Analysis on function spaces of Musielak-Orlicz type 9780429524103, 0429524102, 9780429537578, 0429537573, 9780429552274, 0429552270, 9781498762618, 1498762611

Table of contents :
Content: Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
1: A path to Musielak-Orlicz spaces
1.1 Introduction
1.2 Banach function spaces
1.2.1 The associate space
1.2.2 Absolute continuity of the norm and continuity of the norm
1.2.3 Convexity, uniform convexity and smoothness of a norm
1.2.4 Duality mappings and extremal elements
1.3 Modular spaces
1.3.1 Modular convergence and norm convergence
1.3.2 Conjugate modulars and duality
1.3.3 Modular uniform convexity
1.4 The lpn sequence spaces and their properties
1.4.1 Duality 2.6 Uniform convexity of Musielak-Orlicz spaces2.7 Carathéodory functions and Nemytskii operators on Musielak-Orlicz spaces
2.8 Further properties of variable exponent spaces
2.8.1 Duality maps on spaces of variable integrability
2.9 The Matuszewska-Orlicz index of a Musielak-Orlicz space
2.10 Historical notes
3: Sobolev spaces of Musielak-Orlicz type
3.1 Sobolev spaces: definition and basic properties
3.1.1 Examples
3.2 Separability
3.3 Duality of Sobolev spaces of Musielak-Orlicz type
3.4 Embeddings, compactness, Poincaré-type inequalities
4: Applications

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Analysis on Function Spaces of Musielak-Orlicz Type

Monographs and Research Notes in Mathematics Series Editors: John A. Burns, Thomas J. Tucker, Miklos Bona, Michael Ruzhansky Lineability The Search for Linearity in Mathematics Richard M. Aron, Luis Bernal-Gonzalez, Daniel M. Pellegrino, Juan B. Seoane Sepulveda Iterative Methods and Preconditioning for Large and Sparse Linear Systems with Applications Daniele Bertaccini, Fabio Durastante Monomial Algebras, Second Edition Rafael Villarreal Matrix Inequalities and Their Extensions to Lie Groups Tin-Yau Tam, Xuhua Liu Elastic Waves High Frequency Theory Vassily Babich, Aleksei Kiselev Difference Equations Theory, Applications and Advanced Topics, Third Edition Ronald E. Mickens Sturm-Liouville Problems Theory and Numerical Implementation Ronald. B. Guenther, John. W. Lee Analysis on Function Spaces of Musielak-Orlicz Type Osvaldo Méndez, Jan Lang For more information about this series please visit: https://www.crcpress.com/Chapman--HallCRC-Monographs-and-Research-Notes-in-Mathematics/book-series/CRCMONRESNOT

Analysis on Function Spaces of Musielak-Orlicz Type

Osvaldo Méndez Jan Lang

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20181114 International Standard Book Number-13: 978-1-4987-6260-1 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

To Cristina and Jana.

Contents

Preface

ix

1 A path to Musielak-Orlicz spaces 1.1 1.2

1.3

1.4

1.5

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Banach function spaces . . . . . . . . . . . . . . . . . . . . . 1.2.1 The associate space . . . . . . . . . . . . . . . . . . . 1.2.2 Absolute continuity of the norm and continuity of the norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Convexity, uniform convexity and smoothness of a norm 1.2.4 Duality mappings and extremal elements . . . . . . . Modular spaces . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Modular convergence and norm convergence . . . . . . 1.3.2 Conjugate modulars and duality . . . . . . . . . . . . 1.3.3 Modular uniform convexity . . . . . . . . . . . . . . . The `pn sequence spaces and their properties . . . . . . . . . 1.4.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Finitely additive measures . . . . . . . . . . . . . . . . 1.4.3 Geometric properties of `pn . . . . . . . . . . . . . . . 1.4.4 Applications: Fixed point theorems on `pn spaces . . . 1.4.5 Further remarks . . . . . . . . . . . . . . . . . . . . . Forerunners of the Musielak-Orlicz class: Orlicz spaces, Lp(x) spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Musielak-Orlicz spaces 2.1 2.2

2.3 2.4

2.5 2.6

Introduction, definition and examples . . . Embeddings between Musielak-Orlicz spaces 2.2.1 The ∆2 -condition . . . . . . . . . . . 2.2.2 Absolute continuity of the norm . . Separability . . . . . . . . . . . . . . . . . Duality of Musielak-Orlicz spaces . . . . . 2.4.1 Conjugate Musielak-Orlicz functions 2.4.2 Dual of Lϕ (Ω) . . . . . . . . . . . . Density of regular functions . . . . . . . . Uniform convexity of Musielak-Orlicz spaces

1 2 5 10 21 25 27 30 33 38 40 48 55 61 70 82 84 91

. . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

91 101 110 113 115 118 118 120 124 128 vii

viii

Contents 2.7

Carath´eodory functions and Nemytskii operators on MusielakOrlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Further properties of variable exponent spaces . . . . . . . . 2.8.1 Duality maps on spaces of variable integrability . . . . 2.9 The Matuszewska-Orlicz index of a Musielak-Orlicz space . . 2.10 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . 3 Sobolev spaces of Musielak-Orlicz type 3.1 3.2 3.3 3.4

Sobolev spaces: definition and basic properties . . . 3.1.1 Examples . . . . . . . . . . . . . . . . . . . . Separability . . . . . . . . . . . . . . . . . . . . . . Duality of Sobolev spaces of Musielak-Orlicz type . Embeddings, compactness, Poincar´e-type inequalities

161 . . . .

. . . . .

. . . . .

. . . . .

. . . . .

4 Applications 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

132 137 150 151 155

Preparatory results and notation . . . . . . . . . . . . . . . . Compactness of the Sobolev embedding and the modular setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The variable exponent p-Laplacian . . . . . . . . . . . . . . . 4.3.1 Stability of the solutions . . . . . . . . . . . . . . . . . Γ-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . The eigenvalue problem for the p-Laplacian . . . . . . . . . . Modular eigenvalues . . . . . . . . . . . . . . . . . . . . . . . Convergence properties of the eigenvalues and eigenfunctions Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . .

161 164 165 166 166 181 181 186 190 198 206 210 219 228 243

Bibliography

245

Author index

253

Subject index

255

Notation index

259

Preface

Many years later, facing the need of a mathematical description of advanced physical problems, analysts were to remember that distant work by W. Orlicz which conduced to Musielak-Orlicz spaces. The main focus of this book is to present a unified theory of MusielakOrlicz function spaces while at the same time exploring its ties with other function spaces and its connections with different areas of Analysis. We will place special emphasis on the utility of these spaces in several concrete applications. In particular, we will observe that a wide class of problems arising in nonlinear PDE can be elegantly formulated and efficiently solved if properly translated into the language of Musielak-Orlicz spaces. To the best of the authors knowledge, the first occurrence of what would later be called Musielak-Orlicz spaces is to be traced back to Orlicz’ realization in 1931 [85] of the fact that the exponent p in the definition the Lp spaces introduced by Riesz could be replaced by a function, without losing the essence of the theory. More precisely, in the cited work, Orlicz introduced the class of measurable functions u for which Z

1

|u(x)|p(x) dx < ∞

(0.1)

0

and suggested that “A variety of results about integrable functions with constant p can be generalized to certain classes of functions that are integrable with a function p.” Nineteen years thereafter, Nakano [83] introduced the class of real valued functions u on a domain Ω for which Z ϕ(x, |u(x)|)dx < ∞. Ω

Here ϕ : Ω × [0, ∞) −→ [0, ∞) is a suitable function, which is called a Musielak-Orlicz function (see Chapter 2 for the precise definition). The class introduced by Nakano is plainly a very natural and elegant generalization of that introduced by Orlicz in [85], and will in fact constitute the main focus of this book. ix

x

Preface

It is worth mentioning that Orlicz’ work [85] was mainly motivated by his interest in [8, Satz a, Satz b] in which Banach addresed the question of summability of the coefficients of lacunary Fourier series, i.e., series of the form ∞ a0 X + (an cos kn x + bn sin kn x) , 2 1 where, for all n ∈ N, it holds that kn+1 /kn ≥ r > 1. See also [9] for a remarkable comment by Banach on Orlicz work. It was this endeavor that lead Orlicz to the consideration of spaces of sequences (an ) for which ∞ X |λan |pn < ∞, 1

for some λ > 0 and a given sequence (pn ) ⊂ [1, ∞). In fact, the alert reader will realize that these spaces are nothing but the discrete version of the Lebesgue spaces with variable integrability in [85]; this sequence spaces will in the sequel be denoted by `pn . After the publication of [85] by Orlicz, fundamental research on Lebesgue spaces with variable exponent remained largely dormant until Nakano’s reintroduction of the concept in the general form described above, in 1950. In 1959, Musielak and Orlicz [81] pursued Nakano’s idea from a systematic point of view; the product was laid down in final form by J. Musielak himself in 1983. [82]. Parallel to the above historical developments a new class of problems was gradually emerging in the realm of partial differential equations and variational calculus. These problems involved non-standard growth phenomena and already in the late 1920’s presaged the practical utility of the theoretical developments pursued by Musielak, Orlicz and Nakano and in particular, of the spaces with variable integrability. Notwithstanding the fact that the history of non-standard growth phenomena is convoluted and not easy to follow chronologically, we will mainly elaborate on progress made after 1990. We refer the interested reader to [24] and the references therein for some articles involving variable exponent spaces published between 1959 and 1990. In 1997 V. Zhikov [109], motivated by equations arising in electromagnetism was led to the minimization of integrals of the type Z |∇u(x)|p(x) dx; Ω

these minimization problems conduce to the corresponding Lagrange-Euler equation:   ∆p(·) u := div |∇u|p(·)−2 ∇u = 0. (0.2) Because of the variability of p(x), (0.2) is an example of differential equation

Preface

xi

with non-standard growth. The natural habitat for the solutions of such differential equations must take into consideration the dependence of p(x) on the space variable x. It is at this point obvious that the classical Lp theory is not sufficient in this situation and that a condition such as (0.1) for ∇u seems to be desirable as an a priori requirement on the solutions, i.e.: Z |∇u(x)|p(x) dx < ∞. Ω

Through these applications, then, there inexorably emerged the need for a deeper understanding of these generalized functional spaces with variable integrability. It was moreover observed by Zhikov that the variability of the exponent p(x) might cause the following undesirable situation to occur: Z Z p(x) inf |∇u(x)| dx < inf |∇u(x)|p(x) dx. 1,1 1,∞ u∈W

(Ω)

u∈W

Ω

(Ω)

Ω

The preceding discrepancy of the infima is known as Lavrentiev’s phenomenon, was observed first by Lavrentiev [69] and can occur in the variable exponent setting unless some conditions are assumed on the exponent p(x). This observation furthered the need of a deeper understanding of the structure of spaces of Musielak-Orlicz type (density of smooth functions, duality, weak convergence etc.). Addresing this need is the driving force of this book. The book is organized in the following manner: In Chapter 1 we lay the ground work for the concept of Musielak-Orlicz spaces by introducing notation, terminology and by reviewing some aspects of the abstract theory of function spaces. In the next chapter, Musielak-Orlicz spaces are introduced and exhaustively studied. Chapter 3 focuses on the theory of Sobolev spaces in the context of Musielak-Orlicz spaces. Chapter 4 is intended to explore some applications of the theory developed in Chapters 1 to 3. Chapter 1 is divided into five sections. After a brief note on the notation and terminology to be used in the sequel in Section 1.1, we move to Section 1.2, where the basic abstract theory of function spaces is introduced and some elementary geometric Banach space properties discussed in certain detail. Section 1.3 focuses on the basic general theory of modular spaces. We emphasize for the most part the aspects of the theory that are necessary to undertake the analysis of spaces of Musielak-Orlicz type. Particular focus is placed on the difference between modular and norm-convergence, conjugation and duality. Section 1.4 is exclusively devoted to concrete applications of the foregoing theoretical material. In particular, close attention is paid to those aspects of the variable exponent sequence spaces `pn that are in fact inherent to their modular structure. Specifically, we digress on duality questions of the variableexponent spaces `pn , underlining the special case in which supN pn = ∞, that

xii

Preface

is in the absence of the ∆2 -condition. Close attention is paid to some geometric properties of the modular whose relevance in the theory of fixed point has only recently been recognized. In this regard it is worth mentioning that the uniform convexity of the `pn norm is very cumbersome both to verify and to use under the assumption of a variable exponent (pn ). Nevertheless, under mild assumptions on the exponent sequence (pn ), the modular that defines the variable exponent `pn spaces enjoys some sort of uniform convexity which, while weaker than the uniform convexity property of the norm, is much friendlier to mathematical intuition and it turns out to be sufficient to achieve a generalization of several classical results in fixed point theory, to the variable-exponent setting. It is with this particular type of application in mind that we included Section 1.4. We conclude our discussion in Chapter 1 in Section 1.5 by introducing Orlicz and Lp(x) spaces. The reason for introducing these examples is twofold: they represent concrete instances of the modular structures introduced in Chapter 1 and in addition, could be considered to be forerunners of the Musielak-Orlicz spaces which will be the object of study in Chapter 2. In Chapter 2 we provide the foundations of the fundamental theory of spaces of Musielak-Orlicz type. Except for some modifications that reflect our personal taste, we mainly follow the classical treatment by J. Musielak [82] and [24]. Sections 2.1 to 2.6 deal with the basic functional-analytic properties of Musielak-Orlicz spaces such as embeddings, separability, absolute continuity of the norm and density of regular functions. Because of its relevance to the applications that will be discussed in Chapter 4, a discussion of the action of Nemytskii operators on Musielak-Orlicz spaces is included in Section 2.7. For reasons that will become clear in the next chapter we study in detail some properties of variable exponent spaces and of the Matuszewska-Orlicz index. We conclude Chapter 2 with a brief historical note about the origins of Musielak-Orlicz spaces. In Chapter 3 we introduce the Sobolev spaces of Musielak-Orlicz type. In Sections 3.1 to 3.3 we expound their basic properties. The interested reader is referred to [91] for a survey on Sobolev spaces of Orlicz type. The fundamental functional-analytic properties of Musielak-Orlicz-Sobolev-type spaces were studied in [44, 45, 46, 47, 48, 49, 50, 51]. In Section 3.4, we analyze necessary and sufficient conditions on the Musielak-Orlicz function ϕ, for the validity of the compactness of the Sobolev embedding W01,ϕ (Ω) ,→ Lϕ (Ω).

Chapter 4 is entirely devoted to applications of the results from the previ-

Preface

xiii

ous chapters. We underline the case of the variable exponent Lebesgue spaces and present some stability results for boundary value problems as well as for eigenvalue problems. Most of these results are based on our recent work [64, 65, 66, 67]. We end the chapter by providing a theoretical treatment for the general eigenvalue problem on spaces of Musielak-Orlicz type.

Chapter 1 A path to Musielak-Orlicz spaces

1.1 1.2

1.3

1.4

1.5

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Banach function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The associate space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Absolute continuity of the norm and continuity of the norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Convexity, uniform convexity and smoothness of a norm 1.2.4 Duality mappings and extremal elements . . . . . . . . . . . . . . Modular spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Modular convergence and norm convergence . . . . . . . . . . . . 1.3.2 Conjugate modulars and duality . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Modular uniform convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The `pn sequence spaces and their properties . . . . . . . . . . . . . . . . . . . 1.4.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Finitely additive measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Geometric properties of `pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Applications: Fixed point theorems on `pn spaces . . . . . . 1.4.5 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forerunners of the Musielak-Orlicz class: Orlicz spaces, Lp(x) spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 5 10 21 25 27 30 33 38 40 48 55 61 70 82 84

Introduction

Here we provide basic information about Banach spaces and Banach function spaces. Proofs of absolutely standard results are often eschewed in favor of references to works in which full details are given. To keep technicalities at a reasonable level, this monograph will only deal with real vector spaces. The norm on a normed linear space X will usually be written as k·kX , or even k·k if the context is clear; often k·kp or k·kp,I will be used to represent the norm on Lp (I). The dimension of X is denoted by dim X; and BX (resp. SX ) will stand for the closed unit ball (resp. the unit sphere) in X; B(x, r) will denote the open ball in X with center x and radius r. If A is a subset of a Banach space X with norm k · kX , the closure of A in k · kX will be denoted by Clk·kX (A). By B(X, Y ) will be meant the space of all bounded linear maps from a Banach space X to another such space Y, 1

2

Analysis on Function Spaces of Musielak-Orlicz Type

written as B(X) when Y = X, and as X ∗ (the dual of X) when Y is the space of scalars associated with X; the value of x∗ ∈ X ∗ at x ∈ X is denoted by hx∗ , xiX or hx∗ , xi . Weak convergence in X is represented by a half arrow *, ∗ weak∗ -convergence in X ∗ by * . If X is continuously embedded in Y we write X ,→ Y. The symbol X ,→,→ Y will be used to indicate that X is compactly embedded into Y . The characteristic function of a set S is written as 1S and the Lebesgue n-measure of a subset Ω of Rn will be denoted by |Ω|n , or |Ω| if there is no ambiguity. S∞ will stand for the vector space of all finite linear combinations of characteristic PNfunctions of measurable sets; the vector space of all simple functions w = k=1 wk 1Ek with |EK | < ∞ will be denoted with S.

1.2

Banach function spaces

Much of the mathematical work to be presented in this book relies on the basic notion of Banach function space. For the remainder of this section we fix a measure space (Ω, A, µ) with a σ-finite measure µ, i.e., it is assumed that there exists an increasing sequence (Ωj ) such that µ(Ωj ) < ∞ and Ω=

∞ [

Ωj .

(1.1)

j=1

In particular, if Ω ⊂ Rn is a domain, A will be understood to be the Borel class of subsets of Ω, µ will denote the Lebesgue measure on A and M(Ω) will stand for the set of all Borel measurable, real valued functions on Ω, with the usual identification u = v iff u(x) = v(x) almost everywhere. Definition 1.2.1. A linear space X ⊆ M(Ω) is called a Banach function space if there is a map k·kX : M(Ω) → [0, ∞] with the properties of a norm and such that (i) u ∈ X if and only if kukX < ∞; (ii) kukX = k|u|kX for all u ∈ M(Ω); (iii) if 0 ≤ uk % u, then kuk kX ↑ kukX ; (iv) if E ⊂ Ω and µ(E) < ∞, then k1E kX < ∞; (v) if E ⊂ Ω and µ(E) < ∞, there is a constant c(E) (i.e., independent of u) such that for all u ∈ X, Z |u| dµ ≤ c(E) kukX . E

A path to Musielak-Orlicz spaces

3

In particular, property (iv) implies that any simple function that is a finite linear combination of characteristic functions of sets of finite measure, is in X. In addition, axiom (iii) implies the following lattice property: If u ∈ X and |v| ≤ |u| a.e. in Ω, then necessarily v ∈ X. The distribution function of an element u of a Banach function space X is the map µu : [0, ∞) → [0, ∞] defined, for all λ ≥ 0, by µu (λ) = µ ({x ∈ Ω : |u(x)| > λ}) . Two functions f, g ∈ X are said to be equimeasurable if their distribution functions coincide; X is called rearrangement-invariant if kf kX = kgkX whenever f and g are equimeasurable. Notice that if u is an element of a Banach function space X, then a fortiori |u(x)| < ∞ a.e. which follows straight from axiom (v). Indeed, since |u| is assumed to be σ-finite, (1.1) holds and Z |u(x)| dµ ≤ C(Ωj ∩ {x : |u(x)| = ∞})kukX < ∞. Ωj ∩{x:|u(x)|=∞}

The claim follows at once. We also underline the following observation: Lemma 1.2.1. If a sequence (uj ) of measurable functions converges a.e. to u and if lim inf kuj kX < ∞, then u ∈ X and j→∞

kukX ≤ lim inf kuj kX . j→∞

Proof. Since u is automatically measurable to show that u ∈ X it suffices to prove that kukX < ∞. Writing vn = inf |uk (x)| then vn % lim inf |un | = |u|, n→∞

k≥n

pointwise a.e. Hence, by Axiom (iii) kukX = lim kvn kX ≤ lim k inf |uk |kX ≤ lim inf k|un |kX < ∞. n→∞

n→∞

n→∞

k≥n

The following lemma, though simple, is of paramount importance in the sequel: Lemma 1.2.2. If (un ) ⊂ X is a (norm)-convergent sequence of functions in a Banach function space X, then (un ) is a.e. convergent to its norm limit u ∈ X. Proof. Pick E ⊂ Ω with |E| < ∞; Axiom (v) in Definition 1.2.1 yields: Z µ ({x ∈ E : |un (x) − u(x)| > δ}) ≤ δ −1 |un − u| dµ E

≤ c(E)δ −1 kun − ukX → 0 as n → ∞.

4

Analysis on Function Spaces of Musielak-Orlicz Type

It follows that (un ) converges a.e. in every subset of finite measure; the result follows from the σ-finiteness. Theorem 1.2.3. [11, Theorem 1.6] Every Banach function space X is a Banach space when endowed with the norm k·kX . P∞ Proof. It suffices to show that every absolutely convergent series n=1 un is convergent. To this end, let vm =

m X

|un |;

n=1

it is clear that vm % v =

P∞

n=1

|un | and that, by assumption,

kvm kX ≤

∞ X

kun kX < ∞

n=1

for each m ∈ N. Thus, lim inf kvm kX < ∞ and it follows from Lemma 1.2.1 m→∞ P∞ that v ∈ X and kvkX ≤ lim inf kvn kX . On account of Lemma 1.2.2, n=1 |un | n→∞ P∞ is almost everywhere convergent (to v); therefore, so is n=1 un . We set u=

∞ X

un .

n=1

It will be shown next that u ∈ X (that is, that kukX < ∞) and that

m

X

un − u → 0 as m → ∞.

1

(1.2)

X

To this end, we observe that if wn =

Pn

k=1

uk then for fixed m ∈ N,

wn − wm −→ u − wm as n −→ ∞ pointwise a.e. and since for m < n, kwn − wm kX ≤

n X

kuk kX
n3 and on account of Axiom (ii) in Definition 1.2.1 it is clear that one can choose un ≥ 0. The series ∞ X n−2 un 1

is absolutely convergent (and hence convergent) in X1 , say ∞ X

n−2 un = u ∈ X1 ⊆ X2 .

1

However, since n−2 un ≤ u Axiom (iii) in Definition 1.2.1 yields n ≤ n−2 kun kX2 ≤ kukX2 , which is a contradiction.

1.2.1

The associate space

We next turn to a fundamental concept in the theory of Banach function spaces, namely, that of the associate space. Let (X, k·kX ) be a Banach function space on the measure space (Ω, A, µ). The function k · kX 0 : M(Ω) −→ [0, ∞] Z kvkX 0 := sup |vu| dµ u∈SX

(1.4)

Ω

is easily verified to have the properties of a norm, i.e., it satisfies the triangle inequality, is homogeneous and kvkX 0 = 0 ⇔ v = 0.

6

Analysis on Function Spaces of Musielak-Orlicz Type

Theorem 1.2.5. Given a Banach function space X, the set   Z   X 0 := u ∈ M(Ω) : sup |vu| dµ < ∞   u∈SX Ω

is a Banach function space called the associate space of X. H¨ older’s inequality holds in the form Z |vu| dµ ≤ kvkX kukX 0 Ω 0

for all v ∈ X and u ∈ X . Proof. It is a routine matter to check that X 0 is a linear subspace of M(Ω). The fact that the functional u −→ kukX 0 has the properties of a norm follows from straightforward calculations. It remains to prove the axioms (i) to (v) in Definition 1.2.1. Axiom (i) transpires directly from the definition of X 0 . Property (ii) of Definition 1.2.1 is obvious. For property (iii), let u ∈ M(Ω) and (uj ) ⊂ M(Ω) with 0 ≤ uj % u. One has, for arbitrary v ∈ SX , Z Z uk |v| dµ ≤ uk |v| dµ. Ω

Ω

By virtue of monotone convergence one concludes that for any v ∈ SX , Z Z uk |v|dµ % u|v|dµ. (1.5) Ω

Ω

If kukX 0 = ∞, then for any N ∈ N there exists vN ∈ SX such that Z u|vN | dµ > N Ω

and (1.5) implies that Z uk |vN | dµ > N Ω

for large enough k, whence the arbitrariness of N yields that kuk kX 0 % ∞ as k → ∞. If on the other hand, kukX 0 < ∞, fix δ > 0 and v ∈ SX such that Z kukX 0 − δ < |u(x)||v(x)| dx. Ω

A path to Musielak-Orlicz spaces

7

It follows from the convergence statement (1.5) that for large enough k one has Z kukX 0 − δ < uk |v|dµ ≤ kuk kX 0 , Ω

i.e., kuk kX 0 % kukX 0 as claimed. Property (iv) follows at once, for if E ⊂ Ω and |E| < ∞ and v ∈ SX , one has on account of property axiom (v) for X, that Z (1E v) dµ ≤ c(E), Ω

which immediately yields the desired result. Property (v) is also simple, for if E ∈ A with µ(E) < ∞ and u ∈ X 0 , then: Z Z 1E |u|dµ = k1E kX |u| dµ ≤ k1E kX kukX 0 . k1E kX E

Ω

H¨ older’s inequality follows directly by the definition of kukX 0 . Lemma 1.2.6. Let v ∈ M(Ω). Then v ∈ X 0 if and only if for every u ∈ X, uv ∈ L1 (Ω). In particular, it holds the equality:   Z   X 0 = v ∈ M(Ω) : |vu| dµ < ∞ for all u ∈ X .   Ω

Proof. H¨ older’s inequality shows the necessity of the L1 -condition. Let now v ∈ M(Ω). If Z kvkX 0 = sup |uv| dµ = ∞ u∈SX

Ω

one can extract a sequence (fn ) ⊂ SX such that for each n ∈ N,

R

|vun |dµ >

Ω

n3 . The series

∞ X

n−2 |un |

1

is an absolutely convergent series in X, it is therefore convergent to u ∈ X and by virtue of Lemma 1.2.2 the series converges pointwise a.e. to u. Since |un |n−2 ≤ f pointwise a.e., Z Z u|v|dµ ≥ n−2 |un ||v|dµ > n, Ω

i.e., uv ∈ / L1 (Ω).

Ω

8

Analysis on Function Spaces of Musielak-Orlicz Type

Every Banach function space X coincides with its second associate space 0 X 00 := (X 0 ) , and for any u ∈ X it holds that kukX = kukX 00 . The next lemma aims at the proof of this essential fact (see [11, Theorem 2.7] and [77]). ˜ ⊂ Ω be domains with µ(Ω) ˜ < ∞ and set Lemma 1.2.7. Let Ω n o ˜ . L˜1 = w ∈ L1 (Ω) : supp w ⊆ Ω Then L˜1 is a Banach space with the norm Z kwk = |w|dµ ˜ Ω

and if X ⊆ M(Ω) is a Banach function space, then Bx ∩ L˜1 is a closed, convex subset of L˜1 . Proof. It is routine to verify that L˜1 is a Banach space and that BX ∩ L˜1 is convex. For the closedness, observe that if SX ∩ L˜1 3 wj → w ∈ L˜1 in L˜1 , then it can be assumed without loss of generality that wj −→ w pointwise a.e. in Ω. By assumption lim inf kwj kX ≤ 1. j→∞

On account of Lemma 1.2.1 one has then w ∈ BX . Theorem 1.2.8. Any Banach function space is identical with its second as00 sociate space X 00 . In other words, it holds the set theoretical equality X = X and for any u ∈ X one has kukX = kukX 00 . R Proof. For fixed u ∈ X, |uv|dµ < ∞ for all v ∈ X 0 : this follows from Lemma Ω

00

1.2.6; this yields the set theoretical inclusion X ⊆ X . On the other hand, H¨ older’s inequality implies kukX 00 ≤ kukX . For the remaining inclusion we invoke the σ-finiteness assumption (1.1). For 00 each u ∈ X , k ∈ N set uk = min {|u(x)|, k} 1Ωk . Then 0 ≤ uk % f pointwise a.e. and each uk is dominated by a simple function. Since simple functions belong to any Banach function space, it is 00 clear that uk ∈ X ∩ X . Next, we set about to show that for every k ∈ N it holds that kuk kX ≤ kuk kX 00 . (1.6)

A path to Musielak-Orlicz spaces

9

Once established, inequality (1.6) in conjunction with Axiom (iii) in Definition 1.2.1 will yield kuk kX % kukX ≤ kukX 00 . Clearly, (1.6) holds if kuk kX = 0; it can be supposed in the sequel, therefore, ˜ = Ωk (L˜1 = L1 ) that kuk kX 6= 0. Let R 3 α > 1; Lemma 1.2.7 applies for Ω k and

α uk ∈ L1k \ SX . kuk kX

 ∗ The Hahn-Banach Theorem, then yields the existence of γ ∈ L∞ (Ω) = L˜1 and of a real number β such that, for all a ∈ L1k ∩ SX . Z Z α γa dµ < β < γ uk dµ. kuk kX Ωk

Ωk

It follows easily from here that Z Z sup |γa| dµ ≤ β ≤ |γ| a∈BX ∩L1k

Ωk

α |uk | dµ. kuk kX

Ωk

Next, observe that any w ∈ BX can be written as the pointwise limit w = lim wj (x), j→∞

where wj (x) = min{w(x), j}1Ωj . The monotone convergence theorem implies the equality Z Z sup |γa| dµ = sup |γa| dµ. a∈BX ∩L1k

a∈BX

Ωk

Ωk

In conclusion: Z kγk

X0

|γa| dµ ≤ β

= sup a∈BX

≤

Ωj

α kuk kX

Z |γuk | dµ Ωj

and it is concluded from the last inequality that for any α : α > 1 it holds Z α kuk kX ≤ |γ|uk | dµ kγkX 0 Ωj

α ≤ kγkX 0 kuk kX 00 kγkX 0 = αkuk kX 00 ; i.e., (1.6) holds and the theorem is proved.

