $F\in \mathcal {O}(p(s)^+)$ . By Lemma 3.7, $p(s)=r$ for all $s<r$ . Applying Lemma 3.6(i) puts an end to the proof.

Corollary 3.17 Let $q\in (0,\infty ]$ . Let $R\subseteq (0,\infty ]$ be closed, nonempty, and bounded away from zero. Then $\boldsymbol {\Psi }_q\in \mathcal {O}(R)$ if and only if $q\in R$ .

Proof Suppose that $\boldsymbol {\Psi }_q\in \mathcal {O}(R)$ . By Proposition 3.15, there is $r\in R$ such that $\boldsymbol {\Psi }_q\in \mathcal {O}(r^+)$ . By Lemma 3.7, $q=r$ .

Example 3.18 Let $0<p<\infty $ and the functions $F_{p,a}$ of Example 3.9 with $a<0$ . Then, $F_{p,a}$ does not belong to $\mathcal {O}(R)$ for any set $R\subseteq (0,\infty ]$ . Otherwise, by Proposition 3.15 and Lemma 3.5, there would be $r\in (0,\infty )$ and $(C_s)_{s\ge r}$ in $(0,\infty )$ such that

$$\begin{align*}\frac{1}{C_s} \boldsymbol{\psi}_s(t) \le F_{p,a}(t) \le C_r \boldsymbol{\psi}_r(t), \quad s\in(r,\infty], \, t\in[0,1]. \end{align*}$$

The right-hand inequality would yield $p>r$ . Then, applying the left-hand inequality with $r<s<p,$ we would reach an absurdity.

Theorem 3.19 Let $R\subseteq (0,\infty ]$ be closed, nonempty, and bounded away from zero. Let F be a convex Orlicz function such that $F\in \mathcal {O}(R)$ and $F^* \in \mathcal {O}((0,\infty ])$ . Then, there is $r\in R \cap [1,\infty ]$ such that $F\in \mathcal {O}(r)$ .

Proof By Proposition 3.15, there are $r\in R$ and $q\in (0,\infty ]$ such that $F\in \mathcal {O}(r^+)$ and $F^*\in \mathcal {O}(q^+)$ . Since F and $F^*$ are convex, there are $0<c<1$ and $0<C<\infty $ such that $\max \{F(t), F^*(t)\} \le Ct$ for all $t\in [0,c]$ . By Lemma 3.5, r, $q\in [1,\infty ]$ .

By Lemma 3.5, there are $0<c_1<1$ and $0<C_1<\infty $ such that $F^*(t) \le C_1 \boldsymbol {\Psi }_q(t)$ for all $0 \le t \le c_1$ . Hence,

$$\begin{align*}u v \le F(u) + F^*(v) \le F(u) + C_1 \boldsymbol{\Psi}_q(v), \quad 0 \le v \le c_1. \end{align*}$$

By Lemmas 2.5 and 2.6 and Corollary 3.11, there are $0<c_2<1$ and $0<C_2<\infty $ such that $\boldsymbol {\Psi }_{q'}(t) \le C_2 F(t)$ for all $t\in [0,c_2]$ . By Lemma 3.5, $r\le q'$ .

Pick $s>q$ . By Lemma 3.5, there are $0<c_3<1$ and $0<C_3<\infty $ such that $\boldsymbol {\Psi }_s(t)\le C_3 F^*(t)$ for all $0 \le t \le c_3$ . By Lemma 2.5,

$$\begin{align*}u v \le \boldsymbol{\Psi}_{s'}(u) + \boldsymbol{\Psi}_{s}(v) \le \boldsymbol{\Psi}_{s'}(u) + C_3 F^*(v), \quad 0\le v \le c_3. \end{align*}$$

By Lemma 2.6 and Corollary 3.11, there are $0<c_4<1$ and $0<C_4<\infty $ such that $F(t) \le C_4 \boldsymbol {\Psi }_{s'}(t)$ for all $t\in [0,c_4]$ . By Lemma 3.5, $s'\le s_1$ for all $s_1>r$ . Consequently, $q'\le r$ .

We have proved that $q'=r$ and that $\boldsymbol {\Psi }_{r}(t) \le C_2 F(t)$ for all $t\in [0,c_2]$ . By Lemma 3.5, a reverse inequality holds, that is, there are $0<c_5<1$ and $0<C_5<\infty $ such that $F(t) \le C_5 \boldsymbol {\Psi }_{r}(t)$ for all $t\in [0,c_5]$ .

4 Subsymmetric sequences in variable exponent Lebesgue spaces

A sequence space, say $(\mathbb {S}, \left \lVert \cdot \right \rVert _{\mathbb {S}})$ , will be a function space over $\mathbb {N}$ endowed with the counting measure. Let $(\boldsymbol {e}_n)_{n=1}^\infty $ denote the unit vectors of $\mathbb {F}^{\mathbb {N}}$ . If $\left \lVert \boldsymbol {e}_n\right \rVert _{\mathbb {S}}\approx 1$ for $n\in \mathbb {N}$ , we say that $\mathbb {S}$ is semi-normalized. If the mapping $T_\pi $ given by

$$\begin{align*}(a_n)_{n=1}^\infty \mapsto (a_{\pi(n)})_{n=1}^\infty \end{align*}$$

defines an isometry of $\mathbb {S}$ for any permutation $\pi $ of $\mathbb {N}$ , we say that $\mathbb {S}$ is a symmetric sequence space. In turn, if $T_{\pi }$ defines an isometry from

$$\begin{align*}\{f\in\mathbb{S}\colon \operatorname{\mathrm{supp}}(f)\subseteq\pi(\mathbb{N})\} \end{align*}$$

onto $\mathbb {S}$ for every increasing map $\pi \colon \mathbb {N}\to \mathbb {N}$ , we say that $\mathbb {S}$ is a subsymmetric sequence space. Any symmetric sequence space is subsymmetric.