10

Analysis on Function Spaces of Musielak-Orlicz Type

In general, the associate space of X is (canonically isomorphic to) a closed subspace of the dual X ∗ . It is natural to investigate circumstances under which X 0 coincides with X ∗ . With this in mind, in the forthcoming section (Definition 1.2.3) we shall introduce the concept of absolute continuity of the norm and study its connection to duality. Theorem 1.2.9. The map I : X 0 −→ X ∗ Z hI(u), vi = uv dµ Ω 0

defines an isometry from X to a closed, norm-fundamental subspace of X ∗ , i.e., for any v ∈ X it holds that kvkX =

|hγ, vi|.

sup γ∈BI(X 0 )

Proof. It is clear that I is well defined, linear and injective. By virtue of H¨ older’s inequality, it is also bounded. In addition, for u ∈ X 0 it follows by definition that Z kukX 0 = sup uv dµ = kukX ∗ . (1.7) kvkX ≤1 Ω

0

Hence I is an isometry from X onto its image I(X 0 ). The latter is closed, for if I(vj ) = γj → γ ∈ X ∗ in the topology of X ∗ , then (γj ) is Cauchy in X ∗ and it is apparent from (1.7) that (vj ) is Cauchy in X 0 . Next, we invoke the completeness of X 0 on account of Theorem 1.2.3 and Theorem 1.2.5. Hence, vj → v ∈ X 0 as j → ∞ and it follows easily that γ = I(v). Finally, if v ∈ X Theorem 1.2.8 yields Z kvkX = kvkX 00 = sup vu dµ kukX 0 ≤1 Ω

=

sup

|hγ, vi| .

kγkI(X 0 ) ≤1

The last assertion finished the proof.

1.2.2

Absolute continuity of the norm and continuity of the norm

We begin this section with a note on a measure-theoretic concept. Let (Ω, A, µ) be a measure space as in Section 1.2 we assume in particular that µ is σ-finite (1.1).

A path to Musielak-Orlicz spaces

11

Definition 1.2.2. A sequence (Mn ) of measurable sets is said to converge a.e. to the empty set, written Mn → ∅ a.e., if the indicator function 1Mn −→ 0 pointwise a.e. in X. Lemma 1.2.10. In the preceding notation, the two following conditions are equivalent: (i) Mn → ∅ as n → ∞,

(ii) µ

∞ S ∞ T

! Mj

= 0.

n=1 j=n

Proof. The proof is elementary and follows from the following set-theoretic equality: {x ∈ Ω : 1Mn (x) → 0 as n → ∞} = Ω \

∞ [ ∞ \

Mj .

n=1 j=n

Lemma 1.2.11. Let X be a Banach function space as in Section 1.2, u ∈ X. Then the following conditions are equivalent: (i) ku1Mn kX → 0 as n → ∞ for any sequence (Mn ) ⊆ A such that Mn → ∅ as n → ∞, (ii) ku1Mn kX & 0 as n → ∞ for any decreasing sequence (Mn ) ⊆ A such that Mn → ∅ as n → ∞. Proof. The necessity is obvious from Property (iii) in Definition 1.2.1. Assuming now (ii), consider a sequence of measurable sets (Mn ) with Mn → ∅ as n → ∞. Then [ An = Mk k≥n

is decreasing and the obvious set-theoretic identity ∞ [ \ m=1 n≥m

An =

∞ [ \

Mn

m=1 n≥m

shows that µ(lim sup Mn ) = µ(lim sup An ) = 0. n

n

According to Lemma 1.2.10, then ku1Mn kX −→ 0 as n −→ ∞.

12

Analysis on Function Spaces of Musielak-Orlicz Type

Definition 1.2.3. Let X be a Banach function space. A function u ∈ X is said to have absolutely continuous norm if it satisfies either of the equivalent conditions of Lemma 1.2.11. In the sequel, we set Xa = {u ∈ X : u has absolutely continuous norm } . In the event that Xa = X, X will be said to have absolutely continuous norm. Lemma 1.2.12. If X is a Banach function space, then Xa is a k · kX -closed subspace of X. Moreover, if u ∈ Xa , v ∈ X and |v| ≤ |u| a.e., then necessarily v ∈ Xa . Proof. The algebraic vector space structure of Xa follows easily by definition. If uk −→ u in norm and (Ej ) is a sequence of measurable subsets of Ω such that Ej → ∅, then ku1Ej kX ≤ k(u − un )1Ej kX + kfn 1Ej kX ≤ k(u − un )kX + kun 1Ej kX → 0 as n, j → ∞. The second asseveration is a direct consequence of the remarks following Definition 1.2.3. Example 1.2.1. Lp (Ω): For what is possibly the simplest example, consider the Lebesgue space Lp (Ω), 1 ≤ p < ∞ on (Ω, A, µ). For a sequence of measurable sets Mn & ∅ and f ∈ Lp (Ω), one has, on account of Lebesgue’s dominated convergence: Z p kf 1Mn kLp (Ω) = |f |p 1Mn dµ → 0 as n → ∞. Ω p

Hence, every f ∈ L (Ω) has absolutely continuous norm. Example 1.2.2. L∞ (Ω): If Ω ⊆ Rn is a domain and µ is the Lebesgue measure on the Borel-measurable sets, the only function with absolutely continuous norm in the Lebesgue space L∞ (Ω), however is the a.e. zero function. For, if f ∈ L∞ (Ω), {x ∈ Ω : f (x) > 0} =

∞  [ k=0

x ∈ Ω : f (x) >

1 k



and consequently, if µ({x ∈ Ω : f (x) > 0}) > 0, one has, for some k0 ∈ N   1 µ x ∈ Ω : f (x) > > 0. k0

A path to Musielak-Orlicz spaces 13 n o Let E0 = x ∈ Ω : f (x) > k10 . There exists a compact set S ⊆ E0 with µ)(S) > 0. Consider an increasing sequence of domains (Ωk ), Ωk ⊂ Ω such that ∞ [ Ω= Ωk . k=1

Thus, Sk = S

T

(Ω \ Ωk ) & ∅, however kf 1Sk kL∞ (Ω) ≥

1 . k0

It follows that f does not have absolutely continuous norm, unless f = 0 a.e.. Example 1.2.3. `∞ : Next, consider the space `∞ consisting of all bounded sequence of real numbers endowed with the norm k(an )k∞ = sup |an |.

(1.8)

n∈N

Notice that this space can be viewed as a special instance of the general case discussed in this Section
1.2, namely, as a Banach function space based on the measure space N, 2N , µ , where µ stands for the counting measure and the function norm is given by (1.8). In this context, the class of sequences in `∞ with absolutely continuous norm is precisely the set of sequences that vanish at ∞. More precisely: Lemma 1.2.13. A sequence (an ) ∈ `∞ has absolutely continuous norm if and only if lim an = 0. n→∞

It is customary to denote the space `∞ a with c0 . Proof. If (an ) has absolutely continuous norm consider the sequence Mn = {x ∈ N : x ≥ n}. Evidently, Mn & ∅ and it follows automatically from the definition that for any positive , k(ak )k≥n k∞ <  if n is large enough, which yields (an ) ∈ c0 . Conversely, assume that (an ) ∈ c0 and that Mn & ∅. For any δ > 0, sup |an | < δ for some n0 ∈ N. By n≥n0

assumption, there exists n1 ∈ N such that n ≥ n1 ⇒ 1Mn (j) = 0 for 1 ≤ j ≤ n0 . Consequently, for k ≥ n1 k(an )1Mk k∞ = sup |aj | < δ. j>n0

It follows that (an ) has absolutely continuous norm, as claimed.

14

Analysis on Function Spaces of Musielak-Orlicz Type

Theorem 1.2.14. Let u be a function in a Banach function space X. Then (i) ⇔ (ii) ⇔ (iii) ⇒ (iv). (i) u has absolutely continuous norm. (ii) Given any sequence (un ) of measurable functions on X with un & 0 as n → ∞ and un ≤ |u| a.e., one has kun kX & 0. (iii) For any measurable function v and every sequence of measurable functions (vn ) such that vn → v a.e. and |vn | ≤ |u| a.e., for all n,

(1.9)

one has kvn − vkX −→ 0 as n → ∞. (iv) For each  > 0 there exists δ > 0 such that for any measurable set M with µ(M ) < δ it holds that ku1M kX < . Proof. (i) ⇒ (iv): If (i) holds, (iv) follows by contradiction by taking a sequence (Mn ) of measurable sets with µ(Mn ) < 2−n such that kun kX ≥  and observing that ! ∞ [ µ Mn < 21−k −→ 0 as n −→ ∞. k

Hence (i) ⇒ (iv). Assuming next that (ii) holds, take u ∈ X and consider a sequence (Mn ) of measurable sets such that Mn & ∅. Then, a.e. one has u1Mn → 0 as n → ∞ and u1Mn ≤ |u|. The valididty of (ii) yields ku1Mn kX → 0 as n → ∞, so that u as absolutely continuous norm. This shows that (ii) ⇒ (i). To see that (i) ⇒ (ii), take u ∈ Xa and (un ) be a sequence of measurable functions as in (ii). Since (Ω, M, µ) is assumed to be σ-finite, one can take a sequence of mutually disjoint sets (Ωj ) ⊂ M as in (1.1), i.e., µ(Ωj ) < ∞ for each j ∈ N and ∞ S Ω = Ωj . Obviously, one has (Ω \ Ωj ) & 0 as j → ∞; thus for any  > 0 and 1

sufficiently large k one has ku1Ω\Ωk kX
0, Z 1 µ({x ∈ Ωk : uk (x) > δ}) ≤ uk 1Ωk dµ → 0 as k → ∞. δ Ω

Therefore, for sufficiently large k, property (iv) yields the inequality ku1{x∈Ωk :uk (x)>δ} kX
δ} kX + kuk 1Ω\Ωk kX < .

In all, (i) ⇒ (ii). (i) ⇒ (iii): Assume u has absolutely continuous norm as let (vn ) and v be measurable functions having property (1.9). Setting wk (x) = sup |vj (x) − v(x)| j≥k

it is readily verified that wk & 0 as k → ∞ and that a.e., 0 ≤ wk ≤ 2|u|. Notice that by definition, 2|u| has absolutely continuous norm; a direct application of (ii) yields then kvk − vkX ≤ kwk kX & 0 as k → ∞. (iii) ⇒ (i): For the sufficiency of condition (iii), select u ∈ X, (Mk ), let be a sequence of measurable sets with Mk & ∅ and set vk = u1k . Then |u| ≥ vk & 0 a.e. and the assumed validity of property (iii) implies kvk kX = ku1Mk kX & 0 as k → ∞. For reasons that will naturally emerge in the sequel, another important subspace of a Banach function space X needs to be introduced: Definition 1.2.4. If X is a Banach function space, Xb will stand for the k·kX closure of the set of bounded functions supported in sets of finite measure. In other words Xb = Clk·kX ({u ∈ X : u bounded and |supp u| < ∞}) .

16

Analysis on Function Spaces of Musielak-Orlicz Type The following characterization is almost immediate:

Lemma 1.2.15. Xb is the k·kX -closure of the subspace S consisting of simple functions supported on sets of finite measure. Proof. If v is a bounded function supported on a set of finite measure, let (vn ) be a sequence of simple functions that converges to v uniformly in Ω; such sequence can for example be chosen as vn =

2n−1 2X

2−n j1v−1 (j2−n ,(j+1)2−n ) ,

(1.10)

j=0

if sup v ≤ 2m . Then, for any δ > 0 set  = Ω

δ |Ω|

and for sufficiently large n it

holds kv − vn kX ≤ δ.

Lemma 1.2.16. For any Banach function space X, Xa ⊆ Xb . Proof. If v ∈ Xa it is easy to verify that vn (x) = sgn(v(x)) min {|v(x)|, n1Ωn } , where the sequence (Ωn ) is as in (1.1), defines a sequence of bounded functions that are supported in a set of finite measure with vn → v pointwise a.e. and such that |vn | ≤ |v|. On account of Theorem 1.2.14 (iii) we conclude: kvn − vkX → 0 as n → ∞, i.e., v ∈ Xb . Definition 1.2.5. An order ideal of a Banach function space X is a closed, linear subspace W of X such that if u ∈ W and |v| ≤ |u| a.e., then v ∈ W . Lemma 1.2.17. If X is a Banach function space, then both Xa and Xb are order ideals of X. Proof. As shown in Lemma 1.2.12, Xa is closed. It remains thus to show the order property required for an order ideal. To this end, observe that for any sequence (Ej ) of measurable sets and any u ∈ Xa , if v ∈ X and |v| ≤ |u| a.e., the Banach function space properties of X yield kv1Ej kX ≤ ku1Ej kX → 0 as j → ∞. Next, since Xb is norm-closed by definition, it remains to observe that if u ∈ Xb and |v| ≤ |u| a.e., then for any sequence (un ) ⊆ S that converges to u in norm one can define pointwise the following sequence of bounded functions, supported on sets of finite measure in Ω: vn (x) = (un (x)1un ≤v + v(x)1un >v ) 1v>0 + (v(x)1un >v + un (x)1un ≤v ) 1v≤0 for each n ∈ N. It is straightforward to verify that |vn − v| ≤ |un − u| a.e. and hence that kvn − vkX −→ 0 as n → ∞. The claim follows now by definition of Xb .

A path to Musielak-Orlicz spaces

17

Theorem 1.2.18. Let X be a Banach function space. Then the following conditions are equivalent: (i) Xa = Xb , ∗

(ii) (Xb ) = X 0 , (iii) X ∗ = X 0 , (iv) X = Xb = Xa . Proof. We will proceed with the proof in the real setting. Lemma 1.2.17 and Theorem 1.2.9 guarantee the continuous inclusion ∗

X 0 ,→ (Xb ) . (i) ⇒ (ii) For the necessity statement, it remains to verify that under as∗ sumption (i) the converse inclusion holds. Let γ ∈ (Xb ) and recall that the underlying measure µ is assumed to be σ-finite, i.e., there is a mutually disjoint sequence (Ωj ) with µ(Ωj ) < ∞ such that Ω=

∞ [

Ωj .

j=1

For each j we write Aj for the σ-algebra of measurable subsets of Ωj ; then γ defines a set function δj on Aj in the nearly obvious manner: δj (E) = γ(1E ). The linearity and continuity of γ guarantee the σ-additivity of δj , for if (Ek ) ⊂ Ωj is a collection of mutually disjoint µj -measurable sets then 1∪∞ ∈ Xb = 1 Ek Xa by definition of Xb and by assumption (i). Furthermore, for fixed m ∈ N, 1∪∞ − 1∪m & 0 a.e. as m → ∞ 1 Ek 1 Ek and on account of Theorem 1.2.14, (iii) one has k1∪∞ − 1∪m k & 0, 1 Ek 1 Ek X that is, δj

∞ [

! Ek

 
ME = γ 1∪∞ = γ lim 1 E ∪ k k 1 1 m→∞

1

∞   X = lim γ 1∪M = γ(1Ek ) 1 Ek m→∞

=

∞ X 1

δj (Ek ).

1

18

Analysis on Function Spaces of Musielak-Orlicz Type

Hence, δj is a signed measure on Aj . Moreover if one denotes the restriction of µ to Ωj by µj the inequality (valid for for any E ⊂ Aj ) |δj (E)| ≤ kγk(Xb )∗ k1E kXb shows that µj (E) = 0 ⇒ δj (E) = 0. An application of the Radon-Nikodym theorem yields, for each j ∈ N, a function vj ∈ L1 (Ωj ) such that for any E ∈ Aj : Z δj (E) = γ(1E ) =

vj 1E dµ. Ωj

Define v on Ω as v(x) = vj (x) for x ∈ Ωj . Then for any E ∈ A Z γ(1E ) = v1E dµ. Ω

It follows from the preceding asseveration that for any u ∈ S one has Z γ(u) = vu dµ. (1.11) Ω

In fact, as a consequence of (1.11) one concludes that v ∈ X 0 . In order to see this, observe that given u ∈ X, a sequence (uk ) of simple functions with suppuk ⊆ Ωk can be constructed via (1.10), such that 0 ≤ uk % |u| pointwise in Ω. By virtue of (1.11) and Axiom (iii) in Definition 1.2.1, Z Z |v|uk dµ = v(sgnv)uk dµ Ω

Ωk

= γ((sgnv)uk ) ≤ kγk(Xb )∗ ksgnv)uk kX ≤ kγk(Xb )∗ kukX , i.e., by monotone convergence, Z Z |vu| dµ = lim |v|uk dµ ≤ kγk(Xb )∗ kukX < ∞. Ω

k→∞ Ωk

(1.12)

Lemma 1.2.6 guarantees now that v ∈ X 0 . We set about next to prove that (1.11) holds in fact for any u ∈ Xb , then for (uk ) as in the above paragraph Z Z vuk dµ −→ vu dµ (1.13) Ω

Ω

A path to Musielak-Orlicz spaces

19

on account of (1.12) and dominated convergence. In particular, (seen(1.4), (1.12) and (1.13)): kvkX 0 ≤ kγk(Xb )∗ . We have proved that (i) ⇒ (ii). (ii) ⇒ (iii): Claim (iii) follows from (ii) via Theorem 1.2.9 and Definition 1.2.4. (iii) ⇒ (iv): Now, if (iii) holds, let w ∈ X and consider a sequence (Ej ) ⊂ A with Ej & ∅ as j → ∞. It is enough to consider w ≥ R0 on Ω. For any γ ∈ BX ∗ , by assumption, there exists u ∈ X 0 with γ(v) = v(x)u(x) dx for Ω

any v ∈ Xb and kγk(Xb )∗ = kukX 0 ≤ 1. Since w ∈ Xb and u ∈ X 0 , Lemma 1.2.6 and dominated convergence imply Z wu1Ej dµ → 0 as j → ∞; Ω

it follows that for fixed  > 0 and for sufficiently large j ∈ N, Z wu1Ej dµ < . Ω

Each of the sets  Yj = γ ∈ X ∗ : hγ, 1Ej i <  is weak∗ open (in fact, it is a neighborhood of the origin for the weak∗ topology on X ∗ ) and is in view of the preceding argument, the family (Yj ) constitutes an open cover of the unit ball BX ∗ . On account of the theorem of BanachAlaoglu there is a finite subcover, say (Yjk )1≤k≤N . For J > max{jk , 1 ≤ j ≤ N } and any γ ∈ BX ∗ one thus concludes |hγ, w1EJ i| < . On the other hand Theorem 1.2.8 guarantees that Z kw1EJ kX = kw1EJ kX 00 = sup v(w1EJ ) dµ kvkX 0 ≤1 Ω

≤ . In conclusion, kw1Ek kX → 0 as j → ∞; (iii), thus, implies (iv), as claimed. (iv) ⇒ (i) This implication is trivial. The theorem is thus proved.

20

Analysis on Function Spaces of Musielak-Orlicz Type As a consequence of theorem 1.2.18 we have the following characterization:

Theorem 1.2.19. Let X be a Banach function space. Then (i) X 0 and X ∗ are canonically isometrically isomophic if and only if X has absolutely continuous norm; (ii) X is reflexive if and only if both X and X 0 have absolutely continuous norm. Proof. Assume first that X is reflexive. It was shown in Theorem 1.2.9 that X 0 is isometric to a closed, norm-fundamental subspace of X ∗ . In view of Theorem 1.2.18, to conclude that X has absolutely continuous norm, i.e. that X = Xa , it suffices to show that X 0 cannot be a proper subspace of X ∗ . Assume, on the contrary, that X 0 ( X ∗ . Let 0 6= γ ∈ X ∗∗ be zero on X 0 (the existence of such γ is guaranteed by the Hahn-Banach Theorem). Since X is R reflexive one derives the existence of z ∈ X such that Ω zy dx = 0 for each y ∈ X 0 . By virtue of Theorem 1.2.9 X 0 is identified with a norm-fundamental subspace of X ∗ ; in particular, it holds that Z kzkX = sup | zy dx| = 0, y∈X 0

Ω 0

i.e., z = 0 a.e., which forces γ = 0, a contradiction. Thus, X = X ∗ and Theorem 1.2.9 guarantees that X has absolutely continuous norm. Moreover, the identification X 0 = X ∗ in conjunction with the reflexivity of X implies ∗ (X 0 ) = X ∗∗ ≈ X, ∗

0

i.e., (X 0 ) ≈ (X 0 ) (Theorem 1.2.8). Invoking Theorem 1.2.18 applied to the Banach function space X 0 , one readily concludes that X 0 has absolutely continuous norm, as claimed. 0 Conversely, if both X and its associate X have absolutely continuous norm, then by virtue of Theorem 1.2.18 applied to X one has the equality X ∗ = X 0 ; the dual spaces are hence the same, and it follows that ∗

X ∗∗ = (X 0 ) . 0

∗

Theorem 1.2.18 applies now to X to yield (X 0 ) = X 00 and the latter equals X by Theorem 1.2.9. In all, X ∗∗ ≈ X, as claimed. In particular, we state the obvious consequence of the preceding Theorem: Corollary 1.2.20. Let X be a Banach function space. Then X ∗ = X 0 if and only if X has absolutely continuous norm. In particular, if both X and X ∗ have absolutely continuous norm, X is reflexive. Example 1.2.4. The sequence spaces `p , 1 ≤ p < ∞ have absolutely continuous norm. The preceding Corollary implies then 0

∗

p

(`p ) = (`p ) = ` p−1 ,

A path to Musielak-Orlicz spaces with the convention ∗ `1 ( (`∞ ) .

1.2.3

p p−1

21 0

= ∞ when p = 1. It is easy to check that (`∞ ) =

Convexity, uniform convexity and smoothness of a norm

In this section we present some standard definitions and results pertaining to the geometry of Banach spaces. We commence with the following: Definition 1.2.6. A normed space (X, k · kX ) is said to be strictly convex if whenever u ∈ SX and v ∈ SX and 0 < t < 1, then ktu + (1 − t)vkX < 1. Typical examples of strictly convex spaces include the Lp (Ω, dµ) spaces, 1 < p < ∞, over a measure space (Ω, A, µ). It is well known that L∞ (Ω) the vector space of Lebesgue-measurable, complex valued function on a domain Ω furnished with the supremum norm is not strictly convex. We next state the standard definition of uniform convexity in the context of Banach spaces. Definition 1.2.7. A normed space (X, k · k) is uniformly convex iff for any  : 0 <  ≤ 2 one has

 

x + y

δX () = inf 1 − : kxk ≤ 1 , kyk ≤ 1 , kx − yk ≥  > 0. 2 The number δX () is known as the modulus of uniform convexity of X (see, for example, [17] and [52]). The next result indicates how strict convexity can be used. Proposition 1.2.21. Let K be a closed, convex, non-empty subset of a strictly convex Banach space X. Then there is at most one element x ∈ K such that kxk = inf{kyk : y ∈ K}. If X is reflexive, such an x exists. Proof. Suppose there exist x, y ∈ K with kxk = kyk = inf{kzk : z ∈ K}, x 6= y. Let 0 < λ < 1 : then λx + (1 − λ)y ∈ K, kλx + (1 − λ)yk < kxk and we have a contradiction. Now let X be reflexive and assume that (xk ) is a sequence in K such that limk→∞ kxk k = l := inf{kyk : y ∈ K}. This sequence has a subsequence, again denoted by (xk ), such that xk * x for some x ∈ X; and x ∈ K since K is convex and closed, and hence weakly closed. Since kxk ≤ limk→∞ kxk k = l, the result follows. A characterization of strict and uniform convexity can be given by means of differentiability considerations. Let U be an open subset of a Banach space

22

Analysis on Function Spaces of Musielak-Orlicz Type

X and f : U → R. We say that f is Gˆ ateaux-differentiable at x0 ∈ U if there exists x∗ ∈ X ∗ such that f (x0 + th) − f (x0 ) for all h ∈ X. t→0 t

hx∗ , hi = lim

The limit above is called the derivative of f in the direction h; the functional x∗ is often denoted by grad f (x0 ) and will be referred to as the gradient or Gˆ ateaux derivative of f at x0 . If this limit is uniform with respect to h ∈ SX , we say that f is Fr´ echetdifferentiable at x0 (and then refer to x∗ as the Fr´ echet derivative of f at x0 , written f 0 (x0 )): equivalently, f (x0 + h) − f (x0 ) − hx∗ , hi → 0 as khk → 0. khk Clearly Fr´echet-differentiability implies Gˆateaux-differentiability; in the reverse direction, it can be shown that if grad f (x) exists throughout some neighborhood of x0 and is continuous at x0 , then f is Fr´echet-differentiable at x0 and grad f (x0 ) = f 0 (x0 ). The case when f (x) = kxk (x ∈ X) is of particular interest. The norm k·k is said to be Fr´echet- (resp. Gˆateaux-) differentiable if k·k is Fr´echet- (resp. Gˆ ateaux-) differentiable at every point of X\{0} : the point 0 is excluded because no norm is differentiable at 0. In particular: Lemma 1.2.22. If the norm k · k is Gˆ ateaux- differentiable, then for x0 6= 0,

gradkx0 kX ∗ = 1 and hgradkx0 k, x0 i = kx0 kX . X Given a convex subset C of a Banach space X and a point x ∈ C, a functional f ∈ X ∗ is called a supporting functional of C at x if kf k = 1 and hf, xi = sup {|hf, yi| : y ∈ C} . It emerges that if the norm on X is Gˆ ateaux-differentiable, then grad kxk is a supporting functional of BX at every point x of BX \{0} : for a proof of this and of the characterizations given next of spaces with Gˆ ateaux-differentiable norms we refer to [37], Chapter 8. Theorem 1.2.23. The following statements about a Banach space X are equivalent: (i) The norm on X is Gˆ ateaux-differentiable. (ii) The dual space X ∗ is strictly convex. (iii) Given any x ∈ SX , there is a unique f ∈ SX ∗ such that hf, xi = 1. It turns out that uniform convexity of a space X has significant implications for the dual space X ∗ . To explain this we introduce a new object, the modulus of smoothness of a space.

A path to Musielak-Orlicz spaces

23

Definition 1.2.8. Let X be a Banach space. The modulus of smoothness of X is the function ρX : (0, ∞) → [0, ∞) defined by   kx + τ hk + kx − τ hk ρX (τ ) := sup − 1 : x, h ∈ SX . 2 If limτ →0 ρX (τ )/τ = 0, the space X is said to be uniformly smooth. Note that ρX really is a non-negative function since for all x, h ∈ X we have 2 kxk = kx + τ h + x − τ hk ≤ kx + τ hk + kx − τ hk . Evidently the property of uniform smoothness is preserved on passage to a subspace. For the following theorem see, for example, [37, Chapter 8]. Lemma 1.2.24. Let X be a Banach space. The following statements are equivalent: (i) The space X is uniformly smooth. (ii) The limit kx + τ hk − kxk lim = hgrad kxk , hi t→0 t exists, uniformly for x, h ∈ SX . (iii) The norm of X is Fr´echet-differentiable on SX and the map x 7−→ grad kxk : SX → SX ∗ is uniformly continuous. If k·k satisfies any of the equivalent conditions in the last lemma we shall say that it is uniformly Fr´ echet-differentiable. Connections between the moduli of convexity and smoothness, together with important implications of these connections, are given in the next theorem. Theorem 1.2.25. Let X be a Banach space. Then: (i) for all τ > 0,   1 ρX ∗ (τ ) = sup τ ε − δX (ε) : 0 < ε ≤ 2 2 and

 ρX (τ ) = sup

 1 τ ε − δX ∗ (ε) : 0 < ε ≤ 2 ; 2

(ii) X is uniformly convex if and only if X ∗ is uniformly smooth; (iii) X is uniformly smooth if and only if X ∗ is uniformly convex. Proof. (i) We establish only the first of the two statements, the proof of the second being similar. Let ε ∈ (0, 2] and τ > 0; let x, y ∈ SX be such that kx − yk ≥ ε. By the Hahn-Banach theorem, there exist f, g ∈ SX ∗ such that hf, x + yi = kx + yk and hg, x − yi = kx − yk . Then 2ρX ∗ (τ ) ≥ kf + τ gk + kf − τ gk − 2 ≥ hf + τ g, xi + hf − τ g, yi − 2 = hf, x + yi + τ hg, x − yi − 2 = kx + yk + τ kx − yk − 2.

24

Analysis on Function Spaces of Musielak-Orlicz Type

Thus 2− kx +yk ≥ τ ε − 2δX ∗ (τ ), which gives δX (ε) + ρX ∗ (τ ) ≥ τ ε/2, so that ρX ∗ (τ ) ≥ sup 12 τ ε − δX (ε) : 0 < ε ≤ 2 . To establish the reverse inequality, let τ > 0 and f, g ∈ SX ∗ . Given η > 0, let x, y ∈ SX be such that hf + τ g, xi ≥ kf + τ gk − η and hf − τ g, yi ≥ kf − τ gk − η. Then kf + τ gk + kf − τ gk − 2 ≤ hf, x + yi + τ hg, x − yi − 2 + 2η ≤ kx + yk − 2 + τ kx − yk + 2η ≤ −2δX (kx − yk) + τ kx − yk + 2η   1 ≤ 2 sup τ ε − δX ∗ (ε) : 0 < ε ≤ 2 + 2η, 2 from which the desired inequality follows. (ii) Suppose that X is uniformly convex and let ε0 ∈ (0, 2]. Then δX (ε) ≥ δX (ε0 ) > 0 if ε ∈ [ε0 , 2]. Let τ ∈ (0, δX (ε0 )) . If ε ∈ [ε0 , 2], then ε/2 − δX (ε)/τ ≤ ε/2 − δX (ε0 )/τ ≤ ε/2 − 1 ≤ 0. By (i), ρX ∗ (τ )/τ = sup (ε/2 − δX (ε)/τ ) ≤ sup ε/2 = ε0 /2. 0 hJx, y/ kyki , whence hJx, yi < hJx, xi ; similarly, hJy, xi < hJy, yi . Hence 0 = hJx − Jy, x − yi > hJx, xi + hJy, yi − hJx, xi − hJy, yi = 0. This contradiction completes the proof.