We say that a sequence space $\mathbb {S}$ embeds into a function space $\mathbb {X}$ if there is an isomorphic embedding $T\colon \mathbb {S}\to \mathbb {X}$ . If $(T(\boldsymbol {e}_n))_{n=1}^\infty $ are pairwise disjointly supported, we say that $\mathbb {S}$ disjointly embeds into $\mathbb {X}$ . Note that the following are equivalent:

• $\mathbb {S}$ disjointly embeds into $\mathbb {X}$ .
• There is an isomorphic embedding $T_0\colon \mathbb {S}_0\to \mathbb {X}$ such that $\left (T(\boldsymbol {e}_n)\right )_{n=1}^\infty $ is disjointly supported, that is, the unit vector system of $\mathbb {S}$ is equivalent to a disjointly supported sequence in $\mathbb {X}_0$ .
• There is a pairwise disjointly supported sequence $\mathcal {X}$ in $\mathbb {X}\setminus \{0\}$ such that $\mathbb {S}=\mathbb {S}[\mathbb {X},\mathcal {X}]$ .

Proposition 4.1 Let $\boldsymbol {P}$ be a variable exponent on a $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ . Assume that $\boldsymbol {P}^{-}>0$ . Let $\mathcal {X}=(x_n)_{n=1}^\infty $ be a disjointly supported semi-normalized sequence in $L_{\boldsymbol {P}}$ . Then, $\mathcal {X}$ has a subsequence $\mathcal {Y}$ such that $\mathbb {S}[L_{\boldsymbol {P}},\mathcal {Y}]=\ell _F$ for some Orlicz function $F\in \mathcal {O}(R(\boldsymbol {P}))$ .

Proof We have $L_{\boldsymbol {P}}[\mathcal {X}]=\ell _{\boldsymbol {F}}$ , where $\boldsymbol {F}=(F_n)_{n=1}^\infty $ is given by

$$\begin{align*}F_n(t)=\int_\Omega \boldsymbol{\Psi}_{\boldsymbol{P}(\omega)} \left(t \left\lvert x_n(\omega)\right\rvert\right) \, d\mu(\omega), \quad t\in[0,\infty). \end{align*}$$

We have $b=\sup _n \rho _M^L\left (\left \lvert x_n\right \rvert \right )<\infty $ . For each $n\in \mathbb {N,}$ we define

$$\begin{align*}G_n\colon[0,\infty)\to[0,\infty], \quad G_n(t)=\int_{\Omega\setminus\Omega^{\boldsymbol{P}}_\infty} \frac{t^{\boldsymbol{P}(\omega)}}{\boldsymbol{P}(\omega)} \left(\frac{\left\lvert x_n(\omega)\right\rvert}{b}\right)^{\boldsymbol{P}(\omega)} \, d\mu(\omega). \end{align*}$$

Since $\left \lVert {\left . x_n \right |{}_{{\Omega ^{\boldsymbol {P}}_\infty }}}\right \rVert _\infty \le b$ for all $n\in \mathbb {N}$ , $F_n(t)=G_n(tb)$ for all $t\in [0,1/b]$ . Besides,

$$\begin{align*}c_n:= \int_{\Omega\setminus\Omega^{\boldsymbol{P}}_\infty} \left(\frac{\left\lvert x_n(\omega)\right\rvert}{b}\right)^{\boldsymbol{P}(\omega)} \, d\mu(\omega)\le 1, \quad n\in\mathbb{N}. \end{align*}$$

Consider the following dichotomy:

• $\limsup _n c_n>0$ .
• $\limsup _n c_n=0$ .

In the former case, passing to a subsequence, we can assume that $\inf _n c_n>0$ . By Proposition 3.8 and Lemma 3.3, passing to a further subsequence, we can assume that there exists an Orlicz function $F\in \mathcal {O}(R(\boldsymbol {P}))$ such that

$$\begin{align*}\sum_{n=1}^\infty \sup_{0\le t \le 1}\left\lvert\frac{G_n(t)}{c_n}-F(t)\right\rvert<\infty. \end{align*}$$ sigma summation Underscript n equals 1 Overscript infinity Endscripts sup Underscript 0 less than or equals t less than or equals 1 Endscripts StartAbsoluteValue StartFraction upper G Subscript n Baseline left parenthesis t right parenthesis Over c Subscript n Baseline EndFraction minus upper F left parenthesis t right parenthesis EndAbsoluteValue less than infinity period

By Theorem 2.3, we deduce that $\ell _{\boldsymbol {F}}=\ell _F$ .

In the latter case, passing to a subsequence, we can assume that $\sum _{n=1}^\infty c_n<\infty $ . Since

$$\begin{align*}G_n(t) \le \frac{c_n}{\boldsymbol{P}^-}, \quad 0\le t\le 1, \, n\in\mathbb{N}, \end{align*}$$

$\ell _{\boldsymbol {F}}=\ell _\infty $ by Theorem 2.3. It remains to be shown that $\infty \in R(\boldsymbol {P})$ . Assume by contradiction that it is not the case. Then, $\boldsymbol {P}^{+}<\infty $ and $\Omega ^{\boldsymbol {P}}_\infty $ is a null set. If $0<a<\inf _n \rho _M^L\left (\left \lvert x_n\right \rvert \right )$ ,

$$\begin{align*}1 \le \int_\Omega \left(\frac{\left\lvert x_n(\omega)\right\rvert}{a}\right)^{\boldsymbol{P}(\omega)} \, d\mu(\omega) \le \left( \frac{b}{a}\right)^{\boldsymbol{P}^{+}} c_n, \quad n\in \mathbb{N}. \end{align*}$$

We reach an absurdity, and the proof is over.

The following result generalizes [Reference Hernández and Ruiz11, Proposition 4.5].

Lemma 4.2 Let $\boldsymbol {P}$ be a variable exponent over a $\sigma $ -finite nonatomic measure space $(\Omega ,\Sigma ,\mu )$ . For each $n\in \mathbb {N}$ , let $A_n$ be a finite subset of $R(\boldsymbol {P})$ . Set

$$\begin{align*}\mathcal{N}=\left\lbrace(n,p) \colon (n,p)\in \mathbb{N}\times (0,\infty], \, p\in A_n\right\rbrace. \end{align*}$$

For each $(n,p)\in \mathcal {N}$ , let $U_{n,p}$ be a neighborhood of p. Then, there is a family $(\Omega _{n,p})_{(n,p)\in \mathcal {N}}$ of pairwise disjoint measurable sets such that $\mu (\Omega _{n,p})>0$ and $\boldsymbol {P}(\Omega _{n,p})\subseteq U_{n,p}$ for all $(n,p)\in \mathcal {N}$ .

Proof Assume without loss of generality that $(A_n)_{n=1}^\infty $ is nondecreasing. Set $A_0=\emptyset $ , $A=\cup _{n=1}^\infty A_n$ ,

$$\begin{align*}B=\left\lbrace p\in A \colon \boldsymbol{P}(\mu)\left(\left\lbrace p\right\rbrace\right)=0\right\rbrace, \quad \mathcal{M}=\left\lbrace(n,p)\in\mathcal{N} \colon p\in B\right\rbrace. \end{align*}$$

Let $n_0$ be the smallest $n\in \mathbb {N}$ such that $B_n:=A_{n}\cap B\not =\emptyset $ . We recursively construct for each $n\in \mathbb {N}$ a pairwise disjoint family

$$\begin{align*}(V_{n,p})_{p\in B_n} \end{align*}$$

such that $V_{n,p}\subseteq U_{n,p}$ is a measurable neighborhood of p for all $p\in B_n$ , and the set $D_n:=\bigcup _{p\in B_n} V_{n,p}$ satisfies

$$\begin{align*}\boldsymbol{P}(\mu) \left(D_n\right) <\frac{1}{2} \min_{p\in B_{n-1}} \boldsymbol{P}(\mu) \left(V_ {n-1,p}\right) \end{align*}$$

as long as $n\ge n_0+1$ . Set $E_n=\cup _{k=n}^\infty D_k$ for all $n\in \mathbb {N}$ . Since $\boldsymbol {P}(\mu ) \left (D_{n+1}\right )\le \boldsymbol {P}(\mu ) \left (D_{n}\right )/2$ for all $n\in \mathbb {N}$ , $n\ge n_0$ , $\mu \circ \boldsymbol {P}^{-1}(E_n)\le 2 \mu \circ \boldsymbol {P}^{-1}(D_n)$ for all $n\in \mathbb {N}$ , $n\ge n_0$ . Consequently, $\mu \circ \boldsymbol {P}^{-1}(E_{n+1})<\mu \circ \boldsymbol {P}^{-1}\left (V_ {n,p}\right )$ for all $n\in \mathbb {N}$ and $p\in B_n$ . Set

$$\begin{align*}W_{n,p}=V_{n,p} \setminus E_{n+1}, \quad (n,p)\in \mathcal{M}. \end{align*}$$

The family $(W_{n,p})_{(n,p)\in \mathcal {M}}$ is pairwise disjoint, and $\mu \circ \boldsymbol {P}^{-1}(W_{n,p})>0$ for all $(n,p)\in \mathcal {M}$ . Consider the set

$$\begin{align*}\mathcal{L}=\left\lbrace(n,p)\in \mathcal{M} \colon W_{n,p}\cap (A\setminus B)\right\rbrace=\emptyset. \end{align*}$$

Let $\pi \colon \mathcal {N}\setminus \mathcal {L}\to A\setminus B$ be such that $\pi (n,p)=p$ for all $(n,p)\in \mathcal {N}\setminus \mathcal {M}$ , and $\pi (n,p)\in W_{n,p}$ for all $(n,p)\in \mathcal {M}\setminus \mathcal {L}$ . Since $\mu $ is nonatomic, for each $q\in A\setminus B,$ there is a pairwise disjoint family

$$\begin{align*}(\Omega_{n,p})_{(n,p)\in\pi^{-1}(q) } \end{align*}$$

of nonnull measurable subsets of $\boldsymbol {P}^{-1}(p)$ . If for each $(n,p)\in \mathcal {L,}$ we choose

$$\begin{align*}\Omega_{n,p}=\boldsymbol{P}^{-1}(W_{n,p}), \end{align*}$$

the family $(\Omega _{n,p})_{(n,p)\in \mathcal {N}}$ satisfies the desired conditions.

Proposition 4.3 Let $\boldsymbol {P}$ be a variable exponent on a nonatomic $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ . Assume that $\boldsymbol {P}^{-}>0$ . Let $\mathbb {S}$ be a subsymmetric sequence space. The following are equivalent:

• $\mathbb {S}$ disjointly embeds into $L_{\boldsymbol {P}}$ .
• $\mathbb {S}=\ell _F$ for some $F\in \mathcal {O}(R(\boldsymbol {P}))$ .

Proof Bearing in mind Proposition 4.1, it suffices to prove that for any Orlicz function F such that ${\left . F \right |{}_{{[0,1]}}}\in \mathcal {K}(R(\boldsymbol {P})),$ there exists a disjointly supported sequence $\mathcal {X}$ in $L_{\boldsymbol {P}}^0$ with $L_{\boldsymbol {P}}[\mathcal {X}]=\ell _F$ . Choose a positive sequence $(\varepsilon _n)_{n=1}^\infty $ with $\sum _{n=1}^\infty \varepsilon _n<\infty $ . Let $(F_n)_{n=1}^\infty $ be such that, for all $n\in \mathbb {N}$ , $F_n$ belongs to convex hull of $\left \lbrace \boldsymbol {\Psi }_p\colon p\in R(\boldsymbol {P})\right \rbrace $ , and

$$\begin{align*}\max_{0\le t\le 1} \left\lvert F(t)-F_n(t)\right\rvert\le \varepsilon_n. \end{align*}$$

Let $(A_n)_{n=1}^\infty $ be a sequence of finite subsets of $R(\boldsymbol {P})$ such that for each $n\in \mathbb {N,}$ there is $(a_{n,p})_{p\in A_n}$ in $(0,\infty )$ such that

$$\begin{align*}F_n=\sum_{p\in A_n} a_{n,p} \boldsymbol{\Psi}_p, \quad \sum_{p\in A_n}a_{n,p}=1. \end{align*}$$