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1.3

27

Modular spaces

In this section we fix a real or complex vector space V . We denote the scalar field with K. We define the concept of modular on V , analyze the basics properties and present concrete examples, aiming to emphasize the usefulness of the theory. Definition 1.3.1. An s-convex modular (0 < s ≤ 1) on a real or complex vector space V is a function ρ : X −→ [0, ∞] that satisfies the following conditions: (i) ρ(x) = 0 ⇐⇒ x = 0 (ii) ρ(αx) = |α|ρ(x) for any x ∈ V , |α| = 1 (iii) ρ(αx + (1 − α)y) ≤ αs ρ(x) + (1 − α)s ρ(y) for all x, y ∈ V and α ∈ (0, 1]. In particular, if s = 1 the modular is said to be convex. A convex modular ρ on a vector space V is left-(right-) continuous if for any x ∈ V the map α −→ ρ(αx) is left-(right-) continuous on [0, ∞) (on (0, ∞) for left continuity); if ρ is both left- and right-continuous we refer to it as a continuous modular. Example 1.3.1. If (Ω, A, µ) is a measure space as in Section 1.2 and p ∈ [1, ∞), then Z ρ(u) = |u|p dµ (1.14) Ω p

defines a convex modular on L (Ω, dµ). Example 1.3.2. Any norm on a real or complex vector space X is a convex modular on X. It follows from the above axioms that for each fixed x ∈ the map α −→ ρ(αx) is non-decreasing on [0, ∞), for if 0 ≤ α < β and x ∈ V , then convexity yields: ρ(αx) = ρ(αβ −1 βx) ≤ αβ −1 ρ(βx) ≤ ρ(βx). Consequently: ρ(αx) ρ(αx)

= ρ(|α|x) ≤ |α|ρ(x) if |α| ≤ 1 = ρ(|α|x) ≥ |α|ρ(x) if |α| ≥ 1.

(1.15) (1.16)

To the effect of characterizing the modular space associated to ρ we prove the following lemma:

28

Analysis on Function Spaces of Musielak-Orlicz Type

Lemma 1.3.1. Let ρ be a modular on a vector space V and x ∈ V . Then conditions (i) and (ii) below are equivalent: (i) ρ(λx) < ∞ for some λ > 0. (ii) lim+ ρ(λx) = 0. λ→0

Proof. If (i) holds for λ > 0, and 0 < λj → 0 as j → ∞, then there exists J ∈ N such that j ≥ J ⇒ λj < λ. Since ρ(0) = 0, convexity yields, for j ≥ J: ρ(λj x) ≤ λj λ−1 ρ(λx) → 0 as j → ∞. The arbitrariness of the sequence (λj ) yields (i) ⇒ (ii). Conversely if (ii) holds there must be θ > 0 such that ρ(λ) < 1 for all λ < θ, whence (i) holds. The preceding lemma justifies the equality of the two sets in the next definition: Definition 1.3.2. For a modular ρ on a vector space V we set: Vρ

:= {x ∈ V : ρ(λx) < ∞ for some λ > 0}   = x ∈ V : lim + ρ(λx) = 0 . λ−→0

Proposition 1.3.2. Let ρ be a convex modular on a linear space X and define kxkρ := inf {λ > 0 : ρ(x/λ) ≤ 1} (x ∈ Vρ ). Then:   (i) Vρ , k·kρ is a normed linear space. (ii) If ρ(x) ≤ 1, then kxkρ ≤ 1. (iii) If ρ is left-continuous, then kxkρ ≤ 1 if and only if ρ(x) ≤ 1. If ρ is continuous, then kxkρ < 1 if and only if ρ(x) < 1; and kxkρ = 1 if and only if ρ(x) = 1. Remark 1.3.3. The norm k · kρ is called the Luxemburg norm and was introduced by Luxemburg [77] in 1955. Proof. (i) Let x, y ∈ Vρ and λ ∈ K. From (1.15) and (1.16) it follows that λx ∈ Vρ . Since ρ is convex, ρ (λ (x + y)) ≤

1 {ρ(2λx) + ρ(2λy)} → 0 as λ → 0, 2

and so x + y ∈ Vρ . Thus Vρ is a linear space.

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29

To show that k·kρ is a norm, note first that plainly kxkρ < ∞ for all x ∈ Vρ , and k0kρ = 0. Let x, y ∈ Vρ and λ ∈ K. Then kλxkρ = inf {β > 0 : ρ(λx/β) ≤ 1} = |λ| inf {β > 0 : ρ(x/β) ≤ 1} = |λ| kxkρ . Now let λ > kxkρ and µ > kykρ , so that ρ(x/λ) ≤ 1 and ρ(y/µ) ≤ 1. By the convexity of ρ,     x x+y λ µ y ρ ≤ ρ + ρ ≤ 1. λ+µ λ+µ λ λ+µ µ Thus kx + ykρ ≤ λ+µ, and the triangle inequality follows. Finally, if kxkρ = 0, then ρ(αx) ≤ 1 for all α > 0, and hence, for all λ > 0 and β ∈ (0, 1], ρ(λx) ≤ βρ(λx/β) ≤ β. This shows that ρ(λx) = 0 for all λ > 0, and so x = 0. (ii) If ρ(x) ≤ 1, then from the definition of kxkρ it is immediate that kxkρ ≤ 1. (iii) Suppose that ρ is left-continuous and that kxkρ ≤ 1. Then for all µ ∈ (0, 1) we have ρ(µx) ≤ 1. Then ρ(x) = limµ→1− ρ(µx) ≤ 1. Now assume that ρ is continuous. If kxkρ < 1, then for some λ < 1 we have ρ(x/λ) < 1, so that in view of the convexity of ρ, ρ(x) ≤ λρ(x/λ) ≤ λ < 1. Conversely, if ρ(x) < 1, then by the continuity of ρ, ρ(γx) < 1 for some γ > 1. Thus kγxkρ ≤ 1 and kxkρ ≤ 1/γ < 1. The rest is obvious. Corollary 1.3.4. Let ρ be a left-continuous modular on a modular space V . Then: (i) If kxkρ ≤ 1, then ρ(x) ≤ kxkρ . (ii) If kxkρ > 1, then ρ(x) ≥ kxkρ . (iii) For any x ∈ V , kxkρ ≤ ρ(x) + 1. Proof. For (i), observe that if 0 < kxkρ ≤ 1 then the convexity of ρ yields     kxkρ x ρ(x) = ρ x ≤ kxkρ ρ kxkρ kxkρ ≤ kxkρ ; on the other hand if 1 < λ < kxkρ one has x ρ >1 λ and by convexity one gets 1 < λ−1 ρ(x), which forces ρ(x) ≥ kxkρ , as claimed. (iii) follows immediately from (i) and (ii).

30

1.3.1

Analysis on Function Spaces of Musielak-Orlicz Type

Modular convergence and norm convergence

Though the modular concepts are generally speaking, mathematically weaker than those associated to the norm, they are simpler to handle from the computational point of view and occasionally, sufficiently sharp to handle certain mathematical situations that arise in the treatment of Musielak-Orlicz spaces. Many instances of such a nature will come to light in subsequent chapters; in the meantime we lay the ground work with the pertinent definitions and provide some simple examples to clarify their meaning. Let V be a real or complex vector space and ρ : V −→ [0, ∞] be a convex modular on V . Definition 1.3.3. A sequence (xk ) ⊂ V is said to be modular-convergent to ρ x ∈ V (xn → x) iff there exists λ > 0 for which lim ρ(λ(xk − x)) = 0.

k−→∞

The following terminology is standard and will be used extensively in the sequel: Definition 1.3.4. (see [54]) (i) A sequence {xn } ⊂ Vρ is said to be ρ-convergent to x ∈ Vρ and write ρ xn → x, if and only if ρ(xn − x) → 0. Note that the ρ-limit is unique if it exists. (ii) A sequence {xn } ∈ Vρ is called ρ-Cauchy if ρ(xn − xm ) → 0 as n, m → ∞. (iii) A set C ⊂ Vρ is called ρ-closed if for any ρ-converging-sequence {xn } ⊂ C to x one has x ∈ C. (iv) A set C ⊂ Vρ is defined to be ρ-bounded if δρ (C) = sup{ρ(x − y) : x, y ∈ C} < ∞. Theorem 1.3.5. Let ρ be a convex modular on a vector space V . Then (i) Modular limits are unique. More precisely, if ρ is a convex modular and ρ ρ xn → x and yn → y then x = y. ρ

ρ

ρ

(ii) If xn → x and yn → y, then xn + yn → x + y. ρ

ρ

(iii) If c ∈ R and xn → x, then cxn → cx.

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31

Proof. The proofs are straightforward. We present the details of the proof of (i). By assumption, ρ(λ1 (x − xn )) → 0 and ρ(λ2 (y − xn )) → 0 as n → ∞ for some λ1 > 0, λ2 > 0. It is clear that there is no loss of generality by assuming λ2 > λ1 . Obviously, as n → ∞, convexity yields:   λ1 ρ(λ1 (xn − y)) = ρ λ2 (xn − y) −→ 0. λ2 A further application of convexity yields   1 1 1 ρ λ1 (x − y) ≤ ρ(λ1 (x − xn )) + ρ(λ1 (xn − y)). 2 2 2 Since the right-hand side tends to zero as n → ∞ one must a fortiori have x = y. Theorem 1.3.6. If ρ is a convex, continuous modular on the vector space X, then norm convergence on the space Xρ implies modular convergence. Proof. By convexity and continuity, for any sequence (un ) ⊂ Vρ converging to u in norm, one has:   ku − un kρ |u − un | ρ(|un − u|) = ρ ku − un kρ   |u − un | ≤ ku − un kρ ρ = ku − un kρ , ku − un kρ from which the theorem follows automatically. Theorem 1.3.7. Let ρ be a convex, continuous modular on the vector space X. Then, on the space Xρ , modular convergence is equivalent to (Luxemburg) norm convergence if and only if for any sequence (xk ) ⊂ V , lim ρ(xk ) = 0 ⇒ lim ρ(2xk ) = 0.

k→∞

k→∞

Proof. If the sequence (xk ) converges modularly to 0, then, by assumption, for any m ∈ N one has ρ(2m xk ) ≤ 1 for k ≥ k0 for some k0 depending on m. By definition, this statement implies kxk kρ ≤ 2−m for k ≥ k0 , which in conjunction with the preceding theorem proves the claim. Example 1.3.3. Let X = C([0, 1]) stand for the vector space of all realvalued continuous functions on [0, 1] furnished with the standard real vector space structure. Consider the modular ρ : X → [0, ∞) Z 1 ρ(u) = (e|u(x)| − 1)dx 0

32

Analysis on Function Spaces of Musielak-Orlicz Type

and for n ∈ N set un = 12 n1(2−n ,2−n+1 ) . Clearly (un ) ⊂ Xρ . Then Z

2−n+1

(e2

ρ(un ) =

−1

n

 −1  − 1)dx = 2−n e2 n − 1

2−n

whereas Z

2−n+1

(en − 1)dx = 2−n (en − 1).

ρ(2un ) = 2−n

ρ

It is immediate from the above equalities that un → 0 as n → ∞, while (un ) does not converge in norm. Theorem 1.3.8. Any sequence (xk ) for which ρ(λxk ) → 0 for all λ : λ > 0, is norm-convergent. Proof. For such a sequence (xn )
and any λ > 0 there exists N > 0 such that n n ≥ N implies ρ(λxn ) = ρ λx−1 < 1, which by definition yields kxn k ≤ λ−1 for n ≥ N . Definition 1.3.5. The modular ρ on the vector space V is said to satisfy the ∆2 condition if there exists a constant K ≥ 1 such that for any x ∈ V ρ(2x) ≤ Kρ(x). For example, if 1 < p < ∞, the modular ρp introduced in (1.14) satisfies the ∆2 condition; on the other hand, the modular on C([0, 1]) introduced in Example 1.3.3 does not satisfy the ∆2 condition. In view of Theorems 1.3.6 and 1.3.7 we have the following result: Corollary 1.3.9. If a modular ρ : V −→ [0, ∞] on a vector space V satisfies the ∆2 condition then, on Vρ , norm convergence is equivalent to modular convergence. Theorem 1.3.10. A left-continuous modular on a vector space V is lower semicontinuous on Vρ , i.e. for any sequence (xn ) ∈ Vρ such that xn → x in norm as n → ∞, one has ρ(x) ≤ lim inf ρ(xn ). n→∞

Proof. Let 0 < δ < 1, then on account of convexity:   1−δ ρ((1 − δ)x) ≤ δρ (x − xn ) + (1 − δ)ρ (xn ) . δ Taking the lim inf on both sides and as a direct consequence of Theorem 1.3.6 it is immediate that ρ((1 − δ)x) ≤ (1 − δ) lim inf ρ(xn ). n

The claim follows now directly from the left-continuity assumption on ρ.

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1.3.2

33

Conjugate modulars and duality

In this section we retain the preceding terminolgy. In particular, V stands for a vector space over K = R or K = C and ρ : V −→ [0, ∞] denotes a convex modular on V . A generic element in the dual V ∗ of V will be denoted by x∗ and the action of x∗ on x ∈ V by x∗ (x) = hx∗ , xi. If kxkρ := inf {λ > 0 : ρ(x/λ) ≤ 1} (x ∈ Vρ ) stands for the Luxemburg norm, it as been proved in Theorem 1.3.2 that (Vρ , k · kρ ) is a normed space over K. As usual, let (Vρ , k · kρ )∗ = (Vρ∗ , k · k∗ρ ) stand for the dual space, that is to say, for any x∗ ∈ Vρ∗ the norm is given by the usual definition kx∗ k∗ρ = sup{|hx∗ , xi| : kxkρ ≤ 1}. It will be shortly observed that ρ engenders a modular ρ∗ on the dual Vρ∗ . We will address the natural question as to the relationship between the ρ∗ Luxemburg norm k · kρ∗ and the dual norm k · k∗ρ on the vector space Vρ∗ . We open the discussion with the following theorem: Theorem 1.3.11. Let ρ be a convex modular on a vector space V with corresponding Luxemburg norm k · kρ . Let x∗ be a linear functional on V . Then the following conditions are equivalent: (i) x∗ ∈ Vρ∗ (i.e., x∗ is bounded on Vρ ). (ii) There is c ≥ 0 such that |hx∗ , xi| ≤ c(ρ(x) + 1). Proof. Assume (i) holds and fix x ∈ Vρ . It follows from convexity that   x ρ ≤ 1, 1 + ρ(x) which on account the definition of the Luxemburg norm implies the bound kxkρ ≤ (ρ(x) + 1). Thus, for any x∗ ∈ Vρ∗ it holds that |hx∗ , xi| ≤ kx∗ k∗ρ kxkρ ≤ kx∗ k∗ρ (ρ(x) + 1).

(1.17)

34

Analysis on Function Spaces of Musielak-Orlicz Type

In all, (ii) holds with c = kx∗ k∗ρ . Assume next x∗ ∈ V ∗ satisfies (ii) for some c > 0. Fix δ > 0. For x ∈ Vρ with kxkρ = 1 the definition of the Luxemburg norm yields the existence of λ : 0 < λ < 1 + δ such that ρ(λ−1 x) ≤ 1. Based on the assumption it is concluded that
|hx∗ , λ−1 xi| ≤ c ρ(λ−1 x) + 1 ≤ 2c. In all, |hx∗ , λ−1 xi| ≤ 2(1 + δ)c whence x∗ is actually bounded on Vρ , as claimed. This concludes the proof. Theorem 1.3.12. If ρ : V −→ [0, ∞] is a convex modular on V , then the function ρ∗ : Vρ∗ −→ [0, ∞) ∗

∗

(1.18)

∗

ρ (x ) = sup {|hx , xi| − ρ(x)} x∈Vρ

is a convex, left-continuous semimodular on Vρ∗ , that is, (i) For any x∗ ∈ Vρ∗ , one has ρ∗ (x∗ ) = ρ∗ (−x∗ ). (ii) ρ∗ (sx∗ + (1 − s)y ∗ ) ≤ sρ∗ (x∗ ) + (1 − s)ρ∗ (y ∗ ) for any x∗ ∈ Vρ∗ , y ∗ ∈ Vρ∗ and 0 ≤ s ≤ 1. (iii) For any x∗ ∈ Vρ∗ , limλ→1− ρ∗ (λx∗ ) ≤ ρ∗ (x∗ ). (iv) If ρ∗ (λx∗ ) = 0 for all λ > 0, ∗

(1.19)

∗

then ρ (x ) = 0. Proof. It is immediately seen that by definition, ρ∗ ≥ 0 on Vρ∗ , ρ∗ (0) = 0 and ρ∗ (x∗ ) = ρ∗ (−x∗ ) for any x∗ ∈ Vρ∗ . It follows by definition that hx∗ , xi ≤ ρ∗ (x∗ ) + ρ(x). If condition (1.19) holds for x∗ ∈ Vρ∗ , then for x ∈ Vρ with ρ(αx) < ∞, α > 0, one has λα|hx∗ , xi| = λ|hx∗ , αxi| ≤ ρ∗ (λx∗ ) + ρ(αx) ≤ ρ(αx), which readily yields hx∗ , xi = 0 by letting λ → ∞. Since x is arbitrary, one

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35

concludes that x∗ = 0 and hence that ρ∗ is a semimodular. Next, if 0 ≤ s ≤ 1 it holds that |hsx∗ + (1 − s)y ∗ , xi| − ρ(x) ≤ s|hx∗ , xi| − sρ(x) + (1 − s)|hy ∗ , xi| − (1 − s)ρ(x). Convexity follows immediately from the last inequality. For left-continuity, it is easily observed that for any x∗ as above and any x ∈ Vρ , as λ → 1, 0 < λ < 1, one has |hλx∗ , xi| − ρ(x) % |hx∗ , xi| − ρ(x), the latter easily yields that for arbitrary  > 0 there exists λ0 : 0 < λ0 < 1 such that for 0 < λ0 < λ < 1 it holds: |hx∗ , xi| − ρ(x) < |hλx∗ , xi| − ρ(x) +  that is, ρ∗ (x∗ ) ≤ sup ρ∗ (λx∗ ) = lim ρ∗ (λx∗ ). 0 1 and hence,   x 1 1=ρ ≤ ρ(x). kxkρ kxkρ This yields kxkρ − ρ(x) ≤ 0 if kxkρ > 1.

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37

Consequently, if kx∗ k∗ρ ≤ 1, then {|hx∗ , xi| − ρ(x)} ≤ 0

sup x∈Vρ , ρ(x)>1

and it is readily concluded that ρ∗ (x∗ ) =

{|hx∗ , xi| − ρ(x)}.

sup

x∈Vρ , ρ(x)≤1

In view of the above observation, if 0 6= x∗ ∈ (Vρ )∗ ,  ∗    x 1 ∗ ∗ ρ = sup |hx , xi| − ρ(x) ∗ ∗ kx∗ k∗ρ x∈Vρ , ρ(x)≤1 kx kρ ≤

{kxkρ − ρ(x)}

sup x∈Vρ , ρ(x)≤1

and since kxkρ ≤ 1 if ρ(x) ≤ 1 it follows that  ∗  x ρ∗ ≤ 1. kx∗ k∗ρ By definition of k · kρ∗ it is plain that kx∗ kρ∗ ≤ kx∗ k∗ρ . The last inequality completes the proof of (1.20). Definition 1.3.6. A modular ρ on a vector space V is said to be sequentially weakly lower semicontinuous on Vρ if for any sequence (xn ) ⊂ Vρ that converges weakly to x ∈ Vρ (xn * x) one has ρ(x) ≤ lim inf ρ(xn ). n

Theorem 1.3.16. A modular ρ on a vector space V is sequentially weakly lower semicontinuous on Vρ . Proof. The proof is straightforward. Consider x ∈ Vρ and (xn ) ⊆ Vρ with xn * x as n → ∞. Notice that for any linear functional x∗ ∈ Vρ∗ |hx∗ , xn i| ≤ sup |hy ∗ , xn i|. y ∗ ∈Vρ∗

Hence lim |hx∗ , xn i| ≤ lim inf sup |hy ∗ , xn i|.

n→∞

n

y ∗ ∈Vρ∗

38

Analysis on Function Spaces of Musielak-Orlicz Type

We conclude ρ(x) = ρ∗∗ (x) = sup {|hx∗ , xi| − ρ∗ (x∗ )} x∗ ∈Vρ∗

≤ sup x∗ ∈Vρ∗

n

o lim |hx∗ , xn i| − ρ∗ (x∗ )

n→∞

≤ lim inf sup {|hy ∗ , xn i| − ρ∗ (y ∗ )} n

y ∗ ∈Vρ∗

= lim inf ρ∗∗ (xn ) = lim inf ρ(xn ). n

1.3.3

n

Modular uniform convexity

Bearing some recently discovered applications in mind, this section is devoted to the consideration of some geometric aspects of modular spaces. More precisely we delve into the idea of uniform convexity. It will not have escaped the attention of the reader that due to the lack of homogeneity, it is expectable that even the formulation of the concept of uniform convexity of a modular poses challenges and must necessarily entail technical difficulties beyond those encountered when dealing with the uniform convexity of a norm. Far from constituting a mere abstract mathematical construction, modular uniform convexity has deep consequences and tangible applications, some examples of which will be presented in the sequel (Section 1.4.4). The relevance of the notion of modular uniform convexity is reflected in the fact that it was fully investigated in Orlicz function spaces [24, 82]. The concept of modular uniform convexity was introduced and studied by Nakano [83]. The definition of two different types of modular uniform convexity can be found in [7, 55]. The reader will find it easy to realize that the notions introduced hereafter are natural generalizations of those studied in Section 1.2.3. It is worthwhile to note that if the modular ρ is associated to a MusielakOrlicz function ϕ (Chapter 2) the interplay between the uniform convexity of the Luxemburg norm k · kρ and the function is well understood. The reader is referred to Theorem 2.6.1 in Section 2.6 for the details. In view of the usual concept (Definition 1.2.7) it is natural to state the following definitions: Definition 1.3.7. A convex modular ρ on a vector space V is said to be uniformly convex if for every  > 0 there exists δ = δ() > 0 such that for every u ∈ V
and v ∈ V with ρ(u) = 1 and ρ(v) = 1 and ρ(u − v) >  it holds that ρ u+v < 1 − δ(). 2 Definition 1.3.8. Let ρ be a convex modular on a real vector space V , r > 0,  > 0. Set     u−v D(r, ) = (u, v) ∈ Vρ × Vρ : ρ(u) ≤ r , ρ(v) ≤ r , ρ ≥ r (1.21) 2

A path to Musielak-Orlicz spaces and



1 δ(r, ) = inf 1 − ρ r



u+v 2



39 

: (u, v) ∈ D(r, ) .

(1.22)

If D(r, ) = ∅ we define δ(r, ) = 1. Definition 1.3.9. The modular ρ is said to be type 2-uniformly convex U U C2 (see [55]) if for each s ≥ 0,  > 0, there exists η(s, ) > 0 such that for any r>s>0 δ(r, ) > η(s, ).

Definition 1.3.10. A convex modular ρ on a real vector space V is said to be strictly convex if for any u ∈ V , v ∈ V with ρ(u) = ρ(v)   u+v 1 ρ = (ρ(u) + ρ(v)) ⇐⇒ u = v. 2 2 Lemma 1.3.17. If a convex modular ρ on a vector space V is U U C2, then it is strictly convex. Proof. Assume ρ satisfies the above hypothesis, that u ∈ V , v ∈ V , ρ(u) = ρ(v) = r and that   u+v 1 ρ = (ρ(u) + ρ(v)) . 2 2 Assuming u 6= v one can find  : 0 <  < 1 such that,   u−v ρ ≥ r. 2 By virtue of the property U U C2 one has, for every r0 > r and η(r, ) as in Definition 1.3.9   u+v ρ < xr0 (1 − η(r, )). 2 Letting r0 → r it follows r=

1 (ρ(u) + ρ(v)) = ρ 2



u+v 2

 ≤ r(1 − η(, r)).

Hence, u = v. We next move to yet another definition: Our aim is to consider the modular uniform convexity type-property defined as follows: Definition 1.3.11. Let ρ be a convex modular on a vector space V over R or C.

40

Analysis on Function Spaces of Musielak-Orlicz Type

(i) The modular ρ is said to be strictly convex (in short (SC)) if for every x ∈ V, y ∈ V x such that ρ(x) = ρ(y) and   x+y ρ(x) + ρ(y) ρ = , 2 2 we have x = y. (ii) [83] The modular ρ is said to be uniformly convex in every direction (in short (UCED)) if and only if for any z1 , z2 ∈ Vρ such that z1 6= z2 and R > 0, there exists δ = δ(z1 , z2 , R) > 0 such that    z1 + z2 ρ(x − z1 ) ≤ R =⇒ ρ x − ≤ R(1 − δ), ρ(x − z2 ) ≤ R 2 ¯ for any x ∈ V . V is said to be (UUCED) if δ(z1 , z2 , R) ≥ δ(z1 , z2 , R), ¯ whenever R ≤ R. The idea of uniform convexity in every direction was introduced by Garkavi [41] while studying the concept of Chebyshev centers in Banach spaces. It is a matter of routine to check that if a modular ρ is (U CED) then ρ is also strictly convex.

1.4

The `pn sequence spaces and their properties

In the following discussion we address a concrete example of the mathematical structures introduced in the preceding Sections 1.2 and 1.3 and provide some applications of them. Most of what follows will be further generalized in Section 1.4 in the context of Musielak-Orlicz spaces. Among the first modular spaces to enter the mathematical stage were the variable exponent sequences spaces (see [85]). We fix the measure space (N, 2N , µ), where µ stands for the counting measure on N. The collection of all sequences of real numbers coincides with the vector space M(N) of 2N -measurable real-valued functions on N. Proposition 1.4.1. Consider a sequence

Then the functionals

(pn ) ⊂ (1, ∞).

(1.23)

∞ X |an |pn ρ1,pn ((an )) = pn n=1

(1.24)

A path to Musielak-Orlicz spaces and ρ2,pn ((an )) =

∞ X

|an |pn

41

(1.25)

n=1

are convex, left-continuous modulars on M(N). The functional ρ∞ ((an )) = sup |an |

(1.26)

n∈N

is a convex, continuous modular on M(N). Proof. The proof is straightforward. The fact that the functionals (1.24) and (1.25) are convex modulars can be proved directly from the definition; leftcontinuity follows from the monotone convergence theorem. The modular properties for (1.26) follow by direct calculation. In this case, continuity is obvious from the identity, valid for all λ > 0: sup λ|an | = λ sup |an |. n∈N

n∈N

To easier relate these ideas to the theory in Section 1.3 write V = M(2N ) and introduce the following notation in accordance to Definition 1.3.2: for fixed (pn ) ⊂ (1, ∞) and j = 1, 2,  Vρj,pn = (an ) ∈ Vρj,pn (λ(an )) < ∞ for some λ > 0 . (1.27) On account of Proposition 1.3.2 k(an )kρj,pn := inf {λ > 0 : ρj,pn ((an )/λ) ≤ 1}

(1.28)

defines a norm on Vρj,pn , for j = 1, 2. Lemma 1.4.2. For any sequence of real numbers (an ) and any λ ∈ R, it holds that Vρ1,pn = Vρ2,pn . In fact, for any λ > 0, ρ1,pn (λ(an )) < ∞ ⇔ ρ2,pn (λ(an )) < ∞

(1.29)

and the modulars ρ1,pn and ρ2,pn yield equivalent Luxemburg norms on the space (1.27). Proof. It is straightforward to check that the function 1

x −→ x x maps the interval (1, ∞) onto the interval sequence (pn ) ⊂ [1, ∞), if (an ) ∈ M(N) and  p ∞ ∞ X |an | n X −1 pn = λ 1 1

(1.30) 1

[1, e e ]. Accordingly, for a given λ > 0, it follows from !pn |an | ≤1 −1 ppnn λ

42

Analysis on Function Spaces of Musielak-Orlicz Type

that

pn X ∞  ∞ X |an | ≤ ee−1 λ 1 1

|an |

!pn

−1

ppnn λ

≤ 1.

In other words, by definition of norm, one has: 1

k(an )kρ2,pn ≤ e e λ and invoking the definition of Luxemburg norm again, the above inequality 1 implies k(an )kρ2,pn ≤ e e k(an )kρ1,pn .  pn P∞  ≤ 1 the bounds for the Furthermore, for any λ > 0 such that 1 |aλn | function (1.30) yield: ∞ X 1

p−1 n



|an | λ

 pn ≤1

whence it is clear that k(an )kρ1,pn ≤ k(an )kρ2,pn and the proof of the equivalence k · kρ1,pn ≈ k · kρ2,pn is thus completed. Definition 1.4.1. Let (pn ) ⊂ (1, ∞) be a sequence. In the sequel we will write `pn for the vector space (1.27) endowed with either of the equivalent norms k · kρ1,pn or k · kρ2,pn . When the distinction between the two equivalent norms is unnecessary (as it will be in most of the sequel) we will simply write k · k`pn . We next observe that in the above setting, for j = 1, 2, the functionals k · kj,pn : V −→ [0, ∞] defined by k(an )kρj,pn

( k(an )kρj,pn if (an ) ∈ Vρpn = ∞ otherwise

have the properties of a norm. It is straightforward to verify Axioms (i) − (v) of Definition 1.2.1, whence `pn is a Banach function space for either functional. In summary: Theorem 1.4.3. `pn is a Banach function space for the Luxemburg norms (1.28). Similarly, it can be immediately verified that Vρ∞ coincides with the set of all bounded sequences (an ) and that the Luxemburg norm is given by k(an )kρ∞ = sup |an | = k(an )k∞ n∈N

and that it is a Banach function space with the Luxemburg norm. In particular if the sequence (pn ) is constant (i.e., pn = p for all n) the space `pn defined in (1.27) coincides with the well-known sequence space `p .

A path to Musielak-Orlicz spaces

43

Theorem 1.4.4. If (rn ) is a sequence subject to condition (1.23) and rn ≤ sn for every n ∈ N then the embedding `rn ,→ `sn is continuous.
Proof. If (an ) ∈ `rn , then for some λ > 0 it holds that ρ2,rn aλn ≤ 1, from which it can be plainly observed, since rn ≤ sn for all n, that for any n ∈ N: a sn a rn n n (1.31) ≤ ; λ λ this inequality leads to

a  n

≤ 1. λ It is clear from the latter that (an ) ∈ `sn and that k(an )k`sn ≤ λ. The arbitrariness of λ yields k(an )k`sn ≤ k(an )k`rn , ρ2,sn

as claimed. Alternatively, Theorem 1.4.4 follows directly from Theorem 1.2.4, Theorem 1.4.3 and the set-theoretic inclusion `rn ⊆ `sn implied by inequality (1.31). Lemma 1.4.5. `pn ,→ `∞ boundedly. Proof. For if (an ) ∈ `pn then for some λ > 0 X |λan |pn < ∞. The set-theoretic inclusion follows at once, since λ|an | < 1 for n large enough. P Moreover if kan kpn = 1 then |an |pn ≤ 1 and from here it is easily obtained that kan k`∞ ≤ 1 = kan k`pn . Obviously then the bound kan k`∞ ≤ kan k`pn holds and the boundedness of the embedding follows. The next theorem provides a natural characterization the ∆2 condition of either of the modulars ρj,pn , j = 1, 2, in terms of the behavior of the variable exponent (pn ). Theorem 1.4.6. For a sequence (pn ) ⊂ [1, ∞), the following conditions are equivalent: (i) p+ = sup pn < ∞ n∈N

(ii) The modulars ρ1,pn and ρ2,pn introduced in Lemma 1.4.2 satisfy the ∆2 condition

44

Analysis on Function Spaces of Musielak-Orlicz Type

(iii) ( (an ) ∈ X :

Vρ1,pn =

( (an ) ∈ X :

=

∞ X |λak |pk k=1 ∞ X k=1

pk

) < ∞ for some λ > 0

) |ak |pk 0 )

|ak |

pk

0. Either λ ≥ 1, in which case ρ1,pn ((an )) ≤ ρ1,pn (λ(an )) < ∞ or there exists t ∈ N such that 2t λ > 1. In the latter case, the ∆2 condition yields ρ1,pn ((an )) ≤ ρ1,pn (2t (λan )) ≤ C t ρ1,pn (λ(an )) < ∞. This proves the non-trivial inclusion in (iii). If (iii) holds, then (iv) is a direct consequence of (1.29). Finally, if (iv) is true, the assumption p+ = ∞ leads to the existence of a subsequence (pnj )j of (pn )n with pnj > j; then the sequence (an ) defined as ( 1 if n = nj an = 0 otherwise P∞ is in Vρ2,pn , since n=1 |2−1 an |pn < ∞; yet j

ρ2,pnj ((an )) = ∞. This contradiction finishes the proof of the theorem.

A path to Musielak-Orlicz spaces

45

If the exponent sequence (pn ) increases fast enough, then the variable exponent space `pn is none other than `∞ . In fact, the following lemma characterizes `∞ as a variable exponent space: Lemma 1.4.7. Let (pn ) be a sequence such that pn → ∞ as n → ∞. If there exists α : 0 < α < 1 such that ∞ X

αpn := M < ∞,

n=1

then `pn and `∞ coincide and their norms are equivalent. In other words, `pn = `∞ with equivalent norms if and only if 1N ∈ `pn . Proof. Let an ∈ `pn . Take λ > kan k`pn . Then ∞  X |an | pn n=1

λ

< 1 ⇒ sup

|an | ≤ 1 ⇒ kan kl∞ ≤ λ, λ

which automatically yields the inequality: kan k`∞ ≤ kan k`pn . Conversely, if (an ) ∈ l∞ and λ = M0 kan kl∞ /α, where M0 = max(1, M ) it follows that ∞  ∞ X |an | pn X  |an |α pn = λ M0 kan kl∞ n=1 n=1 ≤

∞  ∞ X α  pn 1 X pn ≤ α = 1. M0 M0 n=1 n=1

Evidently, then, kan klpn ≤

M0 kan kl∞ . α

Example 1.4.1. Consider the sequence pn = log3 log3 n. An elementary calculation shows that for any s ∈ R the series ∞ X

s

(log3 n)

(1.32)

3 log k

diverges. For any k ∈ N one has k pn = (log3 n) 3 and it follows easily from here and from (1.32) that there is no positive constant λ such that ∞ X 1

λpn < ∞;

46

Analysis on Function Spaces of Musielak-Orlicz Type

the space `pn is thus strictly contained in `∞ . We remark in passing the elementary fact that Lemma 1.4.7 shows the equivalence between the statements `pn ≈ `∞ and 1N ∈ `pn . Though in the preceding example 1N ∈ / `pn , the family F = {E ⊂ N : 1E ∈ `pn } is non empty: for example n n o E = 33 , n ∈ N ∈ F. Recall from Section 1.2.2 that given a Banach space X, Xa stands for the subspace of X consisting of the functions of absolutely continuous norm. Lemma 1.2.13 provides a clear-cut description of the subspace `∞ a . It is an elementary fact, on the other hand, that for 1 ≤ q < ∞ the classical sequence spaces `q have absolutely continuous norm, i.e., `qa = `q . This is not the case for the variable exponent sequence spaces. In fact, the existence of functions lacking absolutely continuous norm is equivalent to the absence of the ∆2 -condition. More precisely: Lemma 1.4.8. `pan = `pn if and only if p+ = sup pn < ∞ (that is if and only n

if ρpn satisfies the ∆2 condition, on account of Theorem 1.4.6). In particular, if p+ = sup pn = ∞, then `pan $ `pn . n

Proof. If p+ = sup pn < ∞ then (an ) ∈ `pn if and only if ρpn ((an )) < ∞; n

hence, given  : 0 <  < 1, there exists N > 0 for which n ≥ N implies ∞ X

|an |pn < p+ .

n

Thus

p ∞  X |an | n n



≤

∞ 1 X

p+

|an |pn ≤ 1,

n

that is k(aj )1j≥n kpn < . Assume now that sup pn = ∞. If `pn ≈ `∞ , P pn which is the case if and only if there exists 0 < q < 1 such that q < ∞, ∞ ∞ the conclusion is obvious on account of Lemma 1.2.13 ( ` = c ( ` ). 0 a P pn Otherwise, Lemma 1.4.7 asserts that q = ∞ for each 0 < q < 1. Without loss of generality one can suppose that pn % ∞. Then there exist sequences 0 < an & 0 and 0 < nk % ∞ such that nk+1 −1

1/k ≤

X i=nk

api i ≤ 2/k,

for each k,

(1.33)

A path to Musielak-Orlicz spaces

47

and pnk ≥ k. The existence of such n } and {nk } is a direct consequence P {a from the conditions pn % ∞ and q pn = ∞ for each 0 < q < 1. To see this we observe that there must exist J1 such that J

1 1 X < 2 j=1

  pj 1 2

and that the right-hand side above is at most ∞  p1 +k X 1 k=0

2

 p1 −1 1 = ≤ 1, 2

since 1 < p1 . Set n1 = 1, n2 = J1 + 1, ai = 12 for 1 ≤ i ≤ J1 . Assume the terms nj and ai satisfying (1.33) have been chosen for j = 1, 2, ...k. Then nk+1 can be selected as follows: Since  pj ∞ X 1 = ∞, k+1 j=n k+1

there is Jk+1 ∈ N such that  pj Jk+1  X 1 1 < k + 1 j=n k + 1 k pnk +j ∞  X 1 ≤ k+1 j=0  pnk 1 k+1 2 ≤ ≤ for k > 1, k+1 k k+1 1 and we set nk+1 = Jk+1 + 1 and ai = k+1 for nk ≤ i < nk+1 . Next, take any 0 < d < 1. By virtue of (1.33) we have: ∞ nk+1 ∞ X X−1 X (ai d)pi ≤ dpnk k=1 i=nk

≤

k=1

X

api i

i=nk

k=1 ∞ X

nk+1 −1

nk+1 −1

dk

X i=nk

api i ≤ e

∞ X k=1

dk

2 < ∞. k

P∞ P Consequently, i=k api i = ∞ for each k ≥ 1 and (ai d)pi < ∞ from which follows that (ai ) ∈ `pn \ `pan .

48

1.4.1

Analysis on Function Spaces of Musielak-Orlicz Type

Duality

We next set about to discuss the duality properties of the variable exponent sequence spaces `pn introduced in the previous section. For (pn ) ⊂ (1, ∞), the conjugate variable exponent of (pn ) is the sequence qn defined by pn qn = . pn − 1 Lemma 1.4.9. For 1 < q < ∞,

1 p

+

1 q

= 1 and a fixed s > 0, one has:

  sq 1 = sup ts − |t|p , t ∈ R . q p

(1.34)

Proof. The concavity of the function x → ln x implies the inequality ts ≤

1 p 1 q |t| + |s| , p q

(1.35)

valid for all s ∈ R, t ∈ R. Here, equality holds if and only if t = sq−1 . A straightforward argument then yields the desired result. Lemma 1.4.10. For n ∈ N and s1 , ...., sn ∈ R ( n ) n n X X X |sk |qk |tk |pk = sup tk sk − , t1 , t2 , ..., tn ∈ R . qk pk k=1

k=1

k=1

Proof. It follows from Lemma 1.4.9 and a simple argument. Following the terminology in Section 1.2 we denote the associate space to a Banach function space X with X 0 . n Proposition 1.4.11. If (pn ) ⊂ (1, ∞) set qn = pnp−1 . Then `qn is con0 ∗ tinuously embedded in the associate space (`pn ) and hence in (`pn ) , as a consequence of Theorem 1.2.9, i.e.,

0

∗

`qn ,→ (`pn ) ,→ (`pn ) . Proof. Consider (xn ) ∈ `qn . By definition there exists λ2 > 0 with ∞ X |λ2 xk |qk k=1

qk

< ∞.

Select sequences y = (yn ) ∈ `pn and (λj ) such that 1 ≤ λj < kykρ1,pn + j −1 satisfying the inequality p ∞  X |yk | k −1 ρ1,pn ((yn )λ−1 ) = pk ≤ 1, j λj k=1

A path to Musielak-Orlicz spaces

49

which in concert with Lemma 1.4.9 yields ∞ ∞ X λ λ X j 2 xk yk ≤ |xk ||yk | λj λ2 k=1 k=1   |yk | pk ∞ ∞ X λj X ( λj ) |λ2 xk |qk  ≤ + λ2 pk qk k=1

k=1

k(yn )kρ1,pn + j ≤ λ2

−1

1+

∞ X |λ2 xk |qk k=1

qk

! .

It is thus clear that for any (yn ) ∈ `pn ∞ X

|xk yk | < ∞.

k=1 0

Lemma 1.2.6 yields that (xn ) ∈ (`pn ) and Theorem 1.2.9 asserts then that the linear functional x∗ : `pn −→ R ∞ X x∗ ((yn )) = xk yk

(1.36)

k=1 ∗

is bounded, i.e., x∗ ∈ (`pn ) . Lemma 1.4.12. (Orlicz, 1931, [85]) Let (pn ) be a sequence of exponents subject to (1.23) and (xj ) ⊂ R be a sequence such that ∞ X

xn yn < ∞

(1.37)

n=1

for all sequences (yn ) ∈ `pn . Then the (obviously linear) functional on `pn defined by ∞ X x∗ ((yn )n ) = xn yn (1.38) n=1 ∗

pn

pn ∗

is bounded i.e., x ∈ (` ) . Moreover, (xn ) ∈ ` pn −1 . Proof. We remark the elementaryP fact that assumption (1.37) is equivalent to ∞ the following: for any (yn ) ∈ `pn , n=1 |xn yn | < ∞. Indeed, this follows from the equality: ∞ ∞ X X |xn yn | = xn sgn(xn )|yn |. n=1

n=1

On account of Lemma 1.2.6 it is readily concluded that under the given hy0 pothesis, (xn ) ∈ (`pn ) . As a consequence of Theorem 1.2.9 it follows that the functional defined in (1.38) is bounded, as claimed.

50

Analysis on Function Spaces of Musielak-Orlicz Type

The following corollary is a direct consequence of Lemma 1.2.6, Proposition 1.4.11 and Lemma 1.4.12. Corollary 1.4.13. Let (pn ) ⊂ (1, ∞). Then the associate space of `pn is pn pn 0 ` pn −1 , i.e. (`pn ) = ` pn −1 . We now focus on the question of duality, starting with the following elementary observation: To simplify the notation, the next lemma holds for any of the norms k · kρj,pn , j = 1, 2, thus represented by k · k`pn . Lemma 1.4.14. Let ej ∈ `pn be the sequence whose only nonzero entry is the j th entry, which is equal to 1. Then the linear span of {ej , j ∈ N} is dense in `pan . Proof. Obvious, for if (an ) ∈ `pan , then:

k

X



aj ej = (akn ) `pn −→ 0 as k −→ ∞.

(an ) −

p 1

`

n

Corollary 1.4.15. Let ej be the sequence defined by (ej )n = 0 if j 6= n and (ej )j = 1. If the sequence (pn ) satisfies p+ = sup pn < ∞, then the linear span n

of the set {ej , j ∈ N} is dense in `pn .

Proof. The proof follows immediately from Lemma 1.4.8. ∗

Lemma 1.4.16. Let x∗ ∈ (`pn ) . Then, there exists a sequence (xk ) ∈ `qn such that: (i) For some α ≥ 0: ∞ X |xk |qk 1

qk

≤ αkx∗ k(`pn )∗ .

(ii) For any (an ) ∈ `pn the following bound holds: ∞ X

|xk ak | < kx∗ k(`pn )∗ k(an )k`pn .

k=1

(iii) For any (an ) ∈ `pan , x∗ ((an )) =

∞ X k=1

xk ak .

A path to Musielak-Orlicz spaces

51

(iv) If x∗(xn ) is the functional on `pn defined by `pn 3 (yn ) →

∞ X

yn xn ,

k=1

then x∗ − x∗xn vanishes on `pan . Proof. Let xk = x∗ (ek ). Without loss of generality, to fix ideas one can assume that sup |hx∗ , xi| ≤ 1. kxkρ1,p ≤1 n

Fix N ∈ N and let t1 , t2 , ..., tN be a collection of real numbers. Then the sequence w = (t1 , t2 , ..., tN , 0, 0, 0, .....) is in `pn and plainly x∗ (w) =

N X

xk tk , ρ1,pn (w) =

1

N X |tk |pk 1

pk

.

On account of Theorem 1.3.11 (applied to the modular ρ1,pn ), it follows that for any N ∈ N, N N X X |tk |pk xk t k − ≤1 pk 1 1 and therefore 1 is an upper bound for the set on the right-hand side of Lemma 1.4.10. Consequently, by virtue of Lemma 1.4.10 one has the bound ∞ X |xk |qk 1

qk

≤ 1.

∗

For any x∗ ∈ (`pn ) the preceding reasoning applies to x∗ /kx∗ k(`pn )∗ ; one then has  qk |xk | ∞ X ∗ kx k(`pn )∗ ≤ 1, qk 1 and claim (i) follows immediately by definition of the Luxemburg norm. Statement (ii) follows from (i) and Proposition 1.4.11. Select now a ∈ `pan . Then

∞

! N

X

X ∗

∗ ∗ an en ≤ kx k(`pn )∗ an en → 0 as N → ∞. x ((an )) − x

n=1

It follows that the series

n=N +1

P∞ 1

`pn

ak xk converges and its sum is precisely x∗ ((an )).

52

Analysis on Function Spaces of Musielak-Orlicz Type

Theorem 1.4.17. If 1 < pn ≤ sup = p+ < ∞, then the topological dual of n∈N

`pn is isomorphic to `qn , i.e.

∗

(`pn ) ' `qn .

(1.39)

Proof. It is clear from the preceding paragraph that it remains to show that ∗ under the given assumptions, any x∗ ∈ (`pn ) is of the form (1.36). The pn assumption p+ < ∞ implies that (yk ) ∈ ` if and only if ρpn ((yn )) < ∞. For each j ∈ N, let ej be the sequence defined by ejn = (ejn ) with ejn = 0 if n 6= j ∗ and ejj = 1. For a fixed x∗ ∈ (`pn ) with kx∗ k(`pn )∗ ≤ 1 set x∗ (ej ) = xj , j ∈ N. It transpires from (1.34), that for fixed n ∈ N, ( n ) n n X X X |xk |qk |yk |pk = sup yk xk − , yk ∈ R k = 1, 2, ...n qk pk k=1

k=1

k=1

≤ sup {x∗ (y) − ρp (y) , y ∈ `pn } ≤ 1. The last inequality follows from the fact that kx∗ k(`pn )∗ ≤ 1 and from (1.17). In all, (xk ) ∈ `qn and k(xk )k`qn ≤ 1. Proposition 1.4.14 implies at once that for any (yn ) ∈ `pn , ∞ X x∗ ((yn )) = xk yk . k=1

Claim (1.39) follows immediately from Theorem 1.3.15. A phenomenon not encountered in the constant exponent case is worth exploring at this point, namely, the fact that different sequences (pn ) and (qn ) could give rise to the same variable exponent sequence space `pn . The next series of theorems aims at exploring this question. Theorem 1.4.18. Let (pn ) be a sequence of real numbers with 1 ≤ α ≤ inf pn n

and let (qn ) stand for the sequence of H¨ older conjugates of ( pαn ), i.e., pαn + q1n = pn α 1. Then ` is isomorphic to ` iff and only if there exists γ ∈ R: 0 < γ < 1 for which ∞ X γ qn < ∞. (1.40) n=1

Proof. For any n ∈ N, any 0 < γ < 1 and α, pn and 1n as in the preceding paragraph, in accordance with Young’s inequality it holds that: α

(|an |γ) ≤ assuming, thus that

P∞

∞ X

1

α 1 |an |pn + γ qn , pn qn

γ qn < ∞ it is concluded that

1 |an | ≤ α γ n=1 α

∞ X n=1

pn

|an |

+

∞ X n=1

! |γ|

qn

.

(1.41)

A path to Musielak-Orlicz spaces

53

Inequality (1.41) settles the validity of the set-theoretic inclusion ` α ⊆ ` pn under assumption (1.40). The sufficiency of condition (1.40) in the statement of Theorem 1.4.18 follows from Theorem 1.2.4, Theorem 1.4.3 and Theorem 1.4.4. For the converse, we suppose that `pn ⊆ `α . We first claim that (1.42) implies that ` ∞ X

|λan |

pn α

pn a

(1.42) ⊆ `1 . Indeed, if 0 < λ < 1 and

0 such that n=1 |λan |b < ∞. Inequality (1.35) with t = γ pn , b pn s = (λ|an |) , p = b−pn and q = b/pn applied for each n ∈ N yields: pn

(γλ|an |)

In all

∞ X

(γλ|an |)

pn

n=1

b b − pn b−p pn b γ n + (λ|an |) b b b b ≤ γ b−pn + (λ|an |) .

≤

≤

∞ X

γ

b b−pn

+

n=1

∞ X

b

(λ|an |) < ∞.

n=1

Theorem 1.2.4, Theorem 1.4.3 and Theorem 1.4.4 yield `pn ≈ `b , as claimed. On the other hand, if `b is continuously embedded in `pn , take any sequence b (bn ) ∈ ` pn ; given that pbn < b, it must necessarily hold, (Theorem 1.4.6) that ∞ X

b

|bn | pn < ∞.

n=1 1

It is plain then that (|bn | pn ) ∈ `b ,→ `pn . Consequently, invoking again Theorem 1.4.6 one concludes that (bn ) ∈ `1 . By definition of the sequence (bn ), it is immediate that (an ) ∈ `pn . Conversely, assume that for given (pn ) and b as in the statement of the theb orem, `pn is isomorphic to `b , i.e, `b ⊆ `pn . Consider a sequence (xn ) ∈ ` pn . P∞ b By definition, either n=1 |xn | pn < ∞ or there exists λ0 : 0 < λ0 < 1 for P∞ b which n=1 |λ0 xn | pn < ∞. It is immediately derived from the first case that 1

1

(|xn | pn ) ∈ `b ; since λ0 < λ0pn , in the second case one has ∞  X n=1

1

λ0 |xn | pn

b

≤

∞  X

1

|λ0 xn | pn

b

n=1

< ∞, b

1

i.e., if (xn ) ∈ ` pn , then necessarily (|xn | pn ) ∈ `b ⊆ `pn . In all, we have the b logical implication `b ⊆ `pn ⇒ ` pn ⊆ `1 .

A path to Musielak-Orlicz spaces

55

b

Since 1 ≤ pbn Theorem 1.4.18 applies to ` pn , implying the existence of a real number γ : 0 < γ < 1 such that ∞ X

b

γ b−pn < ∞,

j=1

as claimed.

1.4.2

Finitely additive measures

A digression aiming at preparing the ground for a description of the dual ∗ (`pn ) in the case p+ = sup pn = ∞ is in order. We start with the following n∈N

standard definition. Definition 1.4.2. Let X 6= ∅ and A ⊂ 2X be closed under complements and finite unions. A real valued function µ : A −→ R is said to be a finitely additive measure on (X, A) if for any finite subset {M1 , ...Mn } ⊂ A it holds that (i) µ

N [

! Mk

k=1

=

N X

µ(Mk )

k=1

(ii) µ(∅) = 0

(iii) sup |µ(M )| < ∞. M ∈A

The total variation of µ is (the finitely additive measure) given by: |µ|(A) =   k X  sup |µ(Ak )|, Ai ∩ Aj = ∅ if i 6= j , A 3 Ak ⊆ A for j = 1, 2, ..., k .   j=1

The finitely additive measure µ is said to be of bounded variation if |µ|(X) < ∞. In that case the total variation kµk of µ is given by kµk = |µ|(X).

56

Analysis on Function Spaces of Musielak-Orlicz Type

Recall that we denote the subspace of `∞ consisting of all simple functions of the form N X ϕ= ϕk 1Ek k=1

by S∞ (a word of caution: it is possible that the set Ek ⊆ N is infinite for some k). A finitely additive measure µ with kµk < ∞ defines naturally a linear ∗ functional γ ∈ S∞ , namely ! N N X X γ a k 1 Ek = ak µ(Ek ) k=1

k=1

for aj ∈ R, j = 1, 2, ...N. It follows immediately from this definition that sup

|hγ, ui| ≤ kµk.

u∈S∞ :kuk`∞ ≤1

The functional γ associated to µ in the above fashion can be continuously extended to `∞ on account of the density of S∞ in `∞ ; specifically, if (sk ) ⊂ S∞ is a Cauchy sequence in `∞ , then so is (γ(sk )), by virtue of the finiteness of PNk kµk. Moreover if (sk ) ⊂ S∞ and sk = j=1 ak,j 1Ek,j −→ 0 in `∞ as k → ∞, then X Nk |γ(sk )| = ak,j µ (Ek,j ) (1.43) j=1 ≤ sup |ak,j |kµk → 0 as k → ∞. k,j

Inequality (1.43) shows that if ksk − ak`∞ → 0 as k → ∞, then limk→∞ γ(sk ) exists and is independent of the sequence (sk ). Hence the natural extension (still denoted by γ) defined as γ : `∞ −→ R Z γ(a) = lim γ(sk ) :=

a dµ N

is well defined, linear and bounded; in fact the preceding discussion shows that kγk(`∞ )∗ ≤ kµk. (1.44) Lemma 1.4.20. Any finitely additive measure µ of finite total variation kµk ∗ on 2N induces a naturally defined bounded, linear functional γ ∈ (`∞ ) via (1.43). Moreover (1.44) holds. In addition, if µ vanishes on finite sets, then γ vanishes on c0 .

A path to Musielak-Orlicz spaces

57

Proof. In view of the discussion preceding the statement of Lemma 1.4.20 it remains only to prove the last assertion. To this end, fix a finitely additive measure µ on (N, 2N ) that vanishes on finite sets and consider x ∈ c0 . Fix δ > 0 and let n ≥ N ∈ N imply sup |xj | < δ. Then N3j≥N

|γ(x)| = γ(x1{1≤j≤N } + x1{j>N } ) ≤ γ(x1{1≤j≤N } ) + γ(x1{j>N } ) N X = xj γ(1j ) + |γ(x1{j>N } )| j=1 ≤

N X

|xj ||µ({j})| + kγk(`∞ )∗ kx1{j>N } k`∞

j=1

≤ kγk(`∞ )∗ δ from which the claim is clear. ∗

Lemma 1.4.21. If Λ ∈ (`∞ ) vanishes on c0 , then there exists a finitely additive measure µ on (N, 2N ) such that with finite total variation kµk such that (i) kµk ≤ kΛk(`∞ )∗ (ii) For any sequence x ∈ `∞ one has Z Λ(x) =

x dµ.

(1.45)

N

(iii) µ vanishes on finite sets. Proof. For E ⊂ N set µ(E) = Λ(1E ). It is straightforward to show that, due to the linearity of Λ, µ is a finitely additive measure on N. It remains to prove that the total variation kµk is finite. To this end, consider a finite sequence of mutually disjoint subsets of N, say (Ej ), 1 ≤ j ≤ n and set ξj = sgnΛ(1Ej ). Then n X j=1

|µ(Ej )| =

n X

|Λ(1Ej )| =

j=1

n X j=1

n
X
ξ j Λ 1 Ej = Λ ξj 1Ej j=1

 

n

n X

X

≤ Λ ξj 1Ej  ≤ kΛk(`∞ )∗ ξ 1 j Ej

j=1

j=1

`∞

≤ kΛk

(`∞ )∗

.

58

Analysis on Function Spaces of Musielak-Orlicz Type

Equality (1.45) follows straight from the discussion preceding the statement of Lemma 1.4.20. ∗

Definition 1.4.3. A bounded linear functional x∗ ∈ (`pn ) is said to be singular if it vanishes on `pan . Proposition 1.4.22. In view of Lemma 1.4.16, it is easily seen that x∗ ∈ ∗ (`pn ) is singular if and only if x∗ (ej ) = 0 for all j ∈ N. We now turn to the question of duality in the absence of the ∆2 condition. More precisely, we describe the dual of `pn under the assumption that the exponent sequence (pn ) is unbounded. The first order of business is to underline the fact that if `pn contains sequences of non-absolutely continuous norms, then `pn is “locally” isomorphic to `∞ . Specifically: Lemma 1.4.23. (See [76, 101]) isomorphic with `∞ .

Let `pan $ `pn . Then `pn contains a subspace

Proof. By assumption, one can select a = (an ) ∈ `pn \ `pan with k(an )k`pn = 1. Define the sequence ak as ak := a 1{n:n≥k} = (akn ), that is akn := an for n ≥ k and akn := 0 for n < k, and set ak,l := a 1{n:k≤n≤l} = (ak,l n ).

(1.46)

k,l In other words ak,l n = an for k ≤ n ≤ l − 1 and an := 0 for n < k or l ≤ n. By assumption on a, there exist δ > 0 such that kak k`pn > 2δ for each k ≥ 1. Observing that kak,l k`pn % kak k`pn as l → ∞ one quickly derived the existence of an increasing sequence (ki ) such that ki → ∞ as i → ∞ and that kaki ,ki+1 k`pn > δ for each i. Note that for i 6= j the supports of aki ,ki+1 and aki ,ki+1 are disjoint. Set ( ) ∞ X Za := g ∈ `pn : there exists (ci ) ∈ `∞ such that g = ci aki ,ki+1 . i=1

(1.47) Take c = (ci ) ∈ `∞ and ε > 0. There exists i ∈ N such that kck`∞ ≤ |ci | + ε. Then

∞

X

δkck`∞ ≤ δ|ci | + δε ≤ kci aki ,ki+1 k`pn + δε ≤ cm akm ,km+1 + δε

p m=1 ` n

∞

∞

X

X

km ,km+1 km ,km+1 ≤ kck`∞ a + δε ≤ kck`∞ a + δε



p

p m=1

≤ kck

`∞

`

+ δε.

n

m=1

`

n

A path to Musielak-Orlicz spaces

59

In all, the inequality: δkck`∞

∞

X

km ,km+1 ≤ cm a

m=1

≤ kck`∞

`pn

is derived. In other words, the mapping J : `∞ −→ `pn ∞ X J ((cm )) = cm akm ,km+1

(1.48)

m=1

is an isomorphism from `∞ into J (`∞ ) ⊂ `pn . ∗

The preceding lemma is key to the characterization of the dual space (`pn ) in the absence of the ∆2 condition. More specifically, we have the following result: Theorem 1.4.24. Let (pn ) ⊂ (1, ∞) be a variable exponent and assume that pn → ∞ as n → ∞. ∗ If Λ ∈ (`pn ) and Λ vanishes on `pan then for any b ∈ `pn \ `pan there exists ∗ ˜ ∈ (`∞ ) such that Λ ˜ vanishes on c0 (i) Λ ˜ N ) = Λ(b). (ii) Λ(1 ∗

Conversely, if Λ ∈ (`∞ ) vanishes on c0 , then given any fixed b ∈ `pn \ `pan , ∗ there exists γ ∈ (`pn ) that fulfills the following: (i) γ vanishes on `pan ˜ (ii) γ(1N ) = Λ(b) (iii) If Zb is as in (1.47), then γ|Zb = Λ. ∗

Proof. To prove (i), let Λ ∈ (`pn ) and fix b ∈ `pn . Let Zb denote the subspace ∗ of `pn defined in (1.47); then b ∈ Zb ∼ `∞ . Any Λ ∈ (`pn ) that vanishes on pn ∞ ∗ ˜ `a engenders a functional Λ ∈ (` ) via the isomorphism J : Zb −→ `∞ introduced in (1.48). Specifically ˜ = Λ ◦ J −1 . Λ It is easy to see that Zb ∩ `pan = J −1 (c0 ) ,

(1.49)

˜ vanishes on c0 . Furtheras a consequence of which one readily obtains that Λ more, by the very definition of J one has ˜ N ). Λ(b) = Λ(1

60

Analysis on Function Spaces of Musielak-Orlicz Type ∗

To prove the converse assertion, fix Λ ∈ (`∞ ) and set ˜ ∈ (Zb )∗ Λ ˜ = Λ ◦ J and extending Λ ˜ to `pn via the Hahn-Banach Theorem (we as Λ ˜ continue to denote this Hahn-Banach extension by Λ). In the light of Lemma 1.4.16, (iii), one can write ˜ = γ1 + γ2 , Λ ∗

where γi ∈ (`pn ) for i = 1, 2 and γ1 vanishes on `pan . In fact, for x ∈ `pn γ2 (x) =

∞ X

˜ k ). xk Λ(e

1

The restrictions to Zb of γ1 and γ2 satisfy ˜ = γ1 |Z + γ2 |Z . Λ b b ˜ and γ1 both vanish on J −1 (c0 ). Consequently, By assumption and (1.49), Λ −1 γ2 |Zb also vanishes on J (c0 ). As was observed in the proof of Lemma 1.4.23, the assumption b ∈ `pn \ `pan and the choice of δ > 0 leads to the existence of an increasing sequence (kj ) ⊂ N such that kbkj ,kj+1 k`pn > δ for each j (here bkj ,kj+1 is defined as in (1.46)). P∞ We claim that for any sequence x = 1 cm bkm ,km+1 ∈ Zb one has γ2 (x) = 0. k ,k Indeed, for each m ∈ N, the sequence (bnm m+1 ) = bkn ,km+1 ∈ `pan . Consequently, for each N ∈ N we have: ! N N X X km ,km+1 γ2 cm b = cm γ2 (bkm ,km+1 ) (1.50) m=1

m=1

= 0. Writing x = (xj ) it is immediate by definition that: γ2 (x) =

∞ X

˜ j) xj Λ(e

(1.51)

1

and that for each km as above, km+1

X

˜ j) = xj Λ(e

1

m X j=1

cj

kj X

˜ j ) = 0, bk Λ(e

k=1

on account of (1.50). We conclude that there exists a strictly increasing sequence K1 , K2 , , .... of natural numbers such that for each k ∈ N it holds KN X 1

˜ j) = 0 xj Λ(e

A path to Musielak-Orlicz spaces

61 ∗

and by virtue of (1.51), γ2 (x) = 0. In conclusion, γ1 ∈ (`pn ) vanishes on `pan , ˜ on Zb and satisfies γ1 (b) = Λ(1N ). Theorem 1.4.24 is thus coincides with Λ proved.