By Lemma 3.1, for each $n\in \mathbb {N}$ and $p\in A_n$ , there is a neighborhood $U_{n,p}$ of p such that $\left \lVert \boldsymbol {\psi }_p-\boldsymbol {\psi }_q\right \rVert \le \varepsilon _n$ for all $q\in U_{n,p}$ . Let $(\Omega _{n,p})_{(n,p)\in \mathcal {N}}$ be the family of sets provided by Lemma 4.2. Choose, for each $(n,p)\in \mathcal {N}$ , $f_{n,p}\colon \Omega \to [0,\infty )$ with $\operatorname {\mathrm {supp}}(f_{n,p})=\Omega _{n,p}$ and $\int _\Omega f_{n,p}\, d\mu =1$ . Set

$$\begin{align*}G_{n,p}\colon[0,\infty) \to[0,\infty), \quad t\mapsto \int_{\Omega} \boldsymbol{\Psi}_{\boldsymbol{P}(\omega)}(t) f_{n,p}(\omega)\, d\mu(\omega). \end{align*}$$

Note that, if $0\le t \le 1$ ,

$$\begin{align*}G_{n,p}(t)= \int_{\Omega\setminus \Omega^{\boldsymbol{P}}_\infty} \boldsymbol{\Psi}_{\boldsymbol{P}(\omega)}(t) f_{n,p}(\omega)\, d\mu(\omega). \end{align*}$$

Since

$$\begin{align*}\max_{0\le t \le 1} \left\lvert G_{n,p}(t)-\boldsymbol{\psi}_p(t)\right\rvert\le \varepsilon_n, \end{align*}$$

the Musielak–Orlicz sequence $\boldsymbol {G}=(G_n)_{n=1}^\infty $ given by

$$\begin{align*}G_n=\sum_{p\in A_n} a_{n,p} G_{n,p} \end{align*}$$

satisfies $\max _{0\le t \le 1} \left \lvert G_n(t)-F(t)\right \rvert \le 2 \varepsilon _n$ for all $n\in \mathbb {N}$ . By Theorem 2.3, $\ell _F=\ell _{\boldsymbol {G}}$ . If we set

$$\begin{align*}x_n=\sum_{p\in A_n}\left( a_{n,p} f_{n,p}\right)^{1/\boldsymbol{P}}, \end{align*}$$

then for every $n\in \mathbb {N}$ and $0\le t \le 1,$ we have

$$\begin{align*}\int_\Omega M_{\boldsymbol{P}} \left(t x_n(\omega)\right) d\mu(\omega)=\sum_{p\in A_n} \int_{\Omega\setminus \Omega^{\boldsymbol{P}}_\infty} \boldsymbol{\Psi}_{\boldsymbol{P}(\omega)}(t) a_{n,p} f_{n,p}(\omega) \, d\mu(\omega)=G_n(t). \end{align*}$$

Consequently, $\mathcal {X}=(x_n)_{n=1}^\infty $ is a pairwise disjointly supported sequence with $L_{\boldsymbol {P}}[\mathcal {X}]=\ell _{\boldsymbol {G}}=\ell _F$ .

Corollary 4.4 (See [Reference Hernández and Ruiz11, Theorem 3.5])

Let $q\in (0,\infty ]$ and $\boldsymbol {P}$ be a variable exponent over a nonatomic $\sigma $ -finite measure space. Assume that $\boldsymbol {P}^{-}>0$ . Then, $L_{\boldsymbol {P}}$ has a disjointly supported sequence equivalent to the unit vector system of $\ell _q$ if and only if $q\in R(\boldsymbol {P})$ .

Proof Just combine Proposition 4.3 with Corollary 3.17.

Given a function space $\mathbb {X}$ , the following are equivalent:

• $\mathbb {X}$ fails to be absolutely continuous;
• $\ell _\infty $ embeds into $\mathbb {X}$ ;
• $\ell _\infty $ disjointly embeds into $\mathbb {X}$

(see [Reference Lindenstrauss and Tzafriri20, Proposition 1.a.7]). Since any separable Banach space linearly embeds into $\ell _\infty $ by the Hahn–Banach theorem, the subsymmetric embedding problem for locally convex function spaces splits into two issues. Namely,

• determining which nonseparable subsymmetric sequence spaces embed into a given non absolutely continuous function space, and
• determining which separable subsymmetric sequence spaces embed into a given absolutely continuous function space.

We give now a result for variable exponent Lebesgue spaces framed in the latter problem.

Theorem 4.5 Let $\boldsymbol {P}$ be a variable exponent over a $\sigma $ -finite nonatomic measure space $\mu $ . Suppose that $\boldsymbol {P}^{-}\ge 2$ and $q:=\boldsymbol {P}^{+}<\infty $ . Let $\mathbb {S}$ be a subsymmetric sequence space. Then, $\mathbb {S}$ embeds into $L_{\boldsymbol {P}}(\mu )$ if and only if $\mathbb {S}=\ell _2$ or $\mathbb {S}=\ell _F$ for some $F\in \mathcal {O}(R(\boldsymbol {P}))$ .

Proof Assume without loss of generality that $\mu $ is finite. Suppose that a symmetric sequence space $\mathbb {S}$ other than $\ell _2$ embeds into $L_{\boldsymbol {P}}(\mu )$ . By [Reference Ansorena and Bello2, Theorem 4.2], $L_{\boldsymbol {P}}(\mu )$ contains a disjointly supported sequence equivalent to the canonical basis of $\mathbb {S}$ , and, by Proposition 4.3, $\mathbb {S}=\ell _F$ for some $F\in \mathcal {O}(R(\boldsymbol {P}))$ .

Proposition 4.3 also gives that $\ell _F$ embeds into $L_{\boldsymbol {P}}(\mu )$ for any $F\in \mathcal {O}(R(\boldsymbol {P}))$ . We close the proof by noticing that, since $L_q(\mu )\subseteq L_{\boldsymbol {P}}(\mu )\subseteq L_2(\mu )$ , any Rademacher sequence over $\Omega $ is equivalent to the canonical $\ell _2$ -basis by Khintchine’s inequalities.

The behavior of variable exponent Lebesgue spaces $L_{\boldsymbol {P}}$ in the case where $\boldsymbol {P}^{-}<2$ dramatically opposes that described in Theorem 4.5.

Proposition 4.6 Let $\boldsymbol {P}$ be a variable exponent over a $\sigma $ -finite nonatomic measure space $\mu $ . Suppose that $\boldsymbol {P}^{-}<2$ . Let F be an Orlicz function satisfying (1.1) for some $p\in (\boldsymbol {P}^{-},\infty )\cap [1,2]$ . Then $\ell _F$ isomorphically embeds into $L_{\boldsymbol {P}}$ .

Proof Just combine Proposition 2.9 with Theorem 1.1.

We now pass to the discrete case. To compare the behavior of non-atomic variable exponent Lebesgue spaces with that of atomic ones, we adapt Peirats–Ruiz’s results from [Reference Peirats and Ruiz24] on the latter spaces to our language. Given a sequence $\boldsymbol {P}\colon \mathbb {N}\to (0,\infty ]$ , we set $\boldsymbol {P}_m=(p_{n+m-1})_{n=1}^\infty $ for all $m\in \mathbb {N}$ . Consider the set of Orlicz functions

$$\begin{align*}\mathcal{K}(\boldsymbol{P})=\bigcap_{m=1}^\infty \mathcal{K}(R(\boldsymbol{P}_m)). \end{align*}$$

We denote by $\mathcal {O}(\boldsymbol {P})$ the set of all Orlicz functions for which there is $\varphi \in \mathcal {K}(\boldsymbol {P})$ and $0\le c\le 1$ such that $F(t)\approx \varphi (t)$ for all $0\le t\le c$ .

Denote by $A(\boldsymbol {P})$ the set of all limit points of $\boldsymbol {P}$ , that is,

$$\begin{align*}A(\boldsymbol{P})= \bigcap_{m=1}^\infty R(\boldsymbol{P}_m). \end{align*}$$

Note that any accumulation point of $R(\boldsymbol {P})$ is a limit point of $\boldsymbol {P}$ , but the converse does not hold.

Theorem 4.7 (cf. [Reference Peirats and Ruiz24, Proposition 2.3])

Let $\boldsymbol {P}=(p_n)_{n=1}^\infty $ be a variable exponent with $\boldsymbol {P}^{-}>0$ and $\boldsymbol {P}^{+}<\infty $ . Let $\mathbb {S}$ be a subsymmetric sequence space. Then, $\mathbb {S}$ isomorphically embeds into $\ell _{\boldsymbol {P}}$ if and only if $\mathbb {S}=\ell _F$ for some Orlicz function $F\in \mathcal {O}(\boldsymbol {P})$ . Further, for each $F\in \mathcal {O}(\boldsymbol {P}),$ there is $r\in A(\boldsymbol {P})$ such that $F\in \mathcal {O}(r^+)$ .

Proof Applying [Reference Peirats and Ruiz24, Proposition 2.3] to the case where the Musielak–Orlicz sequence consists of power Orlicz functions yields the wished-for characterization of isomorphic embeddability. Assume that $F\in \mathcal {O}(\boldsymbol {P})$ . By Proposition 3.15, for each $m\in \mathbb {N,}$ there is $r_m\in R(\boldsymbol {P}_m)$ such that $F\in \mathcal {O}(R(\boldsymbol {P}_m),r_m^+)$ . By Lemma 3.7, there is $r\in (0,\infty ]$ such that $r=r_m$ for all $m\in \mathbb {N}$ .

Theorem 4.7 implies that if a subsymmetric sequence space $\mathbb {S}$ isomorphically embeds into the Nakano space $\ell _{\boldsymbol {P}}$ , then there are $r\in A(\boldsymbol {P})$ and $F\in \mathcal {O}(R(\boldsymbol {P}), r^+)$ such that $\mathbb {S}=\ell _F$ . The following example shows that this condition does not characterize isomorphic embeddability.

Example 4.8 Let $\boldsymbol {P}=(p_n)_{n=1}^\infty $ be a variable exponent for which there exists $\lim _n p_n=p\in (0,\infty )$ . Lemma 3.12 implies that, despite $A(\boldsymbol {P})=\{p\}$ , $\mathcal {O}(R(\boldsymbol {P}))$ contains Orlicz functions that are not equivalent to the potential Orlicz function $\boldsymbol {\Psi }_p$ near the origin. However, $\ell _p$ is the unique subsymmetric space that embeds into $\ell _{\boldsymbol {P}}$ . Indeed, if $\varphi \in \mathcal {K}(\boldsymbol {P})$ , then, by Lemma 3.5,

$$\begin{align*}\frac{t^{p_m}}{p_m} \le \varphi(t)\le \frac{t^p}{p}, \quad 0\le t \le 1,\, m\in\mathbb{N}. \end{align*}$$ StartFraction t Superscript p Super Subscript m Superscript Baseline Over p Subscript m Baseline EndFraction less than or equals phi left parenthesis t right parenthesis less than or equals StartFraction t Superscript p Baseline Over p EndFraction comma 0 less than or equals t less than or equals 1 comma m element of double struck upper N period

Letting m tend to infinity, we get $\varphi (t)= \boldsymbol {\Psi }_p(t)$ for all $0\le t \le 1$ .

To close this section, we provide a Nakano space that exhibits a behavior opposed to that of the Nakano space in Example 4.8.

Example 4.9 Let $\boldsymbol {Q}=(q_j)_{j=1}^\infty $ be a sequence in $(0,\infty )$ decreasing to $q\in (0,\infty )$ . We have

$$\begin{align*}R:=R(\boldsymbol{Q})=\{q_j \colon j\in\mathbb{N}\}\cup\{q\}. \end{align*}$$

Let $\sigma \colon \mathbb {N}^2\to \mathbb {N}$ be a bijection. Let $\beta \colon \mathbb {N}\to \mathbb {N}$ be the second component of the inverse of $\sigma $ . Consider the variable exponent $\boldsymbol {P}=(p_n)_{n=1}^\infty $ given by

$$\begin{align*}p_n=q_{\beta(n)}. \end{align*}$$

We have $\mathcal {K}(\boldsymbol {P})=\mathcal {K}(R)$ . Hence, $\ell _{\boldsymbol {P}}$ contains an isomorphic copy of $\ell _F$ for every $F\in \mathcal {O}(R)$ . Further, $A(\boldsymbol {P})=R$ .