1.4.3

Geometric properties of `pn

In this section we investigate the modular geometric properties of the variable exponent sequence spaces `pn . At this point, a digression is in order. It will be shown in a more general context (Corollary 2.6.2) that if the exponent sequence (pn ) satisfies 1 < p− and p+ < ∞ then the Banach space (`pn , k · k`pn ) is uniformly convex. It is however desirable, at least in some mathematical situations to have a concept of “uniform convexity” associated directly to the modular. It is in this spirit that we undertake the discussion presented in this section. Surprisingly, some form of modular uniform convexity is evidenced even in the case when p− = 1 or p+ = ∞. In order to lay the groundwork for the main results, some technical lemmas need to be proved. Lemma 1.4.25. For any p : 1 ≤ p ≤ 2 q(p) = 2 + p(p − 1) − 2p ≤ 0, in fact the strict inequality holds if 1 < p < 2. 000

00

00

Proof. Since q (p) < 0 and q (2) = 2 − 4 ln 22 > 0, q (p) > 0 for all p ∈ [1, 2] and q 0 is increasing on [1, 2]. On the other hand, q 0 (1) = 1 − 2 ln 2 < 0 and q 0 (2) = 3−4 ln 2 > 0. It is clear then that there exists a unique x0 ∈ (1, 2) such that q decreases on (1, x0 ) and increases on (x0 , 2). Since q(1) = q(2) = 0, it is immediate that q < 0 on (1, 2), as claimed. Lemma 1.4.26. If 1 ≤ p ≤ 2 and h : [0, 1] → R be defined as h(s) = sp−1 − (s + 1)p−1 +

p(p − 1) , 2

then h(s) ≤ 0 on [0, 1]. Proof. It is immediately seen that h is increasing and that h(1) = 1 − 2p−1 + p(p−1) = γ(p) < 0 on account of the previous lemma. Hence h(s) < 0 for any 2 x ∈ (0, 1). Lemma 1.4.27. Let a ∈ R, b ∈ R 0 < b < a, 1 ≤ p ≤ 2. Then a − b p p(p − 1) a + b p + ≤ 1 (|a|p + |b|p ) . 2 2 2 2 Proof. Using the transformation s =

t−1 t+1

one has t ≥ 1 ⇐⇒ 0 ≤ s < 1 and

62

Analysis on Function Spaces of Musielak-Orlicz Type

thus, Lemma 1.4.26 implies that for t ∈ [1, ∞) one has: p(t + 1)

p−1



t−1 t+1

p−1

p(p − 1) + − 2p−1 2



t t+1

p−1 !

p2 (p − 1) = p(t − 1)p−1 + (t + 1)p−1 − p2p−1 tp−1 2   d p(p − 1) p p p−1 p = (t − 1) + (t + 1) − 2 t ≤ 0. dt 2 It follows that p(p − 1) (t + 1)p − 2p−1 tp 2   is decreasing on [1, ∞) and since w(1) = p(p−1) − 1 2p−1 < 0 it is plain that 2 w(t) = (t − 1)p +

w(t) ≤ 0 for t ≥ 1. Writing t = ab , the lemma follows at once. Lemma 1.4.28. If 1 < p ≤ 2 and consider G : [0, 1) −→ R
G(s) = 2 2 + p(p − 1)s2 − ((1 + s)p + (1 − s)p ) , then G(s) < 0 for any s : 0 ≤ s < 1. 000

00

Proof. Routine calculations reveal that G (s) < 0, so that G (s) is decreas00 ing. On the other hand, G (0) = 0, and as a consequence, G0 is decreasing on (0, 1), which in conjunction with the fact that G0 (0) = 0 yields that G is decreasing on (0, 1). Finally, G(0) = 0, which proves the claim. Lemma 1.4.29. Let 0 < b < a be real numbers and 1 ≤ p ≤ 2. Then a + b p p(p − 1) a − b 2−p a − b p + ≤ 1 (|a|p + |b|p ) . 2 2 |a| + |b| 2 2 Proof. From Lemma 1.4.28 it follows that 

2 1−s

2 +

p(p − 1) 2



2s 1−s

2

− 2p−1



1+s 1−s

p

 +1

2 1−s

2−p

for 0 ≤ s < 1. Using the transformation s −→

t−1 t+1

the latter implies that for any t ≥ 1, (t + 1)2 +

p(p − 1) (t − 1)2 − 2p−1 (tp + 1)(t + 1)2−p < 0, 2

0 and ab < 0. If a and b have different signs, there is no loss of generality by considering a > 0 and writing b = −c, c > 0. Inequality (1.53) is then, in this case, nothing but the conclusion of Lemma 1.4.27. If ab > 0, on the other hand and with no loss of generality we assume a > b, then inequality (1.53) is Lemma 1.4.29. Theorem 1.4.32. ([7]) Denote the space of all real sequences (an ) by V and let (pj ) ⊂ R be a sequence with 1 < inf pj . Then, the modular ρpn on V j≥1

defined as ρpn : V −→ [0, ∞] a = (an ) −→ ρpn (a) =

∞ X |aj |pj j=1

pj

satisfies the U CC2 condition. Proof. We fix the sequence (pn ) and define ρpn as in Proposition 1.24, Section 1.4. For a subset A ⊆ N we write ρA pn ((an )) =

X |an |pn . pn

n∈A

64

Analysis on Function Spaces of Musielak-Orlicz Type

Let r > 0,  > 0 and consider u = (un ) ∈ `pn , v = (vn ) ∈ `pn such that   u−v ρpn (u) ≤ r , ρpn (v) ≤ r , ρpn ≥ r. 2 Notice that the convexity of ρpn forces  ≤ 1, for   u−v r ≤ ρpn ≤ r. 2 Let A := {n ∈ N : pn ≥ 2}. By assumptions on u and v and on account of the definition of A one must necessarily have either   u−v r ρA ≥ (1.54) pn 2 2 or

 ρN\A pn

u−v 2

 ≥

r . 2

(1.55)

If (1.54) holds, Clarkson’s inequality (1.52) yields    
u−v u+v 1 A A ρA + ρ ≤ ρpn (u) + ρA pn pn pn (v) ; 2 2 2 it is clear then that ρA pn



u+v 2

 ≤


r 1 A ρpn (u) + ρA . pn (v) − 2 2

Thus,  ρ pn

u+v 2



 u+v = + 2 
r 1  N\A 1 A A ≤ ρpn (u) + ρpn (v) − + ρpn (u) + ρN\A pn (v) 2 2 2 1 r = (ρpn (u) + ρpn (v)) − 2 2  ≤r 1− . 2 ρA pn



u+v 2





ρN\A pn

In case (1.55) holds, we define n o  B := n ∈ N \ A : |un − vn | ≤ (|un | + |vn |) . 4 It follows that ρB pn



u−v 2




 B ρ (u) + ρB pn (v) 8 pn  r ≤ (ρpn (u) + ρpn (v)) ≤ . 8 8 ≤

A path to Musielak-Orlicz spaces

65

The validity of (1.55) implies in particular that       u−v u−v u−v N\A B ρN\(A∪B) = ρ − ρ pn pn pn 2 2 2 r r r ≥ − = . 2 4 4 Next, as a direct consequence of inequality (1.53), if n ∈ N \ (A ∪ B), one has pn  u n + vn pn  + inf pn − 1  un − vn N 2 8 2 pn p u n + vn pn (pn − 1)   2−pn un − vn n ≤ + 2 2 4 2 pn 2−pn u n + vn un − vn pn pn (pn − 1) un − vn ≤ + |un | + |vn | 2 2 2 ≤

1 (|un |pn + |vn |pn ). 2

Consequently, integrating the preceding inequality over N \ (A ∪ B) it is plain that       u+v u−v N\(A∪B) ρN\(A∪B) + inf p − 1 ρ n pn N 2 8 pn 2   1 N\(A∪B) ≤ ρ (u) + ρN\(A∪B) (v) pn 2 pn and thus  ρN\(A∪B) pn

u+v 2

 ≤

   2 1  N\(A∪B) ρ pn (u) + ρN\(A∪B) (v) − inf pn − 1 r. pn N 2 32

In fact, then,       u+v u+v u+v (A∪B) N\(A∪B) ρ pn = ρ pn + ρ pn 2 2 2  1  (A∪B) (A∪B) ≤ ρ (u) + ρpn (v) 2 pn    2 1  N\(A∪B) + ρpn (u) + ρN\(A∪B) (v) − inf pn − 1 r pn N 2 32     2   2 ≤ r − inf pn − 1 r = r 1 − inf pn − 1 . N N 32 32 In all, for any r > 0,  > 0 and arbitrary sequences u and v in D(r, ) as specified in Definition 1.3.8, it holds that      2  1 u+v  1 − ρp ≥ min , inf pn − 1 > 0, N r 2 2 32 and `pn is U U C2.

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Analysis on Function Spaces of Musielak-Orlicz Type

Theorem 1.4.32 settles the question of the property U CC2 in the context of variable exponent sequence spaces. The next natural order of business is to consider the U CED property for such spaces. The next theorem points toward this direction. We refer the reader to Definition 1.3.11. Theorem 1.4.33. Consider the vector space `pn . The following statements are equivalent: (i) The set {n ∈ N : pn = 1} has at most one element (property (AO)). (ii) The modular ρpn is (UUCED). (iii) The modular ρpn is (SC). Proof. Note that (ii) easily implies (iii). We next prove that (iii) implies (i). Assume that ρpn is (SC) and (an ) fails the condition (i). There must then exist i, j ∈ N such that i 6= j and pi = p(j) = 1. Set x = (xn ), where xn = 0 if n 6= i and xi = 1, and y = (yn ), where yn = 0 if n 6= j and yj = 1. Then   x+y ρpn (x) = ρpn (y) = ρpn =1 2 and x 6= y. This will contradict our assumption that ρpn is (SC). In order to complete the proof of Theorem 1.4.33, we only need to prove that (i) implies (ii). Assume that condition (i) holds; we will demonstrate that ρ is (U U CED). To this end, let z1 = (zn1 ), z2 = (zn2 ) ∈ `pn such that z1 6= z2 . Let R > 0 and x ∈ `pn such that ρpn (x − z1 ) ≤ R and ρpn (x − z2 ) ≤ R. Set K = {n ∈ N; zn1 6= zn2 }. It is immediate from the assumption z1 6= z2 that K is not empty. We have K = K1 ∪ K2 ∪ K3 , where K1 = {n ∈ K; pn ≥ 2}, K2 = {n ∈ K; 1 < pn < 2} and K3 = {n ∈ K; pn = 1}. Since K is not empty, at least one of the subsets K1 , K2 or K3 is not empty. The proof requires an analysis of the three different scenarios. First case: To start with, we assume that K1 6= ∅. In the light of Theorem 1.4.30 it is clear that 1   1 2 pi 2 pi xi − zi + zi + zi − zi ≤ 1 |xi − zi1 |pi + |xi − zi2 |pi , 2 2 2 for any i ∈ K1 . In addition:   1 2 pn xn − zn + zn ≤ 1 |xn − zn1 |pn + |xn − zn2 |pn , 2 2

A path to Musielak-Orlicz spaces for any n 6∈ K1 , which imply   X z1 + z2 1 ρ pn x − + 2 pi i∈K1

67

1 zi − zi2 pi ρpn (x − z1 ) + ρpn (x − z2 ) ≤ R. 2 ≤ 2

In this case, we take p 1 X 1 zi1 − zi2 i δ(z1 , z2 , R) = > 0. R pi 2 i∈K1

Second case: If K2 is not empty, Theorem 1.4.30 yields   1 2 pi xi − zi + zi + Ai ≤ 1 |xi − zi1 |pi + |xi − zi2 |pi , 2 2 where Ai = =

pi (pi − 1) 2 pi (pi − 1) 21+pi

2−pi 1 zi − zi2 pi zi1 − zi2 |xi − z 1 | + |xi − z 2 | 2 i

1 

|xi −

zi1 |

(1.56)

i

+ |xi −

1 2 2 2−pi |zi − zi | ,

zi2 |

for any i ∈ K2 . On the other hand, the inequalities ρ(x − z1 ) ≤ R and ρ(x − z2 ) ≤ R combined yield: 1 1 |xi − zi1 |pi ≤ R and |xi − zi2 |pi ≤ R, pi pi as a consequence of which it is readily obtained that:  1/pi  1/pi |xi − zi1 | + |xi − zi2 | ≤ 2 pi R ≤2 2R = 21+1/pi R1/pi ≤ 22 R1/pi , for any i ∈ K2 . Hence, 

2−pi  2−pi |xi − zi1 | + |xi − zi2 | ≤ 22 R1/pi ≤ 22 R(2−pi )/pi ;

it follows from here that: 1  21+pi

1

1 1 1 2−pi ≥ 1+pi 2 (2−pi )/pi ≥ 5 (2−pi )/pi . 2 2 R 2 R |xi − zi1 | + |xi − zi2 |

In all, Ai ≥

pi (pi − 1) |z 1 − zi2 |2 , 25 R(2−pi )/pi i

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Analysis on Function Spaces of Musielak-Orlicz Type

for any i ∈ K2 . Therefore, we have: X i∈K2

25

X (pi − 1) |zi1 − zi2 |2 ≤ Ai . (2−p )/p i i R i∈K 2

It follows as a direct consequence of Theorem 1.53 that, if Ai is as defined in (1.56), then   X z1 + z2 (pi − 1) 2 ρK x − + |zi1 − zi2 |2 pn 5 R(2−pi )/pi 2 2 i∈K2   X z1 + z2 2 ≤ ρK x− + Ai pn 2 i∈K2 1 X ≤ (|xpn − zp1n | + |xpn − zp2n |). (1.57) 2 n∈K2

In conjunction with Theorem 1.4.30 and the equality       z1 + z2 z1 + z2 z1 + z2 K2 1 ρpn x − = ρK x − + ρ x − , pn pn 2 2 2 inequality (1.57) yields that   X z1 + z2 (pi − 1) ρ pn x − + |zi1 − zi2 |2 5 (2−pi )/pi 2 2 R i∈K 2

ρp (x − z1 ) + ρpn (x − z2 ) ≤ n ≤ R. 2 One can therefore take X X (pi − 1) (pi − 1) 1 2 2 δ(z1 , z2 , R) = |z − z | = |zi1 − zi2 |2 > 0. i i 5 R1+(2−pi )/pi 5 R2/pi 2 2 i∈K i∈K 2

2

Third case: We finally tackle the case in which K1 = K2 = ∅ and K3 6= ∅. Since condition (i) is fulfilled it follows that K3 is a singleton; say K3 = {i}. For the sake of notational simplicity let us write K3 = K and N \ K = K c . Our assumptions on x, z1 and z2 imply the identities c

c

c

K K ρK pn (x − z1 ) = ρpn (x − z2 ) = ρpn (x − (z1 + z2 )/2) := R(x);

(1.58)

moreover it is clear that for j = 1, 2 j ρK pn (x − zj ) = |xi − zi | ≤ R − R(x).

On account of Theorem 1.4.30, we have 1 1 2 2 2 2 1 2 2 2 xi − zi + zi + zi − zi = |xi − zi | + |xi − zi | ≤ (R − R(x))2 2 2 2

A path to Musielak-Orlicz spaces

69

and from here one concludes that   1 2 2 1 1 2 2 xi − zi + zi ≤ (R − R(x))2 1 − |z − zi | . 2 4(R − R(x))2 i In other words, by the very definition of K: ρK pn

   1/2 z1 + z2 1 1 2 2 x− ≤ (R − R(x)) 1 − |z − zi | . 2 4(R − R(x))2 i

Hence, for R(x) as defined in (1.58) it is easy to derive the inequality:  ρ pn

z1 + z2 x− 2



1/2 1 1 2 2 |z − zi | + R(x) 4(R − R(x))2 i  1/2 1 1 2 2 =R 1− |z − z | i 4(R − R(x))2 i  1/2 ! 1 1 2 2 + R(x) 1 − 1 − |z − zi | 4(R − R(x))2 i  ≤ (R − R(x)) 1 −

= R(1 − B) + R(x)B, where we have introduced the positive quantity s 1 B =1− 1− |z 1 − zi2 |2 . 4(R − R(x))2 i As it transpires from (1.59), we have     z1 + z2 R − R(x) ρpn x − ≤R 1− B . 2 R On the other hand, note that |zi1 − zi2 | ≤ |zi1 − x| + |x − zi2 | ≤ 2(R − R(x)) ≤ 2R; we also underline the elementary fact that the function  1/2 1 s −→ 1 − 1 − 2 |zi1 − zi2 |2 4s is decreasing. It follows from the preceding considerations that s r 1 1 1 2 2 1− 1− |zi − zi | ≥ 1 − 1 − |z 1 − zi2 |2 , 2 4(R − R(x)) 4R2 i

(1.59)

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Analysis on Function Spaces of Musielak-Orlicz Type

which implies s

! 1 = 1− 1− |z 1 − zi2 |2 4(R − R(x))2 i ! r |zi1 − zi2 | 1 1 2 2 ≥ 1− 1− |z − zi | . 2R 4R2 i ! r |zi1 − zi2 | 1 1 2 Setting δ(z1 , z2 , R) = 1− 1− |z − zi |2 , it follows that 2R 4R2 i   z1 + z2 ρ pn x − ≤ R(1 − δ(z1 , z2 , R)). 2 R − R(x) B R

R − R(x) R

Note that δ(z1 , z2 , R) > 0. In all three instances we have δ(z1 , z2 , R) ≥ ¯ whenever R ≤ R, ¯ i.e., the modular ρ is (U U CED) as claimed. δ(z1 , z2 , R),

1.4.4

Applications: Fixed point theorems on `pn spaces

A remarkable fact about the above discussion is that the U U C2 property holds even if sup pn = ∞, that is, in the absence of the ∆2 condition. This n∈N

observation makes the U U C2 condition a valuable tool for dealing with certain applications that have been hitherto heavily ∆2 -dependent. We will present a few applications to problems arising in the theory of fixed points. For an exhaustive treatment of the interplay between the theory of modular spaces and the theory of fixed point of mappings we refer the reader to the monograph [55]. If the sequence of exponents (pn ) satisfies 1 < p− = inf pn and sup pn = n

n

p+ < ∞, there is no substantial difference between dealing with the norm and dealing with the modular in `pn . However in the general case, Theorem 1.3.8 and the example following it show that modular convergence and norm convergence part ways. It is sometimes more convenient to deal with modular convergence. With this aim in mind we introduce some terminology before proceeding any further: A subset W ∈ `pn will be called ρpn -bounded if there exists a constant C ≥ 0 such that the inequality ρpn (u) ≤ C holds for any u ∈ W . W is said to be ρpn -closed if whenever ρp

un →n u one has u ∈ W . The next observation is of particular importance in the sequel: Consider sequences x, y and let (yn ) be ρpn -convergent to y. Fatou’s Lemma yields the following inequality ρpn (x − y) ≤ lim inf ρpn (x − yn ). n→∞

A path to Musielak-Orlicz spaces

71

For obvious reasons, the above is known as the Fatou property of the modular ρ pn . Theorem 1.4.34. ([7]) Let p : N −→ (1, ∞); assume inf pn > 1. Let W ⊂ n∈N

`pn be convex and ρpn -closed and u ∈ `pn satisfy

dρpn (u, W ) = inf {ρpn (u − v) : v ∈ W } < ∞. Then there exists a unique v0 ∈ W for which dρpn (u, W ) = ρpn (u − v0 ). Proof. One can clearly assume that u ∈ / W , otherwise there is nothing to prove. Under this assumption, one must have d(u, W ) > 0, due to the ρpn closedness of W . Let (vn ) ⊆ W be such that   1 ρpn (u − vn ) < d(u, W ) 1 + . n
Then the sequence v2n must be ρpn -Cauchy, i.e. it must necessarily hold that ρpn (2−1 (vn − vm )) → 0 as m, n → ∞. The latter follows by contradiction. Indeed, if otherwise, there would exist δ > 0 and strictly increasing subsequences (nk )k≥1 and (mk )k≥1 with nk > mk for every k, such that   vnk − vmk ρ pn ≥δ (1.60) 2 for each k ∈ N. Since nk > mk , it holds that 

1 max{ρpn (u − vnk ), ρpn (u − vmk )} ≤ d(u, W ) 1 + mk

 := rk .

Together with (1.60) and in by virtue of Definitions (1.3.8) and (1.3.9) and Theorem 1.4.32, there exists η > 0   1 (vmk + vnk ) 1 − ρ pn u − ≥ η > 0, r 2 for any k ∈ N. Though not mentioned explicitly there, it is apparent from the proof of Theorem 1.4.32 that η is independent of rk . Since W is convex by assumption, the last inequality above yields   (vmk + vnk ) d(u, W ) ≤ ρpn u − ≤ r(1 − η) 2   1 = d(u, W ) 1 + (1 − η). mk Letting k tend to ∞ one clearly reaches a contradiction: in conclusion, the

72

Analysis on Function Spaces of Musielak-Orlicz Type
sequence v2n is ρpn -Cauchy, as claimed. Since `pn is ρpn complete, we define v as lim ρpn (v − 2−1 vn ) = 0. n→∞

Notice that

  vk  ρpn 2v − v + → 0 as k → ∞; 2
n and that for fixed k ∈ N, vk +v converges to v2k + v. The convexity and 2 n ρpn -closedness of W implies then that v2k + v ∈ W for each k and invoking again the ρpn -closedness of W we conclude that 2v ∈ W . On account of the Fatou’s property for the modular ρpn one concludes that   vk  d(u, W ) ≤ ρpn (u − 2v) ≤ lim inf ρpn u − v + k→∞ 2    vn + vk ≤ lim inf lim inf ρpn u − n→∞ k→∞ 2 1 ≤ lim inf lim inf (ρpn (u − vn ) + ρpn (u − vk )) n→∞ k→∞ 2 = d(u, W ). It follows that d(u, W ) = ρpn (u − 2v). If w ∈ W and d(u, W ) = ρpn (u − w) it is therfore concluded that   2v + w 1 d ≤ ρ pn u − ≤ (ρpn (u − 2v) + ρpn (u − w)) = d. 2 2 Lemma 1.3.17 guarantees that ρpn is strictly convex and hence that w = 2v, which yields the uniqueness statement. Aiming at presenting further applications of the U U C2 property for `pn we state and prove the following : Theorem 1.4.35. ([7]) Let (Cn )n be a non-increasing sequence of ρpn -closed, convex, nonempty subsets of `pn with inf pn > 1.

n∈N

Assume that there exists v ∈ `pn such that sup d(v, Cn ) < ∞. Then n≥1 ∞ \

Cn 6= ∅.

n=1

Proof. It suffices to assume that for some n0 ∈ N it holds v ∈ / Cn0 , for otherwise there would be nothing to prove. From the ρpn -closedness of Cn0 it

A path to Musielak-Orlicz spaces

73

is easily derived that d(v, Cn0 ) > 0. Since the sequence (Cn )n is non-increasing by assumption, the inequalities ∞ > sup d(v, Cj ) ≥ d(v, Cn ) = inf d(v, u) ≥ u∈Cn

j≥1

inf

u∈Cn−1

d(v, u) = d(v, Cn−1 )

are clear for any n > 1. Thus, the sequence d(v, Cn ) is non-decreasing and bounded. Let L = limn→∞ d(v, Cn ) < ∞; clearly L > 0. For each n ∈ N let un ∈ Cn be chosen so that ρpn (v
− un ) = d(v, Cn ) as in Theorem 1.4.34. It follows that the sequence u2n n is ρpn -Cauchy in `pn and hence it ρpn
converges to, say, u/2 ∈ `pn . Fix k ∈ N. Then the sequence u2n n≥k is contained in Ck and ρpn -converges to u2 , which implies that u2 ∈ Ck , since Ck is ρp -closed. In conclusion, ∞ \ u ∈ Cn , 2 n=1 i.e.,

∞ T

Cn 6= ∅, as claimed.

n=1

Theorem 1.4.36. ([7]) Let inf pn > 1 and ∅ = 6 C ⊂ `pn be a ρpn -closed, ρpn n∈N

bounded, convex set and (Ci )i∈I ⊂ 2C be a family of subsets of C having the finite intersection property, i.e., such that for every finite subset {i1 , ...ik } ⊂ I k T it holds that Cij 6= ∅. Then j=1

\

Ci 6= ∅.

i∈I

Proof. C is ρpn -bounded, it is therefore immediate that for any u ∈ C and i ∈ I, d(u, Ci ) = inf ρpn (u − v) ≤ sup ρpn (u − v) < ∞. v∈Ci

v∈C

For any finite subset A ⊂ I let  dA = d u,

 \

Cj  .

j∈A

If A and B be finite subsets of I, A ⊆ B, then

T

Cj ⊆

j∈B

T

Cj and conse-

j∈A

quently,  d u,

 \ j∈A

Cj  =

v∈

inf T j∈A

Cj

ρpn (u, v) ≤

v∈

inf T j∈B

Cj

ρpn (u, v),

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Analysis on Function Spaces of Musielak-Orlicz Type

i.e., dA ≤ dB . Write (

!

dI = sup d u,

\

) J ⊂ I : and

Ci

i∈J

\

Ci 6= ∅ .

i∈J

Let (An ) be the sequence defined by dI − n S

1 < dAn ≤ dI , n

∞ S

Bn . It is clear then that for each n ∈ N, the set ! T T Ci is ρp -closed, convex and non-empty and that the sequence Ci

write Bn =

k=1

Ak and J =

n=1

i∈Bn

i∈Bn

is non-increasing. Hence, Theorem 1.4.35 applies and we have \ S= Ci 6= ∅. i∈J

By definition, for each n ∈ N, it holds that \ \ Ci ⊆ Ci , i∈J

i∈An

and it follows that for each n one has dI −

1 < dAn ≤ d(u, S) ≤ dI . n

Thus d(u, S) = dI . On account of Theorem 1.4.34, there exists a unique z ∈ S which satisfies ρpn (u − z) = dI and therefore, for any index i0 ∈ I one has \ S ⊇ S ∩ C i0 = Ci 6= ∅; i∈J∪{i0 }

it is seen immediately that dI ≤ d(u, S) ≤ d(u, S ∩ Ci0 ) ≤ dI . In all, d(u, S) = d(u, S ∩ Ci0 ) and by Theorem 1.4.34 there exists a unique w ∈ S ∩ Ci0 for which ρpn (u − w) = d(u, S ∩ Ci0 ) = dI . In particular, w ∈ S, thus, invoking the uniqueness part of Theorem 1.4.34, one must necessarily have w = z. Since i0 is arbitrary, it is concluded that T z∈ Ci and hence, the latter intersection is non-empty, as claimed. i∈I

The following theorem is another consequence of the U U C2 property for `pn .

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75

Theorem 1.4.37. ([7]) Let inf pn > 1 and ∅ 6= C ⊂ `pn be a convex, ρpn n∈N

closed, ρpn bounded and assume that C is not a singleton (i.e., C at least two distinct points). Then there exists u ∈ C for which sup ρpn (u − v) < diam(C), v∈C

where as usual diam(C) = sup ρpn (a − b) stands for the ρpn -diameter of C. a,b∈C

The property established in Theorem 1.4.37 is commonly referred to as the ρpn -normal structure property. Theorem 1.4.37 can thus be rephrased as asserting that if inf pn > 1, then `pn has ρpn -normal structure. n∈N

Proof. The assumptions imply that δ(C) > 0 and that there exist two distinct points u ∈ C, v ∈ C, u 6= v. For any w ∈ C, invoking the U U C2 property it follows at once that, for δ as in the definition of U U C2, (1.3.8),     u+v u−w+v−w ρ pn − w = ρ pn 2 2     ≤ diam(C) 1 − δ diam(C), . diam(C) The arbitrariness of w in concert with the convexity of C yields the claim. Theorem 1.4.38. ([7])

If inf pn > 1, ∅ 6= C ⊂ `pn is convex, ρpn -closed n∈N

and ρpn -bounded, then any map T : C −→ C for which the bound ρpn (T (u) − T (v)) ≤ ρpn (u − v) holds for any u ∈ C, v ∈ C (i.e., T is nonexpansive), has a fixed point, that is to say, there exists w ∈ C such that T (w) = w. Proof. It is obvious that the theorem is true if C is a singleton. Thus, it can be assumed that the cardinality of C is at least 2. Let F = {∅ = 6 K ⊂ C : K is ρpn −closed and T (K) ⊆ K} . Since C ∈ F, F 6= ∅. Moreover, F is partially ordered by the order relation X ≤ Y ⇐⇒ Y ⊆ X.