5 Complemented sequences in variable exponent Lebesgue spaces

Let $\mathbb {X} $ be a function space over a $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ . We say that a sequence space $\mathbb {S}$ complementably embeds into $\mathbb {X}$ if there are bounded linear maps $T\colon \mathbb {S}\to \mathbb {X}$ and $P\colon \mathbb {X}\to \mathbb {S}$ such that $P\circ T=\mathrm {Id}_{\mathbb {S}}$ . If, $\mathbb {S}$ complementably embeds into $\mathbb {X}$ , then there are $\mathcal {X}=(x_n)_{n=1}^\infty $ in $\mathbb {X}$ and $\mathcal {X}^*=(x_n^*)_{n=1}^\infty $ in $\mathbb {X}^*$ , called projecting functionals for $\mathcal {X}$ , such that the unit vector system of $\mathbb {S}$ is equivalent to $\mathcal {X}$ , $(\mathcal {X},\mathcal {X}^*)$ is a biorthogonal system, and the mapping

$$\begin{align*}f\mapsto \sum_{n=1}^\infty x_n^*(f) x_n \end{align*}$$ StartLayout 1st Row f right arrow from bar sigma summation Underscript n equals 1 Overscript infinity Endscripts x Subscript n Superscript asterisk Baseline left parenthesis f right parenthesis x Subscript n EndLayout

defines a bounded operator from $\mathbb {X}$ into $\mathbb {X}$ . Besides, if $\mathbb {S}$ is absolutely continuous or $\mathcal {X}$ is disjointly supported, then the converse also holds. If $\mathcal {X}$ is disjointly supported and $\mathcal {X}^*$ is a disjointly supported sequence in $\mathbb {X}'$ , we say that $\mathbb {S}$ disjointly complementably embeds into $\mathbb {X}$ .

Lemma 5.1 Let $\mathbb {X} $ be a function space and $\mathbb {S}$ be a sequence space. Suppose that $\mathbb {X}$ is L-convex. Suppose that there is a disjointly supported sequence $\mathcal {X}=(x_n)_{n=1}^\infty $ in $\mathcal {X}$ with projecting functionals $\mathcal {X}^*$ in $\mathbb {X}'$ such that $\mathbb {S}=\mathbb {S}[\mathbb {X},\mathcal {X}]$ . Then, $\mathbb {S}$ complementably disjointly embeds into $\mathbb {X}$ .

Proof Let P be the endomorphism of $\mathbb {X}$ given by

$$\begin{align*}f \mapsto \sum_{n=1}^\infty x_n^*(f) x_n. \end{align*}$$ StartLayout 1st Row f right arrow from bar sigma summation Underscript n equals 1 Overscript infinity Endscripts x Subscript n Superscript asterisk Baseline left parenthesis f right parenthesis x Subscript n Baseline period EndLayout

By [Reference Kalton14, Theorem 3.3], there is a constant C such that

$$\begin{align*}\left\lVert\left(\sum_{j\in J} \left\lvert P(f_j)\right\rvert^2\right)^{1/2}\right\rVert\le C \left\lVert\left(\sum_{j\in J} \left\lvert f_j\right\rvert^2\right)^{1/2}\right\rVert \end{align*}$$

StartMetric left parenthesis sigma summation Underscript j element of upper J Endscripts StartAbsoluteValue upper P left parenthesis f Subscript j Baseline right parenthesis EndAbsoluteValue squared right parenthesis Superscript 1 divided by 2 Baseline EndMetric less than or equals upper C StartMetric left parenthesis sigma summation Underscript j element of upper J Endscripts StartAbsoluteValue f Subscript j Baseline EndAbsoluteValue squared right parenthesis Superscript 1 divided by 2 Baseline EndMetric

for every family $(f_j)_{j\in J}$ in $\mathbb {X}$ . Pick a partition $(A_n)_{n=1}^\infty $ of $(\Omega ,\Sigma ,\mu )$ so that $\operatorname {\mathrm {supp}}(x_n)\subseteq A_n$ for all $n\in \mathbb {N}$ . Set $y_n^*=x_n^* \chi _{A_n}$ for all $n\in \mathbb {N}$ . Given $f\in \mathbb {X}$ , applying the above estimate to $\left (f\chi _{A_n}\right )_{n=1}^\infty $ , we obtain

$$\begin{align*}\left\lVert\sum_{n=1}^\infty y_n^*(f) x_n\right\rVert \le C \left\lVert f\right\rVert. \end{align*}$$ StartMetric sigma summation Underscript n equals 1 Overscript infinity Endscripts y Subscript n Superscript asterisk Baseline left parenthesis f right parenthesis x Subscript n Baseline EndMetric less than or equals upper C StartMetric f EndMetric period

Consequently, $(y_n^*)_{n=1}^\infty $ are projecting functionals for $\mathcal {X}$ .

Proposition 5.2 Let $\boldsymbol {P}$ be a variable exponent over a $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ . Suppose that $1\le \boldsymbol {P}^-\le \boldsymbol {P}^+<\infty $ and that a subsymmetric sequence space $\mathbb {S}$ complementably disjointly embeds into $\boldsymbol {L}_{\boldsymbol {P}}$ . Then, $\mathbb {S}=\ell _q$ for some $q\in R(\boldsymbol {P})$ .

Proof There are pairwise disjointly supported sequences $\mathcal {X}=(x_n)_{n=1}^\infty $ in $\boldsymbol {L}_{\boldsymbol {P}}$ and $\mathcal {X}'=(x_n')_{n=1}^\infty $ in $\boldsymbol {L}_{\boldsymbol {P}'}$ such that $\mathcal {X}$ is equivalent to the unit vector system of $\mathbb {S}$ and $\mathcal {X}'$ is equivalent to the unit vector system of $\mathbb {S}'$ . By Proposition 4.1, there are $F\in \mathcal {O}(R(\boldsymbol {P}))$ and $G\in \mathcal {O}(R(\boldsymbol {P}'))$ such that $\mathbb {S}=\ell _F$ and $\mathbb {S}'=\ell _G$ . We have $\ell _{F^*}=\ell _G$ , whence, by Corollary 3.11, $F^*\approx G$ near the origin. Consequently, $F^*\in \mathcal {O}(R(\boldsymbol {P}'))$ . By Theorem 3.19, $F\in \mathcal {O}(q)$ for some $q\in R(\boldsymbol {P})$ . Hence, $F\approx t^q$ .

Lemma 5.3 Let $\boldsymbol {P}$ be a variable exponent over a nonatomic $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ . Suppose that $\boldsymbol {P}^{-}>0$ . Let $\mathbb {S}$ be a subsymmetric sequence space. Set $\boldsymbol {P}_c={\left . \boldsymbol {P} \right |{}_{{\Omega ^{\boldsymbol {P}}_c}}}$ . Then, $\mathbb {S}$ complementably embeds into $L_{\boldsymbol {P}}$ if and only if it complementably embeds into $L_{\boldsymbol {P}_c}$ .

Proof Let $T\colon \mathbb {S}\to L_{\boldsymbol {P}}$ and $P\colon L_{\boldsymbol {P}}\to \mathbb {S}$ be such that $P\circ T=\mathrm {Id}_{\mathbb {S}}$ . Let J be the canonical embedding of $L_{\boldsymbol {P}_c}$ into $L_{\boldsymbol {P}}$ , and Q be the canonical projection from $L_{\boldsymbol {P}}$ into $L_{\boldsymbol {P}_c}$ . By Proposition 2.7, $P=P\circ J\circ Q$ . Consequently, $\mathbb {S}$ complementably embeds into $L_{\boldsymbol {P}_c}$ .

Theorem 5.4 Let $\boldsymbol {P}$ be a variable exponent over a nonatomic $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ . Suppose that $\boldsymbol {P}^{-}>0$ and $q:=\boldsymbol {P}^{+}<\infty $ and that $\boldsymbol {P}^{-1}((1,\infty ))$ is not null. Let $\mathbb {S}$ be a subsymmetric sequence space. Set $\boldsymbol {P}_c={\left . \boldsymbol {P} \right |{}_{{\Omega ^{\boldsymbol {P}}_c}}}$ . The following are equivalent:

(a) $\mathbb {S}$ complementably embeds into $L_{\boldsymbol {P}}$ .
(b) $\mathbb {S}=\ell _2$ or $\mathbb {S}$ complementably disjointly embeds into $L_{\boldsymbol {P}}$ .
(c) $\mathbb {S}=\ell _r$ for some $r\in \{2\} \cup R(\boldsymbol {P}_c)$ .

Proof There is $p>1$ such that $\Omega _0:=\boldsymbol {P}^{-1}([p,q])$ is not null. We can, without loss of generality, assume that $\mu $ is finite and $\mu (\Omega _0)=1$ .

Let $(r_n)_{n=1}^\infty $ be a Rademacher sequence over $\Omega _0$ . Let S be the associated embedding of $\ell _2$ into $L_q(\Omega _0)$ , and Q the associated projection from $L_p(\Omega _0)$ onto $\ell _2$ . Let J be the canonical map from $L_q(\Omega _0)$ into $L_{\boldsymbol {P}}$ , and P be the canonical map from $L_{\boldsymbol {P}}$ into $L_p(\Omega _0)$ . The maps $J\circ S$ and $Q\circ P$ witness that $\ell _2$ complementably embeds into $L_{\boldsymbol {P}}$ .

Suppose a subsymmetric space $\mathbb {S}$ other than $\ell _2$ complementably embeds into $\mathbb {X}:=L_{\boldsymbol {P}}$ . Set $\Omega _1=\boldsymbol {P}^{-1}([1,2))$ and $\Omega _2=\boldsymbol {P}^{-1}([2,\infty ))$ and $\boldsymbol {P}_j={\left . \boldsymbol {P} \right |{}_{{\Omega _j}}}$ , $j=1$ , $2$ . By Lemma 5.3 and [Reference Ansorena and Bello2, Theorem 5.2], there is $a\in \{1,2\}$ such that $\Omega _a$ is nonnull and $\mathbb {S}$ complementably disjointly embeds into $L_{\boldsymbol {P}_a}$ . By Proposition 5.2, $F\in \mathcal {O}(r)$ for some $r\in R(\boldsymbol {P}_a)$ .

In [Reference Hernández and Ruiz11, Proposition 4.4] it is proved that $\ell _r$ complementably embeds into $L_{\boldsymbol {P}}$ for every $r\in R(\boldsymbol {P}_c)$ . For the sake of completeness and clarity, we use our approach to variable exponent Lebesgue spaces to reprove this result. Choose $(\varepsilon _n)_{n=1}^\infty $ in $(0,\infty )$ with $\sum _{n=1}^\infty \varepsilon _n<\infty $ . By Lemma 3.1, there is a sequence $(V_n)_{n=1}^\infty $ of neighborhoods of r such that

$$\begin{align*}\left\lVert\boldsymbol{\psi}_p-\boldsymbol{\psi}_r\right\rVert_\infty\le\varepsilon_n, \quad \left\lVert\boldsymbol{\psi}_{p'}-\boldsymbol{\psi}_{r'}\right\rVert_\infty \le\varepsilon_n \end{align*}$$

for all $n\in \mathbb {N}$ and $p\in V_{n}$ . Use Lemma 4.2 to choose pairwise disjoint measurable sets $(\Omega _n)_{n=1}^\infty $ such that $0<\mu (\Omega _n)<\infty $ and $\boldsymbol {P}(\Omega _n) \subseteq V_n\cap [1,\infty ]$ for all $n\in \mathbb {N}$ . Use Lemma 2.8 to pick sequences $(f_n)_{n=1}^\infty $ and $(g_n)_{n=1}^\infty $ of nonnegative measurable functions such that $\operatorname {\mathrm {supp}}(f_n)=\operatorname {\mathrm {supp}}(g_n)\subseteq \Omega _n$ and

$$\begin{align*}\int_\Omega f_n g_n \, d \mu=\rho_{\boldsymbol{P}}^L(f_n)=\rho^L_{\boldsymbol{Q}}(g_n)=1 \end{align*}$$

for all $n\in \mathbb {N}$ . By Proposition 4.3, there are bounded linear maps

$$\begin{align*}T\colon \ell_r \to L_{\boldsymbol{P}}, \quad S\colon \ell_{r'} \to L_{\boldsymbol{P}'} \end{align*}$$

given by

$$\begin{align*}(a_n)_{n=1}^\infty \mapsto \sum_{n=1}^\infty a_n \, f_n, \quad (a_n)_{n=1}^\infty \mapsto \sum_{n=1}^\infty a_n \, g_n, \end{align*}$$

respectively. The map S is a kernel operator whose conjugate map $S'\colon L_{\boldsymbol {P}} \to \ell _r$ is given by

$$\begin{align*}S'(g)(n)=\int_\Omega g_n(\omega) g(\omega) \, d\mu(\omega) \end{align*}$$

for $n\in \mathbb {N}$ . Since $S'\circ T=\mathrm {Id}_{\ell _r}$ , we are done.