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Analysis on Function Spaces of Musielak-Orlicz Type

If G is a totally ordered subfamily of F, then G possesses the finite intersection property and on account of Theorem 1.4.36 it follows that \ X 6= ∅; X∈G

this clearly implies that

T

X ∈ F and therefore that

X∈G

T

X is an upper

X∈G

bound for G. Zorn’s Lemma yields the existence of a maximal element X0 ∈ F. We set about to prove that X0 contains exactly one point. Denote the intersection of all ρpn -closed, convex subsets of C that contain T (X0 ) by T (X0 ). In particular, since X0 ∈ F, T (X0 ) ⊆ X0 . On the other hand, T (X0 ) ∈ F, for it is convex, ρpn -closed and   T T (X0 ) ⊆ T (X0 ) ⊆ T (X0 ). As a consequence of the maximality of X0 with respect to the indicated inclusion, one has T (X0 ) = X0 . (1.61) Theorem 1.4.37 yields the existence of an element x0 ∈ X0 such that r0 = sup ρpn (x0 − u) < diam(X0 ).

(1.62)

u∈X0

Let Bρpn (a, s) denote the ρpn -ball of radius s centered at a; we remark the obvious fact that the convexity and the Fatou property of the modular ρpn imply that Bρpn (a, s) is ρpn -closed and convex. Set   \ M= Bρpn (v, r0 ) ∩ X0 = u ∈ X0 : sup ρpn (u − v) ≤ r0 ; v∈X0

v∈X0

then M is ρpn -closed and convex and M ⊂ X0 . Moreover, if x ∈ M , then for any v ∈ X0 ρpn (T (x) − T (v)) ≤ ρpn (x − v) ≤ r0 . In other words, if v ∈ X0 , ρpn (T (v)−T (x)) ≤ r0 , i.e., T (X0 ) ⊆ Bρpn (T (x), r0 ). By definition of T (X0 ) it is plain that: T (X0 ) ⊆ Bρpn (T (x), r0 ); and from (1.61) it follows that X0 = T (X0 ) ⊆ Bρpn (T (x), r0 );

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77

that is, for any v ∈ X0 , ρpn (T (x) − v) ≤ r0 , i.e., T (x) ∈ Bρpn (v, r0 ). It follows from the definition of M that T (M ) ⊆ M, so that M ∈ F. and since M ⊆ X0 and X0 is maximal, one has a fortiori: X0 = M. By definition, then, if w ∈ X0 , ρpn (w − x0 ) ≤ r0 ; this forces the inequality diam(X0 ) ≤ r0 , which contradicts (1.62) unless diam(X0 ) = 0. Hence, diam(X0 ) = 0 and X0 = {a} is a singleton. Since also T (X0 ) ⊆ X0 , necessarily T (a) = a we conclude that T has a fixed point, as claimed. From the modular uniform convexity proved in [7] the authors were able to derive an intersection property similar to that of weak-compactness in Banach spaces. Since, in general, the new uniform convexity discussed in the previous section does not imply any good intersection property, we will need the following definition. Definition 1.4.4. Let C be a nonempty ρpn -closed and ρpn -bounded convex subset of `pn . We will say that C satisfies the property (R), if for any decreasing sequence {Cn }n≥1 of ρpn -closed convex nonempty subsets of C, we have \ Cn 6= ∅. n≥1

The following technical proposition will be useful throughout. Proposition 1.4.39. Let C be a nonempty ρpn -closed and ρpn -bounded convex subset of `pn . (i) Assume that C satisfies the property (R) and let K be a nonempty ρpn closed convex subset of C. Then K is ρpn -proximinal in C, i.e. for any x ∈ C, the set Pρpn ,K (x) = {y ∈ C; ρpn (x − y) = inf ρpn (x − z)} is z∈K

not empty. Moreover if ρpn is (SC) (Definition 1.3.11), then K is a ˇ sev subset, i.e. Pρ ,K (x) is a singleton for any x ∈ C. Cebyˇ pn (ii) Assume that C satisfies the property (R) and that ρpn is (SC) (Definition 1.3.11). Then for any family T {Cα }α∈Γ of ρpn -closed convex nonempty subsets of C such that Cα 6= ∅ for any finite subset α∈Γf T Γf ⊂ Γ, we have Cα 6= ∅. α∈Γ

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Analysis on Function Spaces of Musielak-Orlicz Type

(iii) Assume that C satisfies the property (R) and ρpn is (UUCED). Let K be a nonempty ρpn -closed convex subset of C. Then K has a unique ˇ sev center x ∈ K, i.e., ρpn -Cebyˇ   sup{ρpn (x − y); y ∈ K} = inf sup{ρpn (z − y); y ∈ K} . z∈K

Proof. Assume that C satisfies the property (R). Let K be a nonempty ρpn closed convex subset of C. For x ∈ C, we have dρpn (x, K) = inf{ρpn (x − y); y ∈ K} < ∞, since C is ρpn -bounded. Moreover, from the equality   \ Pρpn ,K (x) = Bρpn x, dρpn (x, K) + 1/n ∩ K, n≥1

where Bρpn (x, r) is the ρpn -ball centered at x with radius r, it follows that the property (R) implies that Pρpn ,K (x) is not empty. It is clear that if ρpn is (SC), then Pρpn ,K (x) must be reduced to a single point, which completes the proof of (i). In order to prove (ii), assume that C satisfies the property (R) and that ρpn is (SC). LetT{Cα }α∈Γ be a family of ρpn -closed convex nonempty subsets of C such that Cα is not empty, for any finite subset Γf ⊂ Γ. Let x ∈ C. Then α∈Γf

sup dρpn (x, Cα ) < ∞ holds since C is ρpn -bounded. For any subset F ⊂ Γ, α∈Γ T set dF = dρpn (x, Cα ). Note that if F1 ⊂ F2 ⊂ Γ are finite subsets, then α∈F

dF1 ≤ dF2 . Set n  \  o \ dΓ = sup dρpn x, Cα , J ⊂ Γ such that Cα 6= ∅ . α∈J

α∈J

For any n ≥ 1, there exists a subset Fn ⊂ n ΓTsuch that o dΓ − 1/n < dFn ≤ dΓ . ∗ Set Fn = F1 ∪ · · · ∪ Fn , for n ≥ 1. Then Cα is a decreasing sequence α∈Fn∗

of nonempty ρpn -closed convex subsets of C. The property (R) satisfied by C implies that for [ [ J= Fn∗ = Fn n≥1

n≥1

it holds that \

Cα 6= ∅.

α∈J

Set K =

T α∈J

Cα . Note that dρpn (x, K) = dΓ because, for any n ≥ 1, dΓ − 1/n < dFn ≤ dρpn (x, K) ≤ dΓ .

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79

Because of (i), there exists a unique y ∈ K for which ρpn (x − y) = dρpn (x, K) = dΓ . For fixed α0 ∈ Γ, one has: K ∩ Cα0 =

\

Cα 6= ∅

α∈J∪{α0 }

because of the same argument based on the property (R). It follows then that dρpn (x, K) ≤ dρpn (x, K ∩ Cα0 ) ≤ dΓ . Hence dρpn (x, K ∩ Cα0 ) T= dρpn (x, K) = dΓ which implies y ∈ K ∩ Cα0 . Therefore, we have y ∈ Cα ; in turn this proves that α∈Γ

\

{Cα } = 6 ∅.

α∈Γ

In order to prove (iii), assume that C satisfies the property (R) and ρpn is (U U CED). Let K be a nonempty ρpn -closed convex subset of C. Set rρpn (x, K) = sup ρpn (x − y), for any x ∈ K, and y∈K

Rρpn (K) = inf {rρpn (x, K); x ∈ K}. All the above numbers are finite since C is ρpn -bounded. Note that the set Kn = {x ∈ K; rρpn (x, K) ≤ Rρpn (K) + 1/n}, is non-empty for any n ≥ 1 and that   \ Kn = Bρpn y, Rρpn (K) + 1/n ∩ K, n ≥ 1, y∈K

which shows that (Kn ) is a decreasing sequence Tof ρpn -closed convex nonempty subsets of K. The property (R) implies that Kn is nonempty. Clearly, any n≥1 T x∈ Kn will satisfy rρpn (x, K) = Rρpn (K). We next set out to show that T n≥1 Kn is reduced to a single point. Assume that there exists z ∈ K such that n≥1

z 6= x and rρpn (z, K) = Rρpn (K). Since ρpn (x − z) ≤ rρpn (x, K) = Rρpn (K) and x 6= z, we conclude that Rρpn (K) > 0. From the (U U CED) property of ρpn one derives the existence of δ = δ(x, z, Rρpn (K)) > 0 such that    x+z ρpn (y − x) ≤ Rρpn (K) =⇒ ρpn y − ≤ Rρpn (K)(1 − δ), ρpn (y − z) ≤ Rρpn (K) 2

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Analysis on Function Spaces of Musielak-Orlicz Type

for any y ∈ K, which implies  Rρpn (K) ≤ rρpn

x+z ,K 2

 ≤ Rρpn (K)(1 − δ).

This contradiction finishes the proof of (iii), which in turn completes the proof of Proposition 1.4.39. We aim at utilizing the above ideas for proving an analogue to Kirk’s fixed point theorem in `pn . The following definition will be needed to that effect: Definition 1.4.5. Consider a nonempty set C ⊂ `pn and a mapping T : C → C. T is said to be ρpn -Lipschitzian if for some constant K ≥ 0 one has ρpn (T (x) − T (y)) ≤ K ρpn (x − y),

for any x, y ∈ C.

In particular T is called ρpn -nonexpansive if K = 1 and x ∈ C is called a fixed point of T if T (x) = x. The collection of all fixed points of T will be denoted by F ix(T ). The following modular version of Kirk’s fixed point theorem is an improvement from the one discovered in [7] since it does not require that inf pn > 1. n∈N

Theorem 1.4.40. Assume that ρpn satisfies the condition (AO) (Theorem 1.4.33). Let C be a nonempty ρpn -closed convex ρpn -bounded subset of `pn which has the property (R). Let T : C → C be a ρ-nonexpansive mapping. Then F ix(T ) is a nonempty ρpn -closed convex subset of C. Proof. Let ∅ = 6 C ⊂ `pn be ρpn -closed, convex and ρ-bounded and consider T : C → C to be a ρpn -nonexpansive mapping. Assume that C is not a singleton: it is clear that no generality is lost with this assumption. Consider the family F = {K ⊂ C; K 6= ∅, K ρpn -closed, convex and T (K) ⊂ K}. The family F is not empty since C ∈ F. Our assumption on the sequence (pn ) implies that ρpn is (U U CED). Zorn’s Lemma in concert with Proposition 1.4.39 yields the existence of a minimal element of F, which we denote by K0 . We contend that K0 consists of exactly one point. For otherwise we set co(T (K0 )) to be the intersection of all ρpn -closed, convex subsets of C which contain T (K0 ). In particular, co(T (K0 )) ⊂ K0 since T (K0 ) ⊂ K0 and we readily conclude that   T co(T (K0 )) ⊂ T (K0 ) ⊂ co(T (K0 )).

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81

The minimality of K0 yields K0 = co(T (K0 )). Next, set x ∈ K0 to be the ˘ unique ρpn -Ceby˘ sev center of K0 , that is:   rρpn x(x, K0 ) = sup{ρpn (x−y); y ∈ K0 } = inf sup{ρpn (z−y); y ∈ K0 } , z∈K0

which exists according to (iii) of Proposition 1.4.39. Since T is ρpn nonexpansive and K0 ⊂ Bρpn (x, rρpn (x, K0 )), it is concluded that T (K0 ) ⊂ Bρpn (T (x), rρpn (x, K0 )). Since all ρpn -balls are ρpn -closed and convex, it is easily obtained that K0 = co(T (K0 )) ⊂ Bρpn (T (x), rρpn (x, K0 )). One concludes, therefore, that rρpn (T (x), K0 ) ≤ rρpn (x, K0 ) = sup{ρpn (x − y); y ∈ K0 }   = inf sup{ρpn (z − y); y ∈ K0 } , z∈K0

˘ i.e., T (x) is also ρpn -Ceby˘ sev center of K0 . Therefore it must a fortiori hold that T (x) = x. (1.63) Assertion (1.63) implies K0 = {x}; this is a contradiction to our assumption that K0 contains more than one point. Hence any minimal element of F is reduced to one point: Consequently, F ix(T ) is not empty. Since T is ρpn -nonexpansive, F ix(T ) is ρpn -closed. In fact, the set F ix(T ) is also convex. Indeed, set z1 , z2 ∈ F ix(T ) with z1 6= z2 . Let α ∈ [0, 1]. Then: ρpn (zi − T (α z1 + (1 − α)z2 )) = ρpn (T (zi ) − T (α z1 + (1 − α)z2 )) ≤ ρpn (zi − (α z1 + (1 − α)z2 )), for i = 1, 2. Since ρpn is (U U CED), then ρpn is (SC). It is therefore concluded that T (αz1 + (1 − α)z2 ) = α z1 + (1 − α)z2 , i.e., α z1 + (1 − α)z2 ∈ F ix(T ), which completes the proof of Theorem 1.4.40.

As a consequence of the properties of the fixed point set proved in Theorem 1.4.40 we present the following common fixed point result. Theorem 1.4.41. Consider the vector space `pn . Assume that ρpn satisfies the condition (AO). Let C be a nonempty ρ-closed convex ρpn -bounded subset of `pn which satisfies the property (R). Let {Tα }α∈Γ be a commutative famT ily of ρpn -nonexpansive self-mappings defined on C. Then F ix(Tα ) is a α∈Γ

nonempty ρpn -closed convex subset of C.

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Analysis on Function Spaces of Musielak-Orlicz Type

Proof. Let S, T : C → C be two commutative ρpn -nonexpansive mappings. Theorem 1.4.40 implies that F ix(T ) is a nonempty ρpn -closed convex subset of C. Since S and T commute, then we have S(F ix(T )) ⊂ F ix(T ). A further application of Theorem 1.4.40 yields that the restriction of S to F ix(T ) has a fixed point, i.e., F ix(T ) ∩ F ix(S) 6= ∅. T It follows from this argument that for any finite subset Γf of Γ, F ix(Tα ) α∈Γf

is a nonempty ρpn -closedTconvex subset of C. Since C satisfies the property (R), we conclude that F ix(Tα ) 6= ∅. The fact that this intersection is α∈Γ

ρpn -closed and convex follows easily.

1.4.5

Further remarks

The (modular) spaces `pn were used by W. Orlicz in 1931 [85] to clarify some questions on lacunary series that had been studied by Banach [8]. To get at the bottom of the problem in point we recall the well-known fact that if ∞ a0 X + (an cos nx + bn sin nx) 2 n=1 is the Fourier series of f ∈ L2 ((0, 2π)), then ∞ X


|an |2 + |bn |2 < ∞.

(1.64)

n=1

By virtue of the Riesz-Fischer theorem [111, Theorem 1.1], the power 2 in (1.64) is optimal, that is, given any δ > 0, there must exist g ∈ L2 (0, 2π) such that the sequence of its Fourier coefficients is not in `2−δ . In this regard, Carleman [16] constructed a continuous function g on [0, 2π], the numerical series of whose Fourier coefficients satisfies ∞ X


|an |2−δ + |bn |2−δ = ∞

n=1

for any δ > 0. Thus the power 2 cannot be lowered even under the continuity assumption on f . A special class of trigonometric series has appealed the attention of mathematicians since the nineteenth century, namely, the class of lacunary series. As early as 1872, the famous example of a continuous, nowhere differentiable function provided by Weierstrass involved a series of this type. Far from attempting any exhaustive treatment of this deep topic, we present some isolated facts as an example of how, almost inadvertently, spaces with variables exponents start creeping into the center stage of mathematics through this theme. Specifically:

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83

Definition 1.4.6. The trigonometric series ∞

a0 X + (an cos αn x + bn sin αn x) 2 n=1

(1.65)

is said to be lacunary if for each n ∈ N, αn+1 > q > 1. αn As far back as 1872, the famous example of a continuous, nowhere differentiable function provided by Weierstrass involved a series of this type. It is well known that the Fourier series of an integrable function might diverge pointwise almost everywhere. However, Kolmogorov proved that a lacunary Fourier series converges almost everywhere [59]. The following two theorems due to Sidon [97, 98] and Zygmund [110], respectively. Theorem 1.4.42. (Sidon, 1927) If the series (1.65) is the Fourier series of a measurable, bounded function, then ∞ X

(|an | + |bn |) < ∞.

n=1

Theorem 1.4.43. (Zygmund, 1930) If the series (1.65) is the Fourier series of a function f ∈ L1 ((0, 2π)), then in fact f ∈ Lp ((0, 2π)) for every p : p ∈ (0, ∞) and in particular ∞ X


|an |2 + |bn |2 < ∞.

n=1

Enlightened by the preceding two theorems and influenced by the aforementioned work by Carleman, in 1930, Banach elaborated on the work done by Zygmund in [110]. More specifically, he undertook the task of proving the following: Theorem 1.4.44. (Banach, 1930) There exist a sequence (εn ) with 0 < εn < 1 and lim εn = 0 and a continuous function f ∈ C([0, 2π]) with lacunary n→∞

Fourier series (1.65) for which ∞ X


|an |2−n + |bn |2−n = ∞.

(1.66)

n=1

As a consequence of the preceding theorem one has the following: Theorem 1.4.45. (Orlicz, 1931) There exist sequences (an ), (bn ) such that ∞  X n=0

2−n

2−n

|an | 1−n + |an | 1−n



0, (ii) lim ϕ(t) = ∞, lim ϕ(t) = 0, t→∞

t→0+

(iii) ϕ is nondecreasing and convex, (iv) ϕ is continuous. The definition of Orlicz function is more or less stringent than ours depending on the authors. For an exhaustive treatment of this topic, which is here presented only as an example, we refer the reader to [2, 11, 62, 91, 92]. In this setting, the functional defined on V = M(Ω) by Z M(Ω) 3 v −→ ρϕ (v) = ϕ(|v(x)|) dx Ω

is a convex, left-continuous modular. The Orlicz class is the collection of all measurable functions v ∈ M(Ω) such that ρϕ (v) < ∞. The vector space Vρϕ consists, in this example of those v ∈ M(Ω) such that, for some λ > 0 Z ϕ(λ|v(x)|) dx < ∞ Ω

and the Luxemburg norm on Vρϕ is given by   Z  
kvkρϕ = inf λ > 0 : ϕ λ−1 |v(x)| dx ≤ 1 ;   Ω

the Banach space (Vρϕ , kvkρϕ ) is called the Orlicz space corresponding to ϕ and is usually denoted by Lϕ (Ω). Examples include the classical Lp -spaces n n−1

for 1 ≤ p < ∞ (ϕ(t) = tp ) and the space Lξ (Ω) with ξ(t) = et − 1, n = 2, 3, 4... encountered naturally in the formulation of one of the classical Sobolev embedding theorems [42, Theoremn 7.15]. We select yet another example before embarking in the abstract theory of Musielak-Orlicz spaces. Our motivation to present this paradigmatic case separately stems in part from its intrinsic importance in current applications and from the fact that, though in a primitive stage, this is the second example of Musielak-Orlicz space ever mentioned in the literature [85]. Specifically let P(Ω) stand for the class of all measurable, extended real-valued function p : Ω −→ [1, ∞].

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Analysis on Function Spaces of Musielak-Orlicz Type

It is easy to see that ϕ : Ω × [0, ∞) −→ [0, ∞] ϕ(x, t) = tp(x) is also measurable. For each such p define the sets: Ω0 = {x ∈ Ω : 1 < p(x) < ∞} Ω1 = {x ∈ Ω : p(x) = 1} Ω∞ = {x ∈ Ω : p(x) = ∞} and set: p− = ess inf p(x) and p+ = ess sup p(x) if |Ω0 | > 0, x∈Ω0

x∈Ω0

p− = p+ = 1 otherwise. Theorem 1.5.1. The function ρp : M(Ω) −→ [0, ∞] Z ρp (u) = |u(x)|dµ + ess sup |u(x)| x∈Ω∞

Ω0 ∪Ω1

defines a convex, continuous modular on M(Ω). Proof. Properties (i)−(iii) of Definition 1.3.1 have to be checked. It is obvious that ρp (u) = 0 ⇐⇒ u = 0 a.e. and that for |λ| = 1 and measurable u Z ρp (λu) = |λu(x)|dµ + ess sup |λu| = ρp (u). x∈Ω∞ Ω\Ω∞

Convexity and continuity follow at once. Notice that the corresponding Luxemburg norm defined on M(Ω)ρp (i.e., on the space of all Borel-measurable functions v on Ω for which ρp (λv) < ∞ for some λ > 0, see Definition 1.3.2) is given by  kukp = inf λ > 0 : ρp (λ−1 u) ≤ 1 . In particular, if Ω ⊆ Rn , A is the σ-algebra of Borel subsets of Ω and µ is the Lebesgue measure, Lp (Ω) is referred to as the Lebesgue space with variable exponent. If p(x) is constant in Ω, the space Lp (Ω) coincides with the classical Lebesgue space Lp (Ω) and the Luxemburg norm is identical to the Lebesgue norm   p1 Z 1 u −→ kukp =  |u(x)|p dx = (ρp (u)) p . Ω

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87

Proposition 1.5.2. Let p ∈ P(Ω). Then kukp ≤ 1 if and only if ρp (u) ≤ 1; moreover, if kukp ≤ 1, then ρp (u) ≤ kukp . If u ∈ Lp (Ω)\{0} and ess sup p(x) < ∞, {x:u(x)6=0}

then 

|u(x)| kukp

Z



ρp u/ kukp = Ω0 ∪Ω1

!p(x) dx + ess sup x∈Ω∞

|u(x)| = 1. kukp

Proof. The first assertion is contained in Proposition 1.3.2, Section 1.3, the second in Corollary 1.3.4 of Section 1.3. Now suppose that u ∈ Lp (Ω)\{0} and that ess sup p(x) < ∞, {x:u(x)6=0}



assume that K := ρp u/ kukp  a kukp /λ K ≤ 1. Then Z



ρp (u/λ) =

|u(x)| λ



< 1 and let λ ∈

p(x) dx + ess sup x∈Ω∞

Ω0 ∪Ω1





0, kukp



be such that

|u(x)| λ

 |u(x)|  = kukp /λ λ/ kukp  dx + ess sup λ x∈Ω∞ Ω0 ∪Ω1   Z   a a  |u(x)| p(x) |u(x)|  ≤ kukp /λ  λ/ kukp dx + ess sup λ x∈Ω∞ kukp Ω0 ∪Ω1   !p(x) Z  a |u(x)| |u(x)|  ≤ 1. ≤ kukp /λ  dx + ess sup kukp x∈Ω∞ kukp 

a 

a

Z



|u(x)| λ

p(x)

Ω0 ∪Ω1

Since λ < kukp , we have a contradiction, and so K ≥ 1. Let (λk ) be a decreasing sequence with limit kukp . Then p(x) Z  |u(x)| |u(x)| ρp (u/λk ) = dx + ess sup ≤ 1, λk λk x∈Ω∞ Ω0 ∪Ω1

R

so that the integrals

p(x)

(|u(x)| /λk )

dx are bounded above. Since

Ω0 ∪Ω1

|u(x)| /λk % |u(x)| / kukp , it follows from monotone convergence that !p(x) Z |u(x)| |u(x)| K= dx + ess sup ≤ 1. kukp x∈Ω∞ kukp Ω0 ∪Ω1

Thus K = 1 and the proof is complete.

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Analysis on Function Spaces of Musielak-Orlicz Type

As a matter of fact Lp (Ω) has a richer structure than that or a mere modular space: we next show that it is a Banach function space. Theorem 1.5.3. If Ω ⊆ Rn , A is the σ-algebra of Borel subsets of Ω, µ is the Lebesgue measure and p ∈ P(Ω), then Lp (Ω) is a Banach function space. In particular, Lp (Ω) is a Banach space. Proof. Properties (i), (ii) and (iv) of a Banach space are clearly  function  satisfied. For (iii), let 0 ≤ uk % u a.e.: then kuk kp is non-decreasing. If kukp < ∞, suppose that there exists λ > 0 such that kuk kp % λ < kukp . Then p(x) Z u (x) uk (x) k 1≥ dx + ess sup kuk kp ku k k p x∈Ω∞ Ω0 ∪Ω1

Z ≥ Ω0 ∪Ω1

Z % Ω0 ∪Ω1

uk (x) p(x) uk (x) dx + ess sup λ λ x∈Ω∞ u(x) p(x) u(x) > 1, dx + ess sup λ λ x∈Ω∞

and we have a contradiction. The case in which kukp = ∞ is handled analogously. In order to address Property (v), let E ⊂ Ω be such that |E| < ∞, let u ∈ X and put M = {x ∈ E ∩ (Ω0 ∪ Ω1 ) : |u(x)| < 1} and N = {x ∈ E ∩ (Ω0 ∪ Ω1 ) : |u(x)| ≥ 1}. Then:  1   kukp

 Z  |u(x)| dx + ess sup |u(x)| x∈E∩Ω∞

E∩(Ω0 ∪Ω1 )

Z = M

|u(x)| dx + kukp Z

≤ |M | + N

Z

|u(x)| |u(x)| dx + ess sup kukp x∈E∩Ω∞ kukp

N p(x)

|u(x)|

p(x) kukp

|u(x)| x∈E∩Ω∞ kukp

dx + ess sup

≤ |M | + 1,

Sharper relationships between the modular and the norm can be obtained under stricter conditions on p.

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89

Proposition 1.5.4. Let p ∈ P(Ω) be such that 1 < p− ≤ p+ < ∞. Then for all u ∈ Lp (Ω), n o n o p p p p min kukp− , kukp+ ≤ ρp (u) ≤ max kukp− , kukp+ . Proof. First note that by Proposition 1.5.2, ρp (u/ kukp ) = 1. Thus if kukp = λ > 1, λ−p+ ρp (u) ≤ ρp (u/λ) = 1 ≤ λ−p− ρp (u), and so p

p

kukp− ≤ ρp (u) ≤ kukp+ . The case kukp < 1 is handled in a similar manner. Theorem 1.5.5. If p+ < ∞, then the modular ρp satisfies the ∆2 condition. Proof. Let f ∈ Lp (Ω). Clearly 2p(x) ≤ 2p+ < ∞ in Ω, which yields Z ρp (2u) = |2u|p dµ + ess sup |2u| Ω1 ∪Ω0 ≤ 2p+ ρp (u).

x∈Ω∞

Corollary 1.5.6. If p+ < ∞, then ρp modular convergence is equivalent to k · kp convergence in Lp (Ω). Proof. The corollary follows immediately from Theorem 1.5.5 and Corollary 1.3.9. Spaces with variable exponent also resemble classical Lebesgue spaces with regard to Lebesgue points. We recall that if p is a constant and 1 ≤ p < ∞, then given any v ∈ Lp (Rn ), a point x ∈ Rn such that Z −1 p lim |B(x, r)| |v(y) − v(x)| dy = 0 r→0

B(x,r)

is called a Lebesgue point of v. It is a very familiar fact (see [36, Corollary 1, p. 44]), that almost every point of Rn is a Lebesgue point of v. By a simple adaptation of the Corollary just mentioned it can be shown that if p ∈ P(Rn ), p+ < ∞ and v ∈ Lp (Rn ), then Z −1 p(y) lim |B(x, r)| |v(y) − v(x)| dy = 0 r→0

B(x,r)

for a.e. x ∈ Rn . We finish this section with a useful technical lemma reminiscent of a similar property in the classical setting.

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Analysis on Function Spaces of Musielak-Orlicz Type

Proposition 1.5.7. Suppose p ∈ P(Ω) is such that p+ < ∞. Then ρp (uk ) → 0 if and only if kuk kp → 0; moreover, if kuk kp → 0, then uk → 0 in measure. Proof. Suppose that ρp (uk ) → 0, let ε ∈ (0, 1) and let N ∈ N be such that ρp (uk ) < ε if k > N. Then   −1/p+ ρp ρp (uk ) uk Z −1 p(x) −1/p+ ≤ ρp (uk ) |uk (x)| dx + ρp (uk ) kuk k∞,Ω∞ Ω\Ω∞ −1

≤ ρp (uk )

ρp (uk ) = 1.

Thus kuk kp ≤ ρp (uk )

1/p+

∗

< ε1/p ,

so that kuk kp → 0. That kuk kp → 0 implies ρp (uk ) → 0 is clear. Finally, suppose that kuk kp → 0 and uk does not converge to 0 in measure. Then there exist ε > 0, δ ∈ (0, 1) and a subsequence (ukl )l∈N such that for all l ∈ N, inf |{x ∈ Ω : |ukl (x)| > ε}| ≥ δ. Thus ρp (ukl ) ≥ δεp+ for all l ∈ N. But by the first part of the proposition, ρp (ukl ) → 0 : contradiction. The variable exponent Lebesgue spaces constitute one important example of Banach functions spaces; their ubiquity in various areas of theoretical and applied mathematics [24, 60] justifies their careful study. We will continue to direct our efforts to the study of these spaces in the light of a more general mathematical framework, namely the theory of Musielak-Orlicz spaces which in particular, encompasses the variable exponent Lebesgue scale. Further properties of the variable exponent Lebesgue scale Lp (Ω) will be investigated in the general context of Musielak-Orlicz spaces in Chapter 2.