Theorem 5.5 Let $\boldsymbol {P}$ be a variable exponent over a nonatomic $\sigma $ -finite measure space $(\Omega ,\Sigma ,\mu )$ . Suppose $0<\boldsymbol {P}^{-}\le \boldsymbol {P}^{+}\le 1$ . Given a semi-normalized subsymmetric sequence space $\mathbb {S}$ , the following are equivalent:

• $\mathbb {S}$ complementably embeds into $L_{\boldsymbol {P}}$ .
• $\mathbb {S}$ complementably disjointly embeds into $L_{\boldsymbol {P}}$ .
• $\mathbb {S}=\ell _1$ and $\Omega _1:=\boldsymbol {P}^{-1}(1)$ is not null.

Proof Set $\mu _1={\left . \mu \right |{}_{{\Omega _1}}}$ and $\boldsymbol {P}_1={\left . \boldsymbol {P} \right |{}_{{\Omega _1}}}$ . If $\Omega _1$ is nonnull, then $L_{\boldsymbol {P}_1}=L_1(\mu _1)$ .

Suppose that $\mathbb {S}$ complementably embeds into $L_{\boldsymbol {P}}$ . By Lemma 5.3, $\Omega _1$ is nonnull and $\mathbb {S}$ complementably embeds into $L_1(\mu _1)$ . By [Reference Lindenstrauss and Pełczyński17], $\mathbb {S}=\ell _1$ .

Conversely, if $\Omega _1$ is not null, it is well-known that $\ell _1$ complementably disjointly embeds into $L_1(\mu _1)$ .

Lemma 5.6 Suppose a subsymmetric sequence space $\mathbb {S}$ complementably embeds into an absolutely continuous sequence space $\mathbb {X}$ over $\mathbb {N}$ . Then, $\mathbb {S}$ complementably disjointly embeds into $\mathbb {X}$ .

Proof Let $\mathcal {X}=(x_n)_{n=1}^\infty $ in $\mathbb {X}$ with projecting functionals $\mathcal {X}^*=(x_n^*)_{n=1}^\infty $ be equivalent to the unit vector system of $\mathbb {S}$ . By the Cantor diagonal technique, passing to a subsequence, we can assume that $\mathcal {X}$ converges pointwise. Since $(x_{2n-1})_{n=1}^\infty $ is equivalent to $(x_{2n})_{n=1}^\infty $ , the mapping

$$\begin{align*}f \mapsto \sum_{n=1}^\infty x_{2n-1}^*(f) x_{2n-1} - \sum_{n=1}^\infty x_{2n-1}^*(f) x_{2n} \end{align*}$$

defines an endomorphism of $\mathbb {X}$ . Consequently, we can replace $\mathcal {X}$ by

$$\begin{align*}\left(x_{2n-1}-x_{2n}\right)_{n=1}^\infty \end{align*}$$

and $\mathcal {X}^*$ by

$$\begin{align*}\left(x_{2n-1}^*\right)_{n=1}^\infty. \end{align*}$$

In this way, we can assume that $\mathcal {X}$ converges to zero pointwise. By the gliding-hump technique (see [Reference Lindenstrauss and Tzafriri19, Proposition 1.a.9] or [Reference Ansorena and Bello2, Lemma 3.5]), passing to a subsequence, we can assume that $\mathcal {X}$ is finitely disjointly supported. We conclude the proof by applying Lemma 5.1.

Our latest result improves upon a Peirats–Ruiz’s achievement from [Reference Peirats and Ruiz24].

Theorem 5.7 (cf. [Reference Peirats and Ruiz24, Theorem 2.9])

Let $\boldsymbol {P}\colon \mathbb {N}\to (0,\infty )$ be a variable exponent with $\boldsymbol {P}^{-}>0$ and $\boldsymbol {P}^{+}<\infty $ . Let $\mathbb {S}$ be a subsymmetric sequence space. The following are equivalent:

(i) $\mathbb {S}$ complementably embeds into $\ell _{\boldsymbol {P}}$ .
(ii) $\mathbb {S}=\ell _r$ for some $r\in A(\boldsymbol {P})$ .
(iii) There is a subsequence $\boldsymbol {Q}$ of $\boldsymbol {P}$ such that $\mathbb {S}=\ell _{\boldsymbol {Q}}$ .

Proof (i) implies (ii) by Lemma 5.6 and Proposition 5.2. In turn, (ii) implies (iii) by Corollary 4.4 and Lemma 3.1. Since it is obvious that (iii) implies (i), we are done.

Data availability statement

Data sharing does not apply to this article as no datasets were generated or analyzed during the current study.

Competing interests

The authors have no competing interests to declare that are relevant to the content of this article.

Footnotes

Both authors acknowledge the support of the Spanish Ministry for Science and Innovation under Grant PID2022-138342NB-I00 for Functional Analysis Techniques in Approximation Theory and Applications (TAFPAA). G.B. has also been partially supported by PID2022-137294NB-I00, DGI-FEDER and by Project E48_23R, D.G. Aragón

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Article contents

On the basic sequence structure of variable exponent Lebesgue spaces

Abstract

Keywords

MSC classification

Information

1 Introduction

Theorem 1.1 (See [Reference Bretagnolle and Dacunha-Castelle8])

2 Musielak–Orlicz function spaces

Theorem 2.2 (See [Reference Ansorena and Marcos3])

2.1 Musielak–Orlicz sequence spaces

Theorem 2.3 [Reference Musielak21, Theorem 8.11]

2.2 Orlicz sequence spaces

2.3 Variable exponent Lebesgue spaces

3 Orlicz functions generated near zero by power functions

4 Subsymmetric sequences in variable exponent Lebesgue spaces

Corollary 4.4 (See [Reference Hernández and Ruiz11, Theorem 3.5])

Theorem 4.7 (cf. [Reference Peirats and Ruiz24, Proposition 2.3])

5 Complemented sequences in variable exponent Lebesgue spaces

Theorem 5.7 (cf. [Reference Peirats and Ruiz24, Theorem 2.9])

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