Chapter 2 Musielak-Orlicz spaces

2.1 2.2

2.9 2.10

Introduction, definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . Embeddings between Musielak-Orlicz spaces . . . . . . . . . . . . . . . . . . . . 2.2.1 The ∆2 -condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Absolute continuity of the norm . . . . . . . . . . . . . . . . . . . . . . . . Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Duality of Musielak-Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Conjugate Musielak-Orlicz functions . . . . . . . . . . . . . . . . . . . . 2.4.2 Dual of Lϕ (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density of regular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniform convexity of Musielak-Orlicz spaces . . . . . . . . . . . . . . . . . . . . Carath´eodory functions and Nemytskii operators on Musielak-Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further properties of variable exponent spaces . . . . . . . . . . . . . . . . . . 2.8.1 Duality maps on spaces of variable integrability . . . . . . . . The Matuszewska-Orlicz index of a Musielak-Orlicz space . . . . . . Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

Introduction, definition and examples

2.3 2.4

2.5 2.6 2.7 2.8

91 101 110 113 115 118 118 120 124 128 132 137 150 151 155

The modular spaces `pn introduced in Section 1.4 and the space Lp (Ω) defined in Section 1.5 are but particular instances of the same mathematical concept. The key observation to study their commonalities rather than emphasizing their differences is that the modulars ρpn and ρp introduced in Section 1.4 and in Section 1.5, respectively entail a higher degree of complexity than that inherent to the mere definition of modular given in Section 1.3. Specifically, from now on we will focus on a detailed exploration of the intrinsic measure-theoretic character of both ρpn and ρp . To the effect of precisely describing the specialness of the particular class of modulars we have in mind, some terminology must be introduced. Definition 2.1.1. For a given measure space (Ω, A, µ) we say that ϕ is a Musielak-Orlicz (M O if no confusion arises) function on (Ω, A, µ) (or on Ω if no confusion arises from such simplification) as the collection of all functions ϕ : Ω × [0, ∞) −→ [0, ∞) 91

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Analysis on Function Spaces of Musielak-Orlicz Type

such that (i) ϕ(·, t) is measurable for each t ∈ [0, ∞), (ii) For a.e. x ∈ Ω, ϕ(x, ·) is non-decreasing, convex, continuous and ϕ(x, 0) = 0. (iii) ϕ(x, t) > 0 for t > 0, (iv) ϕ(x, t) −→ ∞ as t −→ ∞. For example if Ω ⊆ Rn , A is the Borel σ-algebra of subsets of Ω, µ is the Lebesgue measure and p is a measurable function p : Ω → [1, ∞), then ϕ(x, t) = tp(x) is a Musielak-Orlicz function on Ω in this setting. In particular the function ϕ in Definition 2.1.1 is said to be an Orlicz function if it is independent of x ∈ Ω. Definition 2.1.2. The Musielak-Orlicz function ϕ is said to be locally integrable if for any t > 0 and any subset W ⊆ Ω with µ(W) < ∞ one has Z ϕ(x, t) dx < ∞. W

Remark 2.1.1. The condition introduced in Definition 2.1.2 is referred to as local integrability of ϕ. It is emphatically not an artificial technical condition. As we shall see (for example, in Corollary 3.1.2 of Chapter 3), local integrability of ϕ is required to have a meaningful definition of the Sobolev spaces of Musielak-Orlicz type. Definition 2.1.3. If (Ω, A, µ) is a measure space, a simple function is a (finite) linear combination of characteristic functions of sets of finite measure. The class of simple functions will be denoted by S. Example 2.1.1. If Ω = N, µ is the counting measure, A = 2N and p : N −→ ∞, the function ϕ : N × [0, ∞) −→ [0, ∞) ϕ(n, t) = tp(n) is a proper, locally integrable Musielak-Orlicz function on N. On the other hand, it is easy to produce a Musielak-Orlicz function that is not locally integrable: Consider the Lebesgue measure on the interval (0, 1) and for any x ∈ (0, 1) let nx be the unique natural number for which x ∈ (2−(nx +1) , 2−(nx ) ]. Define p as p : (0, 1) → [1, ∞) p(x) = nx + 1

Musielak-Orlicz spaces

93

and define the Musielak-Orlicz function ψ by φ(x, t) = tp(x) . Then Z

1

1

Z

p(x)

φ(x, 3)dx = 0

3

dx =

0

=

n=0

∞  n+1 X 3 n=0

∞ Z X

2

2−n

3n+1 dx

2−(n+1)

= ∞.

Let the elementary fact be observed at this point that if u : Ω −→ [−∞, ∞] is a A-measurable function on Ω, then so is x −→ ϕ(x, |u(x)|). Let M(Ω) stand for the set of all extended real valued A-measurable functions defined on Ω. Theorem 2.1.2. The functional ρϕ : M(Ω) −→ [0, ∞] Z ρϕ (u) = ϕ(x, |u(x)|) dx

(2.1)

Ω

is a convex, left-continuous modular. Proof. If ρϕ (u) = 0 then necessarily ϕ(x, |u(x)|) = 0 a.e.; by virtue of (ii) in Definition 2.1.4, u(x) = 0 a.e. in Ω. If |α| = 1. ϕ(x, |αu(x)|) = ϕ(x, |u(x)|) for any u ∈ M(Ω). Convexity follows immediately from the assumed convexity of ϕ. If λ → λ0 , u ∈ Lϕ (Ω) it is plain, on account of the monotonicity and continuity of ϕ and the monotone convergence theorem that Z Z ρϕ (λu) = ϕ(x, λ|u(x)|)dx → ϕ(x, λ0 |u(x)|)dx = ρϕ (λ0 u). Ω

Ω

Let the fact be noticed that the above reasoning holds even if one assumes ρϕ (λ0 u) = ∞.

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Definition 2.1.4. Let ϕ be an Musielak-Orlicz function on Ω. When the σalgebra A and the measure µ are clear from the context, which will be the case throughout this monograph, we set   Z   Lϕ (Ω) = u ∈ M : ϕ(x, λ|u(x)|) dx < ∞ for some λ > 0 .   Ω

It is easy to see that under the prescribed hypothesis Lϕ (Ω) is a vector space. When endowed with the Luxemburg norm (Proposition 1.3.2)    Z    |u(x)| kukϕ = inf λ > 0 : ϕ x, dx ≤ 1   λ Ω

ϕ

L (Ω) becomes a normed space (in fact, a Banach space if µ is σ-finite, as will be shown in Theorem 2.1.6), and will be called the Musielak-Orlicz space generated by ϕ (M O space for short, if there is no room for confusion). Lemma 2.1.3. Let ϕ be an Musielak-Orlicz function on the measure space (Ω, A, µ), (uk ) ⊆ M(Ω) and u ∈ M(Ω). Then: (i) If uk → u a.e. in Ω, then ρϕ (u) ≤ lim inf ρϕ (uk ). k→∞

(ii) If 0 ≤ uk % u a.e. in Ω, then ρϕ (u) = limk→∞ ρϕ (uk ). Proof. The proof relies on the fact that a.e. x ∈ Ω the function t −→ ϕ(x, t) is lower semicontinuous. For, if ε > 0, then by virtue of the monotonicity of ϕ, ϕ(t0 − ε) ≤ inf ϕ(x, t) x,t0 −ε 2−kλ . Then Z 1 Z 1 ∞ Z 2−n X −nλ φ(x, λ)dx ≥ φ(x, 2 )dx = φ(x, 2−kλ )dx 0

0

=

n=0

∞ X

2−(n+1)

2−(n+1) 22n−kλ = ∞.

n=0

It is apparent that 1(0,1) ∈ / Lφ ((0, 1)). If ϕ is locally integrable Musielak-Orlicz function it easily seen that Lϕ (Ω) contains the class S of simple functions. However, it is not hard to verify that in Example 2.1.1, Lφ (Ω) contains the class S in spite of φ not being locally integrable. Definition 2.1.5. Let (Ω, A, µ) be a measure space. The Musielak-Orlicz function ϕ (or the space Lϕ (Ω)) is said to be proper iff: (i) Lϕ (Ω) contains the class of simple functions. (ii) If F ∈ A and µ(F ) < ∞, then there exists a positive constant c(F ) such that for any u ∈ Lϕ (Ω) it holds that Z 1F (x)u(x)dµ ≤ c(F )kukϕ . Ω

Example 2.1.5. The Musielak-Orlicz space `pn already introduced in Section 1.4 is easily seen to be proper. Example 2.1.6. Let Ω ⊆ Rn , A the Borel σ-algebra of subsets of Ω, µ be the Lebesgue measure and p a measurable function on Ω with 1 ≤ p < ∞. Then it is easy to verify that the Musielak-Orlicz function ϕ : Ω × [0, ∞) −→ [0, ∞) ϕ(x, t) = t

(2.2)

p(x)

is locally integrable if p+ < ∞ but not if p+ = ∞. It follows that ϕ is proper PM in the former case and not proper in the latter, for if w = k=1 sk 1Ek is a simple function on Ω then, if p+ < ∞ it is clear that ρϕ (w) =

M Z X

p(x)

sk

dx < ∞,

k=1E k

so that the class of simple functions is contained in Lϕ (Ω).

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97

For the remaining part of this work, the notation ρp := ρϕ and Lp (Ω) := Lϕ (Ω) will be used in the particular case of ϕ given by (2.2). The corresponding norm will be denoted by k · kLp (Ω) or k · kp if there is no room for confusion or ambiguity. Invoking Theorem 1.5.3 and Axiom (v) in the definition of Banach function p space it is concluded that Lt (Ω) = Lp (Ω) is proper under the assumption p+ < ∞. It was shown in Theorem 1.5.3 that Lp (Ω) is a Banach function space. If the Musielak-Orlicz function ϕ is proper, the theorem extends to the more general case of Lϕ (Ω). More precisely: Theorem 2.1.4. Let (Ω, A, µ) be a measure space and ϕ be a proper MusielakOrlicz function on Ω. Then, when endowed with the Luxemburg norm k · kϕ , Lϕ (Ω) is a Banach function space. Proof. Axioms (i) − (v) of Definition 1.2.1 need to be checked. It is a straightforward matter to verify that the functional u −→ k · kϕ has the properties of a norm on M(Ω). Axiom R (i) follows simply by observing that if u ∈ Lϕ (Ω) then for some λ > 0 ϕ(x, λ|u(x)|) dµ = γ < ∞, then Ω

either γ ≤ 1 or Z ϕ(x, γ

−1

Z λ|u(x)|) dµ ≤ γ

Ω

ϕ(x, λ|u(x)|) dµ < ∞; Ω

from both cases one derives kukϕ < ∞. Axiom (ii) follows automatically from Definition 2.1.4. Properties (iv) and (v) are being assumed in the statement of the theorem. To prove property (iii) it suffices to consider the case kukϕ 6= 0. We observe first that if 0 ≤ uk % u pointwise a.e. in Ω, then the continuity and the monotonicity of ϕ yield     uk (x) u(x) ϕ x, dµ % ϕ x, a.e., kukϕ kukϕ from which we conclude   Z  Z  uk (x) u(x) ϕ x, dµ % ϕ x, dµ ≤ 1, kukϕ kukϕ Ω

Ω

by virtue of the monotone convergence theorem, Theorem 2.1.2 and Proposition 1.3.2, (iii). Hence kuk kϕ ≤ kukϕ .

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As a consequence of the monotonicity of ϕ, if ρϕ x, uk+1 ≤ 1 for λ > 0, then λ
also ρϕ x, uλk ≤ 1, which readily yields kuk kϕ ≤ kuk+1 kϕ . Now, given δ > 0 select λδ such that kukϕ − δ < λδ < kukϕ and  Z  u(x) ϕ x, > 1, λδ Ω

and since as k → ∞, Z

   Z  uk (x) u(x) ϕ x, % ϕ x, , λδ λδ

Ω

Ω

for sufficiently large k one has   Z  Z  uk (x) uk (x) ϕ x, > 1 ≥ ϕ x, , λδ kuk kϕ Ω

Ω

monotonicity yields kukϕ − δ < λδ < kuk kϕ ≤ kukϕ for k large enough. In all, kuk kϕ % kukϕ .

Corollary 2.1.5. Since every Banach function space is a Banach space by Theorem 1.2.3, the preceding result implies that the Musielak-Orlicz space Lϕ (Ω) is a Banach space if ϕ is a proper Musielak-Orlicz function. Next, we present the proof of the completeness of Lϕ (Ω) without the restriction imposed on ϕ in Corollary 2.1.5. Theorem 2.1.6. Let ϕ be as in Definition 2.1.4. If µ is σ-finite, Lϕ (Ω) is a Banach space. Proof. Let (Ωk ) be a sequence of pairwise disjoint, measurable sets, µ(Ωk ) < ∞ for any k ∈ N such that ∞ [ Ω= Ωk . k=1

Write Ak = A ∩ 2Ωk . We remark in passing that for every k ∈ N, (Ωk , Ak , µk ), where µk is the standard restriction of µ to Ak , is a measure space itself. Further, we observe

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99

the obvious fact that for any k ∈ N and an arbitrary λ > 0, µk is absolutely continuous with respect to the measure ηkλ defined on Ak by Z ηkλ (Y ) = ϕ(x, r) dµk . Y

Accordingly, since µk (Ωk ) < ∞, for any  > 0 there exists δ > 0 such that for any A ∈ Ak one has ηkλ (A) < δ ⇒ µk (A) < . (2.3) Fix λ > 0 and for  > 0, let δ : 0 < δ < 1 be selected to satisfy (2.3) for k = 1 and r = λ and consider a (norm) Cauchy sequence (uj ) ⊂ Lϕ (Ω). Fix N > 0 such that kun − um kϕ < δ holds for m, n ≥ N , let λ be an arbitrary positive number and for fixed m, n ≥ N set Yλ = {x ∈ Ω1 : |un (x) − um (x)| ≥ λ} . Then η1λ (Yλ ) =

Z ϕ(x, λ) dµ1 Yλ

Z ≤

ϕ(x, |un (x) − um (x)|) dµ1 ≤ ρϕ (|un − um |)   kun − um kϕ = ρϕ |un − um | ≤ kun − um kϕ < δ. kun − um kϕ Yλ

Consequently µ1 (Yλ ) <  and the sequence of restrictions (un |Ω1 ) is Cauchy in measure, hence µ1 -convergent in measure to an A1 -measurable function v1 (1) defined in Ω1 . Thus, there exists a subsequence (unj )j that converges µ1 −a.e. to v1 . Assume now that for 1 < k ∈ N an Ai -measurable function vi defined on Ωi is given for each 1 ≤ i ≤ k and a subsequence (u(k) nj ) j of the original sequence (un ) has been found such that (u(k) nj (x))j →

k X

vi (x)1Ωi (x)

µ − a.e. in

i=1

k [

Ωi .

k=1

Almost verbatim, the preceding argument implies the existence of a Ak+1 (k+1) measurable function uk+1 and a subsequence (unj )j of (un ) that converges Pk+1 µ-a.e. to i=1 vi 1Ωi . It is obvious that the function u defined on Ω as u=

∞ X k=1

vk (x)1Ωk

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Analysis on Function Spaces of Musielak-Orlicz Type

is A- measurable, that the original sequence (un ) converges to u in measure and that (uknk ) converges to u in measure in Ω. For arbitrary λ > 0, 0 <  < 1, let N0 be chosen so large that k, j ≥ N0 (j) implies kλ(uk − unj )kϕ < . Then Z ρϕ (λ(uk − u)) = ϕ(x, λ|uk (x) − u(x)|) dµ Ω

Z ≤

lim ϕ(x, λ|uk (x) − u(j) nj (x)|) dµ

j→∞ Ω

Z ≤ lim inf j→∞

ϕ(x, λ|uk (x) − u(j) nj (x)|) dµ

Ω

≤ lim inf ρϕ (λ(uk − u(j) nj )) j→∞

(j)

= kλ(uk −

u(j) nj )kϕ ρϕ

λ(uk − unj )

!

(j)

kλ(uk − unj )kϕ

= kλ(uk − u(j) nj )kϕ < . By virtue of Theorem 1.3.8 the arbitrariness of λ yields uk → u in Lϕ (Ω) as k → ∞.

Theorem 2.1.6 can be extended to the case in which the Musielak-Orlicz function ϕ satisfies the condition ϕ(x, t) ≥ 0 a.e. in Ω, see [24]. Other variants of the theme are possible, such as the consideration of a different norm [82]. We emphasize the following corollary to Theorem 2.1.6: Corollary 2.1.7. If (un ) ⊂ Lϕ (Ω) and Lϕ (Ω)

un −→ u, then there exists a subsequence of (un ) that converges to u µ − a.e. in Ω. Corollary 2.1.8. If (Ω, A, µ) is σ-finite, in particular, if Ω ⊆ Rn is a Borel set and the underlying measure is the Lebesgue’s measure, then, for 1 ≤ p− ≤ p+ < ∞, Lp (Ω) is a Banach space.

Musielak-Orlicz spaces

2.2

101

Embeddings between Musielak-Orlicz spaces

In this section we shall discuss necessary and sufficient conditions on the Musielak-Orlicz functions ϕ and ξ for the validity of the embedding Lϕ ⊂ Lξ . The main result in this section is Theorem 2.2.1, which was proved in [82]. We follow the proof in [24]. Definition 2.2.1. Let ϕ and ξ be Musielak-Orlicz functions on the same measure space (Ω, A, µ). We use the notation ϕ-ξ if there exists a positive constant B and 0 ≤ h ∈ L1 (Ω) such that ϕ(x, t) ≤ ξ(x, Bt) + h(t)

(2.4)

for all t ∈ [0, ∞) and a.e x ∈ Ω. Theorem 2.2.1. Let ϕ and ξ be Musielak-Orlicz functions on the measure space (Ω, M, µ). Assume that µ is non-atomic. Then the embedding Eξϕ : Lξ (Ω) ,→ Lϕ (Ω)

(2.5)

is continuous if and only if ϕ - ξ. Proof. Assume first that ϕ - ξ; the set-theoretical inclusion Lξ (Ω) ⊆ Lϕ (Ω) is obvious and it is also clear from (2.4) that if λ > 0 and ρξ (λ−1 u) ≤ 1 then ρϕ (λ−1 B −1 u) ≤ 1 + khk1 . Convexity quickly yields
ρϕ (λ(1 + khk1 )B)−1 u ≤ 1, whence kukϕ ≤ (1+khk1 )Bkukξ . This settles the sufficiency of condition (2.4). For the necessity of condition (2.4), let Eξ,ϕ be continuous and set ( ϕ(x, t) − ξ(x, kEξ,ϕ kt) if ξ(x, t) < ∞ α(x, t) = ∞ otherwise. Set Q ∩ [0, ∞) = (rk )k∈N with r1 = 0 and hk (x) := max α(x, rj ). 1≤j≤k

Clearly, each hk is measurable and for each x ∈ Ω, hk (x) ≥ α(x, r1 ) = α(x, 0) = 0.

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Analysis on Function Spaces of Musielak-Orlicz Type

Moreover, for x ∈ Ω the sequence (hk (x))k∈N is nondecreasing. Since ξ and ϕ are Musielak-Orliczs-functions, α is left-continuous, from which it is easily derived that h(x) = sup hk (x) = sup α(x, t) t≥0

k∈N

and that a.e. x ∈ Ω it holds the inequality ϕ(x, t) ≤ ξ(x, kEξ,ϕ kt) + h(x). Next, we set about to prove that h ∈ L1 (Ω) and that khkL1 (Ω) ≤ 1. We first show that |{x ∈ Ω : h(x) = ∞}| = 0. A contradiction will be reached by assuming the contrary. If |I| = |{x : h(x) = ∞}| > 0, we set 

2 x ∈ I : α(x, rk ) ≥ |I|   k [ = Vk+1 \  Vj  .



Vk : = and Wk+1

j=1

Clearly I=

∞ [

Wk =

k=2

Let f :=

∞ [

Vk .

k=1 ∞ X

rk 1Wk .

k=2

Notice that the condition ξ(x, |f (x)|) < ∞ holds a.e. x ∈ I. Indeed, ξ(x, |f (x)|) = ξ(x, rk ) for some k ∈ N and α(x, rk ) > 0, which by definition of α implies ξ(x, |f (x)|) < ∞. ∞ S Next, we observe that for x ∈ I = Wk k=2

∞ = h(x) = sup α(x, t), t≥0

from which one immediately derives the existence of k ∈ N for which α(x, rk ) >

2 . |I|

Musielak-Orlicz spaces

103

It follows straight from the definition of α and of Wk that for any x ∈ I, ϕ(x, |f (x)|) ≥ ξ (x, |f (x)|kEξ,ϕ k) +

2 . |I|

In all, one concludes Z Z ϕ(x, |f (x)|) dx ≥ ξ (x, |f (x)|kEξ,ϕ k) dx + 2. I

(2.6)

(2.7)

I

On the other hand the assumed continuity of the embedding Eξ,ϕ and Proposition 1.3.2 yield Z ξ(x, |f (x)|kEξ,ϕ k) dx ≤ 1 ⇔ kf kEξ,ϕ kkLξ (Ω) ≤ 1. Ω

On account of the assumed boundedness of the embedding (2.5) the preceding statement yields kf kLϕ (Ω) ≤ 1. R Therefore, ϕ(x, |f (x)|) dx ≤ 1, which together with (2.7) allows us to conΩ

clude that

Z ξ(x, |f (x)|kEξ,ϕ k) dx > 1. Ω

Since ξ is finite a.e. and Lebesgue’s measure is non-atomic, there exists a measurable set M ⊆ I with Z ξ(x, kEξ,ϕ kf (x)1M (x)) dx = 1. I

In particular, the last inequality implies that Z ϕ(x, |f (x)|1M (x)) dx ≤ 1.

(2.8)

Ω

Notice that |M | > 0. Integrating inequality (2.6) on M Z |M | ϕ(x, |f (x)|1M (x)) dx ≥ 1 + 2 > 1, |I| I which contradicts (2.8). Hence h is a.e. finite. Next we prove that in fact, khk1 ≤ 1. Suppose, otherwise that is, assume that for some  one has Z h(z) dz >  > 1. Ω

(2.9)

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Analysis on Function Spaces of Musielak-Orlicz Type

In this case we set 

 2 x ∈ I : α(x, rk ) ≥ h(x) 1+   k [ = Vk+1 \  Vj  .

Vk : = Wk+1

j=1

Let f be as in the preceding argument. We claim that the inequality ξ(x, |f (x)|) < ∞ ∞ S

holds a.e. x ∈ Ω. Indeed, for x ∈ Ω \

Wk , one clearly has f (x) = 0 and it

k=2

follows at once that ξ(x, |f (x)|) = 0. Otherwise, for any k ∈ N, k > 1 one has α(x, rk ) > 0, thus, ξ(x, |f (x)|) = ξ(x, rk ) < ∞, as claimed. By definition of α and Wk , it holds the inequality ϕ(x, |f (x)|) ≥ ξ (y, |f (x)|kEξ,ϕ k) +

2 h(x). 1+

(2.10)

Indeed, (2.10) is obvious if x ∈ Wk for k ≥ 2; on the other hand if x ∈ ∞ S Ω\ Wk then clearly k=2

α(x, rk ) ≤

2 h(x), 1+

which automatically extends to α(x, t) ≤

2 2 h(x) = sup α(x, s) 1+ 1 +  s≥0

for any t ∈ [0, ∞). In turn, the latter is only possible if h(x) = 0. Hence (2.10) holds trivially. Integrating inequality (2.10) in Ω it follows at once that Z Z Z 2 ϕ(x, |f (x)|) dµ ≥ ξ (x, |f (x)|kEξ,ϕ k) dµ + h(x) dµ 1+ Ω Ω Ω Z 2 2 ≥ h(x) dµ ≥ > 1. (2.11) 1+ 1+ Ω

Inequality (2.11) together with the continuity of Eξ,ϕ forces Z ξ(x, |f (x)|kEξ,ϕ k) dµ > 1, Ω

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105

for otherwise it would be concluded from the continuity of kEkξ,ϕ that kf kLϕ (Ω) ≤ 1, a contradiction to (2.11) by virtue of Proposition 1.3.2. As in the preceding argument, it is concluded that there exists a measurable set M ⊂ Ω for which Z ξ(x, kEξ,ϕ k|f (x)|1M (x)) dµ = 1. (2.12) Ω

Invoking again Proposition 1.3.2 and the continuity of Eξ,ϕ , the last equality yields Z ϕ(x, |f (x)|) dµ ≤ 1. (2.13) M

Notice that M

\

{z : f (z) > 0} = M

∞ \[

Wk = M

\ {z : h(z) > 0},

k=2

which in concert with equality (2.12) yields \ M {z : h(z) > 0} > 0 and thus

Z h(z) dµ > 0.

(2.14)

M

In all, assumption (2.9) yields inequalities (2.10) and (2.14). Integrating inequality (2.10) on M : Z Z Z 2 ϕ(x, |f (x)|) dµ ≥ ξ (x, |f (x)|kEξ,ϕ k) dµ + h(x) dµ 1+ M M M Z 2 ≥1+ h(x) dµ > 1, 1+ M

which contradicts (2.13). This contradiction stems from assumption (2.9). Hence Z h(x) dx ≤ 1, Ω

as claimed. The proof of Theorem 2.2.1 is thus complete.

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Example 2.2.1. To illustrate the preceding results, we consider the case of embeddings among the variable exponent Lebesgue spaces. It is well known that when Ω has finite measure, the classical Lebesgue spaces are ordered: if p < q, then Lq (Ω) ,→ Lp (Ω). That the same holds for spaces with variable exponent was first shown by Kov´a˘cik and R´akosn´ık in Theorem 2.8 of [60]. Their argument is given below. Recall that in this example we fix a domain Ω ⊆ Rn and consider the measure space (Ω, B, µ), where B is the Borel σalgebra of subsets of Ω and µ stands for the Lebesgue measure. We continue to use the notation of Section 1.5. Theorem 2.2.2. Let |Ω| < ∞ and suppose that p, q ∈ P(Ω). Then Lq (Ω) is continuously embedded in Lp (Ω) if and only if p(x) ≤ q(x) a.e. in Ω. When the embedding id exists, kidk ≤ 1 + |Ω| . Proof. First suppose that p(x) ≤ q(x) a.e. in Ω. Note that Ωp∞ ⊂ Ωq∞ . Let u belong to the closed unit ball of Lq (Ω). Then by Proposition 1.5.2, Z q ρq (u) = |u| dx + ess supΩq∞ |u| ≤ 1. Ω\Ωq∞

Thus |u(x)| ≤ 1 a.e. in Ωq∞ . Hence ρp (u) ≤ |{x ∈ Ω\Ωq∞ : |u(x)| ≤ 1}| Z q + |u| dx + |Ωq∞ \Ωp∞ | + ess supΩq∞ |u(x)| Ω\Ωq∞

≤ |Ω| + ρq (u) ≤ |Ω| + 1. As ρp is convex, ρp (u/ (|Ω| + 1)) ≤ (|Ω| + 1)

−1

ρp (u) ≤ 1,

from which we see that kidk ≤ 1 + |Ω| . Now suppose that it is not the case that p(x) ≤ q(x) a.e. in Ω. Then there is a subset Ω∗ of Ω, with positive measure, such that p(x) > q(x) for all x ∈ Ω∗ . First assume that |Ωp∞ ∩ Ω∗ | > 0. Then there exist A ⊂ Ωp∞ ∩ Ω∗ , with 0 < |A| < ∞, and r ∈ (1, ∞) such that 1 ≤ q(x) ≤ r < ∞ = p(x) for all x ∈ A. There are pairwise disjoint sets Ak such that A = ∪∞ k=1 Ak and ∞ P −k r |Ak | = 2 |A| for all k ∈ N; define u = (3/2) 1Ak . For this function u we k=1

have kukp ≥ ku1A k∞ = ∞, so that u ∈ / Lp (Ω). However, Z ∞ Z X q ρq (u) = |u| dx = (3/2)kq/r dx

≤

A ∞ X k=1

k=1A k

(3/2)k |Ak | = |A|

∞ X k=1

(3/4)k < ∞,

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107

and hence u ∈ Lq (Ω). On the other hand, if |Ωp∞ ∩ Ω∗ | = 0, then 1 ≤ q(x) < p(x) < ∞ for a.e. x ∈ Ω∗ and there is a set A ⊂ Ω∗ , with 0 < |A| < ∞, and numbers a > 0, r ∈ (1, ∞) such that q(x) + a ≤ p(x) ≤ r whenever x ∈ A. Choose sets ∞ P Ak as before and define v(x) = (2k k −2 )1/q(x) (x ∈ Ω). Then k=1

ρq (v) =

∞ X

2k k −2 |Ak | = |A|

k=1

∞ X

k −2 < ∞,

k=1

q

which shows that v ∈ L (Ω). But since for all λ ∈ (0, 1], ∞ Z ∞ X X ρp (λv) ≥ λr (2k k −2 )p/q dx ≥ λr (2k k −2 )1+a/r |Ak | k=1A k ∞ X

= λr |A|

k=1

2ak/r k −2(1+a/r) = ∞,

k=1

we see that v ∈ / Lp (Ω).

Corollary 2.2.3. Suppose that p ∈ P(Ω) and let (uk ) be a sequence in Lp (Ω) that converges to u in Lp (Ω). Then there is a subsequence of (uk ) that converges pointwise a.e. in Ω to u. Proof. First suppose that |Ω| < ∞. By Theorem 2.2.2, Lp (Ω) ,→ L1 (Ω) and so ku − uk k1 → 0. As the result is true for L1 (Ω), there is nothing more to do. If |Ω| = ∞, apply what has already been proved to Lp (Ω ∩ B(0, m)) (m ∈ N) and use diagonalization. Corollary 2.2.4. Let p ∈ P(Ω) be such that p+ < ∞ and let u, uk ∈ Lp (Ω) (k ∈ N). Then uk → u in Lp (Ω) if and only if (uk ) converges to u in measure on Ω and ρp (uk ) → ρp (u). Proof. Suppose that uk → u in Lp (Ω). By Proposition 1.5.7, (uk ) converges to u in measure on
Ω and ρp (uk − u) → 0. By Corollary 2.2.3, there is a subsequence ul(k) k∈N of (uk ) that converges pointwise a.e. on Ω to u. The inequality   p(x) p(x) p(x) |uk (x)| ≤ 2p+ −1 |uk (x) − u(x)| + |u(x)| shows that given ε > 0, there exist δ > 0 and k0 ∈ N such that for all B ⊂ Ω with |B| < δ and all k ≥ k0 , Z p |uk | dx ≤ ε. B

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Analysis on Function Spaces of Musielak-Orlicz Type

Hence by Vitali’s theorem (see [63, 2.1.4]), lim ρp (ul(k) ) = ρp (u).

k→∞

In fact, limk→∞ ρp (uk ) = ρp (u); for if not, there would be a subsequence of (ρp (uk )) whose elements would be uniformly bounded away from ρp (u), and an application of the preceding argument to this subsequence would give a contradiction. For the converse, suppose that (uk ) converges to f in measure on Ω and ρp (uk ) → ρp (u). Now use the inequality   p(x) p(x) p(x) |uk (x) − u(x)| ≤ 2p+ −1 |uk (x)| + |u(x)| , and argue as in the first part to obtain ρp (uk −u) → 0 and hence kuk − ukp → 0.

We now show (following [34]) that for embedding maps, greater precision in norm estimation can be obtained when p and q are near to one another in the sense that for some ε ∈ (0, 1), p(x) ≤ q(x) ≤ p(x) + ε a.e. in Ω.

(2.15)

Lemma 2.2.5. Suppose that |Ω| < ∞ and p, q satisfy (2.15). If u ∈ M(Ω) and ρq (u) ≤ 1, then ρp (u) ≤ ε |Ω| + ε−ε . Proof. Put Ω1 = {x ∈ Ω : |u(x)| < ε} , Ω2 = {x ∈ Ω : ε ≤ |u(x)| ≤ 1} , Ω3 = {x ∈ Ω : 1 < |u(x)|} . Then ρp (u) =

3 Z X j=1

p

|u| dx =

3 X

Aj .

j=1

Ωj

It is clear that Z A1 ≤ Ω1

εp dx ≤

Z

Z εdx ≤ ε |Ω| , A3 ≤

Ω1

Ω3

On Ω2 , q(x)−p(x)

εε ≤ εq(x)−p(x) ≤ |u(x)| and so 1 ≤ |u(x)|

p(x)−q(x)

≤ ε−ε .

q

|u| dx.

≤ 1,

Musielak-Orlicz spaces Hence

Z

q

p−q

|u| |u(x)|

A2 =

109 Z

dx ≤ ε−ε

Ω2

q

|u| dx. Ω2

Thus ρp (u) ≤ ε |Ω| + ε−ε

Z

Z

q

q

|u| dx + Ω2

|u| dx Ω3

  Z Z q q ≤ ε |Ω| + ε−ε  |u| dx + |u| dx Ω2

≤ ε |Ω| + ε−ε

Z

Ω3 q

|u| dx ≤ ε |Ω| + ε−ε .

Ω

Lemma 2.2.6. Suppose that |Ω| < ∞ and p, q satisfy (2.15). Then kidkLq →Lp ≤ ε |Ω| + ε−ε . Proof. Clearly K := ε |Ω| + ε−ε > 1. For each u such that ρq (u) ≤ 1 we have, by Lemma 2.2.5,
ρp (u/K) ≤ K −1 ρp (u) ≤ ε |Ω| + ε−ε /K = 1, and the result follows. The required estimate from below is established in two stages. Lemma 2.2.7. Suppose that 1 ≤ |Ω| < ∞ and p, q satisfy (2.15). Then kidkLq →Lp ≥ 1. Proof. Let v(x) = |Ω|

−1/q(x)

(x ∈ Ω). Then ρq (v) = 1. Since |Ω|

−p(x)/q(x)

−1

≥ |Ω|

we have, for each λ ∈ (0, 1), Z ρp (v/λ) = Ω

−p/q

|Ω| λp

Z dx ≥

−1

|Ω| dx ≥ λp

Ω

Z

−1

|Ω| λ

dx = λ−1 > 1.

Ω

Hence kidk ≥ λ for each λ ∈ (0, 1), which shows that kidk ≥ 1. Lemma 2.2.8. Suppose that 0 < |Ω| < 1 and p, q satisfy (2.15). Then ε

kidkLq →Lp ≥ |Ω| .

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Analysis on Function Spaces of Musielak-Orlicz Type

Proof. Let v be as in the last lemma. Since 1−p(x)/q(x)

|Ω|

ε/q(x)

≥ |Ω|

ε

≥ |Ω| ,

we see that Z

−p/q

|Ω|

ρp (v) =

dx = |Ω|

−1

Ω

Z

1−p/q

|Ω|

ε

dx ≥ |Ω| .

Ω ε

Thus for each positive λ < |Ω| , Z Z v p ρp (v/λ) > ε dx = |Ω| Ω

Ω −ε

Z

v p ε/p dx |Ω| −ε

p

|v| dx ≥ |Ω|

= |Ω|

ε

|Ω| = 1.

Ω ε

Hence kidk ≥ λ for each positive λ < |Ω| , and the result follows. From these lemmas we immediately have the following theorem and corollary. Theorem 2.2.9. Suppose that 0 < |Ω| < ∞ and p, q satisfy (2.15). Then the norm of the embedding id of Lq (Ω) in Lp (Ω) satisfies ε

min (1, |Ω| ) ≤ kidkLq →Lp ≤ ε |Ω| + εε . Corollary 2.2.10. Suppose that 0 < |Ω| < ∞ and p ∈ P(Ω) is such that 1 < p− ≤ p+ < ∞; suppose that for each k ∈ N, qk ∈ P(Ω) and εk > 0 are such that limk→∞ εk = 0 and p(x) ≤ qk (x) ≤ p(x) + εk (x ∈ Ω, k ∈ N). Let idk be the natural embedding of Lqk (Ω) in Lp (Ω). Then lim kidk kLqk →Lp k = 1.

k→∞

2.2.1

The ∆2 -condition

It is evident from the theory presented so far that the ∆2 condition plays an essential role in the treatment of modular spaces. In this section a more detailed analysis is undertaken. Definition 2.2.2. An Musielak-Orlicz function ϕ is said to fulfill the ∆2 condition if there exist constants C > 1 and S0 > 0 such that, uniformly in Ω, the following inequality holds: ϕ(x, 2s) ≤ Cϕ(x, s) for all s ≥ S0 , x ∈ Ω.

(2.16)

Musielak-Orlicz spaces

111

Lemma 2.2.11. Let ϕ be a Musielak-Orlicz function. If there exists a natural number K > 1 and constants C > 1 and S0 > 0 such that, uniformly in Ω, it holds that ϕ(x, Ks) ≤ Cϕ(x, s) for all s ≥ S0 , x ∈ Ω, (2.17) then ϕ satisfies the ∆2 condition. Proof. If K > 1, then K j > 2 for some j ∈ N. Inequality (2.17) implies thus that uniformly in Ω one has ϕ(x, 2s) ≤ ϕ(x, K j s) ≤ C j ϕ(x, s) for all s ≥ S0 .

Example 2.2.2. Given a measure space (Ω, A, µ) and 1 ≤ p < ∞ the Lebesgue space Lp (Ω) is the Musielak-Orlicz space arising from the consideration of the Musielak-Orlicz function ϕ : Ω × [0, ∞) −→ [0, ∞) ϕ(t) = tp . It is routine to verify the validity of the ∆2 condition for Lp (Ω). On the other hand, setting ϕ : Ω × [0, ∞) −→ [0, ∞) ( t if 0≤t≤1 ϕ(t) = et−1 if t>1 a simple calculation reveals that ϕ does not satisfy the ∆2 condition. We emphasize the fact that if ϕ is a Musielak-Orlicz function on a measure space (Ω, A, µ) and ϕ satisfies the ∆2 condition, then   Z   u ∈ M(Ω) : ϕ(x, |u(x)|)dx < ∞ = Lϕ (Ω). (2.18)   Ω

Proposition 2.2.12. Let ϕ be a Musielak-Orlicz function on a measure space (Ω, A, µ) and assume that ϕ satisfies the ∆2 condition (2.16). Then, there exist constants α = α(ϕ, C) ≥ 0 and β = β(ϕ, C) ≥ 0 such that for 0 ≤ t ≤ 1 and all s ≥ S0 , C −1 tα ϕ(x, s) ≤ ϕ(x, ts) ≤ tϕ(x, s) and that for 1 < t and all s ≥ S0 , tϕ(x, s) ≤ ϕ(x, ts) ≤ Ctβ ϕ(x, s).

(2.19)

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Analysis on Function Spaces of Musielak-Orlicz Type

Proof. For any s ≥ S0 , x ∈ Ω and 0 < t < 1, let 2n ≤ 1t < 2n+1 , n ∈ N0 .   − ln t 1 ϕ(x, s) = ϕ x, t s ≤ C n+1 ϕ (x, ts) ≤ CC ln C ϕ(x, ts) t   lnlnC2 1 ≤C ϕ(x, ts); t the right-hand inequality follows from convexity. The left-hand side inequality for t ≥ 1 follows from the convexity assumption on ϕ; the remaining inequality is proved as the one above: ln t

ln t

ϕ(x, ts) ≤ ϕ(x, 2[ ln K ]+1 s) ≤ CC [ ln K ] ϕ(x, s) ln C

≤ Ct ln 2 ϕ(x, s).

Lemma 2.2.13. If ϕ satisfies the ∆2 condition, then the modular ρϕ given in (2.1) is continuous. Proof. Fix u ∈ Lϕ (Ω), hence, for some γ > 0 it holds Z ϕ(x, γ|u(x)|) dx < ∞. Ω

If (λj ) ⊂ (0, ∞) converges to λ, the monotonicity of ϕ yields   3 ϕ(x, λj |u(x)|) ≤ ϕ x, λ|u(x)| ≤ ϕ(x, γ|u(x)|) 2 if 32 λ ≤ γ. The ∆2 condition comes into play when observing that for 32 λ > γ, inequality (2.19) implies that for |u(x)| ≥ S0 one has      α 3 3 3λ ϕ x, λ|u(x)| = ϕ x, λγ|u(x)| ≤ C ϕ(x, γ|u(x)|). 2 2γ 2γ A straightforward application of Lebesgue’s dominated convergence theorem now yields Z Z ρϕ (λj u) = ϕ(x, λj |u(x)|) dx + ϕ(x, λj |u(x)|) dx |u|≤S0

−→ ρϕ (λu) as j −→ ∞ i.e., the continuity claim.

|u|≥S0

Musielak-Orlicz spaces

113

Lemma 2.2.14. If ϕ is an Musielak-Orlicz function satisfying the ∆2 condition, then for any 0 6= u ∈ Lϕ (Ω), one has  Z  |u(x)| ϕ x, dx = 1. kukϕ Ω

Proof. Take a sequence (λj ) ⊂ (0, ∞) convergent to kukϕ with  Z  |u(x)| ϕ x, ≤ 1. λj Ω

By virtue of Lemma 2.2.13, one must necessarily have  Z  |u(x)| ϕ x, dx ≤ 1, kukϕ

(2.20)

Ω

with the same token it is apparent that the assumption of strict inequality in (2.20) leads to a contradiction.

2.2.2

Absolute continuity of the norm

Recall from Section 1.2.2, if X be a Banach function space, two subspaces of X are to be distinguished, namely: (i) The set of all elements of X with absolutely, continuous norm, denoted by Xa ; (ii) The closure of the set of all bounded functions supported in sets of finite measure, denoted by Xb . In the next theorem we characterize the space Lϕ a (Ω). Theorem 2.2.15. Let ϕ be a Musielak-Orlicz function on the σ-finite measure space (Ω, A, µ). Then ϕ Lϕ a (Ω) = {u ∈ L (Ω) : ρϕ (λu) < ∞ for all λ > 0}

(2.21)

Proof. Let (Ωk ) be a sequence of mutually disjoint open sets with µ(Ωk ) < ∞, whose union is Ω. If u ∈ Lϕ (Ω) and ρϕ (λu) = ∞ for λ > 0 then ∞ X

ρϕ (u1Ωk ) =

k=1

∞ Z X

ϕ(x, λ|u(x)|) dµ = ∞.

k=1Ω k

Let N1 ∈ N satisfy N1 X n=1

ρϕ (λu1Ωk ) > 1;

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Analysis on Function Spaces of Musielak-Orlicz Type

there exists then N2 > N1 with N2 X

ρϕ (λu1Ωk ) > 1;

n=N1 +1

in this fashion one obtains a strictly increasing sequence (Nk ) ⊂ N0 satisfying N0 = 0 and Nk+1 X ρϕ (λu1Ωk ) > 1 n=Nk +1

˜k = for k ≥ 0. Set Ω

NS k+1

˜ k) Ωj for k > 1; it is clear that the sequence (Ω

j=Nk +1

consists of mutually disjoint sets and that 1Ω˜ k → ∅ as k → ∞. By construction
ρϕ λu1Ω˜ k > 1 and it follows straight from the definition of the Luxemburg norm that ku1Ω˜ k kϕ > λ−1 , whence it is clear that u ∈ / Lϕ a (Ω). Thus, ϕ Lϕ a (Ω) ⊆ {u ∈ L (Ω) : ρϕ (λu) < ∞ for all λ > 0} .

Conversely, for any u in the right-hand side set of (2.21), any sequence (Ek ) of measurable sets such that 1Ek → ∅ and any ε > 0, one has
lim ρϕ ε−1 u1Ek = 0 k→∞

on account of the dominated convergence theorem; it is readily derived from here that ku1Ek kϕ −→ 0 as k → ∞. Corollary 2.2.16. If a Musielak-Orlicz function ϕ on the σ-finite measure space (Ω, A, µ) satisfies the ∆2 condition, then ϕ ϕ Lϕ a (Ω) = L (Ω) = Lb (Ω).

Conversely, if ϕ is an Musielak-Orlicz function on (Ω, A, µ) and Lϕ a (Ω) = Lϕ (Ω), then there exist C > 0 and 0 ≤ h ∈ L1 (Ω) such that for each t > 0: ϕ(x, 2t) ≤ Cϕ(x, t) + h(t) a.e. x ∈ Ω.

(2.22)

Proof. The first assertion follows immediately from the fact that the ∆2 condition for ϕ implies that ρϕ (λu) < ∞ for any u ∈ Lϕ (Ω) and any λ > 0, from Theorem 2.2.15 and Lemma 1.2.16. For the remaining claim, observe that setting ξ(x, t) = ϕ(x, 2t), the assumption and Theorem 2.2.15 yield the continuity of the embedding E : Lϕ (Ω) ,→ Lξ (Ω) and on account of Theorem 2.2.1 the latter implies (2.22).

Musielak-Orlicz spaces

2.3

115

Separability

The study of separability of Musielak-Orlicz spaces is tied to both the structure of the underlying measure and of the Musielak-Orlicz function. Throughout this section we will consider the measure µ to be separable and the Musielak-Orlicz function ϕ on the measure space (X, A, µ) to be locally integrable, that is (Definition 2.1.2) for any t ≥ 0 and any W ∈ A with µ(W ) < ∞, it holds that Z ϕ(x, t) dµ < ∞.

(2.23)

W

Remark We observe the useful fact that if ϕ is locally integrable and (Bk ) is a sequence of measurable sets for which µ(Bk ) → 0 as k → ∞, then necessarily, for any λ > 0, it holds that Z ρϕ (λ1Bk ) = ϕ(x, λ) dµ → 0 as k → ∞. Bk

For otherwise, there would exist δ > 0 and a sequence (Bk ) ⊂ A with µ(Bk ) < 2−k and ρϕ (λ1Bk ) > δ. Then if Dj =

S

Bk for j ∈ N, µ(D1 ) = 1 and pointwise,

k≥j

1D1 ≥ 1Dj → 0 as j → ∞. On account of Lebesgue’s theorem the latter would imply Z Z ϕ(x, λ) dµ ≤ ϕ(x, λ) dµ → 0 as j → ∞, Bj

Dj

a contradiction. Next, we recall the standard notation A 4 B = (A \ B)

[

(B \ A)

to denote the symmetric difference between the sets A and B. Definition 2.3.1. The measure space (X, A, µ) is said to be separable if there exists a sequence (Ek ) ⊂ A, with µ(Ek ) < ∞ for all k ∈ N, such that for any E ∈ A with µ(E) < ∞ and any  > 0, there exists k0 ∈ N with µ(E 4 Ek0 ) < .

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Analysis on Function Spaces of Musielak-Orlicz Type

If Ω ⊆ Rn is Lebesgue measurable, then the (restriction of the) Lebesgue measure to Ω is separable. Recall that S0 stands for the class of simple functions of the type w=

J X

wk 1Ak with µ(Ak ) < ∞ and wk 6= wj for k 6= j.

k=0

Lemma 2.3.1. Given a separable measure space (Ω, A, µ) as in Definition 2.3.1 and a locally integrable Musielak-Orlicz function ϕ on Ω, the (countable) set C ⊂ S0 consisting of simple functions of the type s(t) =

M X

sk 1Ek , sk ∈ Q, 1 ≤ k ≤ M

k=0

is k · kϕ -dense in the class S0 . PJ Proof. Let ε > 0 and u = k=1 uk 1Ak ∈ S0 and set max{s1 , ..., sJ } := 2M . On account of the separability of µ, the local integrability of ϕ and the Remark following Definition 2.23 there is a finite collection (Fk )Jk=1 of the sequence (En ) given in Definition 2.3.1 such that for 1 ≤ k ≤ J one has the inequality Z ϕ(x, 4λJM ) dµ < ε/2. Fk 4Ak

Invoking again the local integrability of ϕ it is immediate that if λ > 0 is J S arbitrary and A = Ai , then pointwise in Ω, i=1

L1 (Ω, µ) 3 ϕ(x, 1A (x)) ≥ ϕ(x, 2λr1A (x)) → 0 as r → 0. By virtue of Lebesgue’s theorem there exists δ > 0 such that Z ϕ(x, 2λδ1A (x)) < ε/2. Ω

Then one can select rational number r1 , ..., rJ with |rj −uj | < δ and |rj | ≤ 2M for j = 1, 2, ..., J. Let J X v= rk 1Fk . j=1

Clearly X X J J X J |u − v| = (uj − rj )1Aj + |rj ||1Aj \Fj | ≤ δ1A + 2M |1Aj \Fj |. j=1 j=1 j=1

Musielak-Orlicz spaces

117

In all, for any λ > 0  1 ρϕ (λ(u − v)) = 2

Z

1 ϕ(x, 2λδ) dµ + 2

A

≤

≤

Z ϕ x, 4M λ Ω

1 ε + 2 2J

J Z X

J X

 |1Aj \Fj | dµ

j=1


ϕ x, 4M Jλ|1Aj \Fj | dµ

j=1 Ω

J 1 1 Xε + = ε. 2 2J j=1 2

The arbitrariness of λ implies the claim. Theorem 2.3.2. Let (X, A, µ) be a separable measure space and ϕ : Ω × [0, ∞) −→ [0, ∞) ϕ be locally integrable. If Lϕ (Ω) = Lϕ a (Ω), then L (Ω) is separable.

Proof. We notice first that as an immediate consequence of condition (2.23), the class S of simple functions of the type w=

J X

wk 1Ek , µ(Ek ) < ∞

k=0

is contained in Lϕ (Ω). It is well known that for an A-measurable function u ≥ 0 in Ω there exists a non-decreasing sequence of simple non-negative functions (un ) which converges pointwise to u in Ω. In particular, if u ∈ M(Ω), n∈N Z Z ϕ(x, 2n ) dµ ≤ ϕ(x, |u(x)|) dµ {x:|u(x)|≥2n }

Ω

the sequence can be chosen to be of the type S. Select two nondecreasing sequences in S, (an ) and (bn ) converging pointwise to u+ = max{u, 0} and u− = max{−u, 0}. Hence, pointwise un = an − bn → u+ − u− = u as n → ∞. In particular, if u ∈ Lϕ (Ω), for any x ∈ Ω ϕ(x, λ(un (x) − u(x))) ≤ ϕ(x, λ(an (x) − u+ (x) − bn (x) + u− (x))) ≤ ϕ(x, λ(2(u+ (x) + u− (x))) ≤ ϕ(x, 2λ|u(x)|) < ∞,

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Analysis on Function Spaces of Musielak-Orlicz Type

on account of Theorem 2.2.15. Thus, for any λ > 0 and by direct application of Lebesgue’s dominated convergence theorem it follows that Z ρϕ (λ(un − u)) = ϕ(x, λ(un (x) − u(x))) dµ → 0 as n → ∞. Ω

Thus (Theorem 1.3.8) un converges to u in Lϕ (Ω). Separability is implied by the preceding result in conjunction with Lemma 2.3.1. The remark following Definition 2.3.1 in concert with the preceding theorem yields Corollary 2.3.3. Let Ω ⊆ Rn be open, consider the measure space (Ω, B, µ) where B is the Borel σ-algebra and µ is the Lebesgue measure. If ϕ is a locally ϕ integrable Musielak-Orlicz function on Ω and Lϕ (Ω) = Lϕ a (Ω), then L (Ω) is separable. Remark 2.3.4. Corollary 2.3.3 will be obtained by different means in Theorem 2.5.6. Corollary 2.3.5. If Ω ⊆ Rn is a Borel set, µ is the Lebesgue’s measure on Rn and p : Ω −→ [1, ∞) is Borel measurable with p+ < ∞, then the variable exponent Lebesgue space Lp (Ω) is separable. Proof. If p+ < ∞, then the Musielak-Orlicz function tp(x) satisfies the ∆2 condition (Theorem 1.5.5) which in turn implies that Lp (Ω) has absolutely continuous norm. Corollary 2.3.3 yields the desired result.

2.4

Duality of Musielak-Orlicz spaces

Spaces of Musielak-Orlicz type are modular spaces; it is thus in order to relate the study of their duality to the setting of Section 1.3. The following discussion will facilitate the connection we wish to establish.

2.4.1

Conjugate Musielak-Orlicz functions

Fix an Musielak-Orlicz function ϕ on Ω. We observe the fact that for each fixed x ∈ Ω the functional ( ϕ(x, r) if r≥0 ρx (r) = ϕ(x, −r) if r 0, ν > 0 and tν > 0 such that ϕ∗ (x, |u(x)|) −

 < |u(x)|tν − ϕ(x, tν ). 3

It follows immediately that for any j ∈ N, ϕ∗ (x, |u(x)|) −

 < |u(x)|tν − ϕ(x, tν ) 3 = |u(x)|(tν − qj ) − ϕ(x, tν ) + ϕ(x, qj ) + |u(x)|qj − ϕ(x, qj )

and selecting M such that qM < tν and max {tν − qM , ϕ(x, qM ) − ϕ(x, tν )}
ϕ∗ (x, |u(x)|) − .

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Analysis on Function Spaces of Musielak-Orlicz Type

For each k ∈ N and x ∈ Ωk , let jk (x) be the smallest natural number such that wk (x) = qjx (x) |u(x)| − ϕ(x, qjk (x) ). With this definition, for each k ∈ N let Bj,k = {x ∈ Ωk : j = jk (x)} and observe that for x ∈ Ωk k X

wk (x) =

(qj |u(x)| − ϕ(x, qj )) 1Bj,k

j=1

= sk (x)|u(x)| − ϕ(x, sk (x)), where sk =

k X

qj 1Bj,k

j=1

is a simple function and hence belongs to Lϕ (Ω) by assumption. Define Λu ∈ ∗ (Lϕ (Ω)) as Z ϕ L (Ω) 3 u −→ Λu (v) = u(x)v(x)dµ Ω

ρ∗ϕ

and recall the definition of the modular given in Theorem 1.3.12. We conclude from the above discussion that Z Z wk (x)dµ = (sk (x)|u(x)| − ϕ(x, sk (x))) dµ Ω

Ω

Z sk (x)|u(x)|dµ − ρϕ (sk )

= Ω

≤ =

sup

 Z

v∈Lϕ (Ω)  Ω ∗ ρϕ (Λu ) .

  u(x)v(x)dµ − ρϕ (v) 

A straightforward application of the monotone convergence theorem now yields Z ρϕ∗ (u) = ϕ∗ (x, |u(x)|)dµ ≤ ρ∗ϕ (Λu ) . Ω

On the other hand, it is clear from Young’s inequality (2.25) that   Z  ρ∗ϕ (Λu ) = sup u(x)v(x)dµ − ρϕ (v)  v∈Lϕ (Ω)  Ω

≤ ρ (u). ϕ∗

Musielak-Orlicz spaces

123

In all, it is concluded that ρϕ∗ (u) = ρ∗ϕ (Λu ) . Consequently, if kukϕ∗ 6= 0,     u Λu ρϕ∗ = ρ∗ϕ = 1, kΛu k(Lϕ (Ω))∗ kΛu k(Lϕ (Ω))∗ whence kukϕ∗ ≤ kΛu k(Lϕ (Ω))∗ = kuk(Lϕ (Ω))0 , which completes the proof of the inequalities (2.27). The next theorem reveals the depth of the idea of conjugation of MusielakOrlicz function and generalizes the well-known Riesz representation theorem for the classical Lp scale. Theorem 2.4.4. Let (Ω, A, µ) be a σ-finite measure space and let ϕ be a proper, locally integrable Musielak-Orlicz function on Ω. Denote the conjugate of ϕ by ϕ∗ (see (2.24)). Then the map ∗

∗

M : Lϕ (Ω) −→ (Lϕ (Ω)) Z hM (u), φi = u(x)φ(x)dµ Ω

is an isomorphism if and only if for any λ > 0 and any u ∈ Lϕ (Ω) , ρϕ (λu) < ∞. Proof. The proof follows immediately from Theorem 1.2.18, Theorem 2.2.15 and Theorem 2.4.3. Corollary 2.4.5. In the notation of Theorem 2.4.4 if (Ω, A, µ) is a σ-finite measure space and ϕ and its conjugate ϕ∗ are proper, then Lϕ (Ω) is reflexive. Proof. If M is as in Theorem 2.4.4, then ∗
∗  ∗ ∗∗ M −1 : Lϕ (Ω) −→ (Lϕ (Ω)) is an isomorphism. A further application of Theorem 2.4.4 in conjunction ∗∗ with the fact that ϕ∗∗ = ϕ reveals that, given Λ ∈ (Lϕ (Ω)) there exists ϕ uλ ∈ L (Ω) such that Z −1 hM (Λ), φi = uλ (x)φ(x)dµ, Ω ∗

for any φ ∈ Lϕ (Ω). Reflexivity follows immediately.

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Analysis on Function Spaces of Musielak-Orlicz Type

Corollary 2.4.6. If Ω ⊆ Rn is open and p : Ω −→ (1, ∞) is Borel measurable with p+ < ∞, then if p1 + 1q = 1 in Ω then the following isomorphism result holds: ∗ 0 (Lp (Ω)) ≈ (Lp (Ω)) ≈ (Lq (Ω)) . If, in addition p− > 1, then Lp (Ω) is reflexive.

2.5

Density of regular functions

In this section we consider the measure space (Ω, B, µ) where Ω ⊆ Rn is an open set, B is the σ-algebra of the Borel subsets of Ω and µ is the Lebesgue’s measure on Ω. Furthermore, we asume that ϕ is a locally integrable MusielakOrlicz function on Ω satisfying the ∆2 condition (Definition 2.2.2). It is further assumed that there exists γ > 0 such that γ ≤ inf ϕ(x, 1). x∈Ω

(2.28)

Lemma 2.5.1. Under the preceding conditions, the class S of simple functions supported on sets of finite measure on Ω is dense in Lϕ (Ω). Proof. Consider first a non-negative function u ∈ Lϕ (Ω) and a nondecreasing sequence (sn ) of nonnegative simple functions converging pointwise to u: for any n ∈ N, ϕ(x, |sn (x) − u(x)|) ≤ ϕ(x, 2|u(x)|); equality (2.18) and the dominated convergence theorem imply that ρϕ (sn − u) % 0 as n → ∞. Thus ksn − ukϕ −→ 0 as n → ∞ by Corollary 1.3.9. This proves the claim in the case u ≥ 0. Otherwise, we write as usual u = u+ − u− ; by the first part, given  > 0 there are simple functions s and t such that ku+ − skϕ < 2 and ku− − skϕ < 2 respectively; this proves the claim. Lemma 2.5.2. Let u ∈ M(Ω) with kuk∞ ≤ A and such that µ ({x : u(x) 6= 0}) < ∞. If M = {x : u(x) 6= 0}, then u ∈ Lϕ (M ) and given  > 0 there exists v ∈ C(Ω), ||v||∞ ≤ A and ku − vkϕ < . Proof. Certainly u ∈ Lϕ (M ). Indeed, due to the local integrability assumption on ϕ, it is obvious that Z Z ϕ(x, |u(x)|)dx ≤ ϕ(x, A)dx < ∞. M

M

Musielak-Orlicz spaces 125 R Next, fix η > 0 such that ϕ(x, 2A η )dx ≤ 1. Lusin’s Theorem yields a function Ω

v ∈ C(Ω), compactly supported such that if Ωη = {x ∈ Ω : u(x) 6= v(x)} , then µ(Ωη ) < η. Moreover, Lusin’s Theorem asserts that kvk∞ ≤ kuk∞ ≤ A. Consequently, the local integrability of ϕ yields Z Z ϕ(x, |v(x)|)dx ≤ ϕ(x, A) < ∞. Ω

Ω ϕ

The above inequality shows that v ∈ L (Ω). Furthermore,   Z  Z  |v(x) − u(x)| |v(x) − u(x)| ϕ x, dx = ϕ x, dx η η Ω

Ωη

Z ≤



2A ϕ x, η

 ≤ 1;

Ωη

it is obvious then that ku − vkϕ ≤ η.

A direct consequence of the previous lemma is that simple functions on Ω can be approximated in the Lϕ (Ω) norm, by continuous functions. More specifically, Lemma 2.5.3. Let Ω and ϕ be as specified above and s ∈ S be a simple function in Lϕ (Ω). Then, given  > 0 there exists a function v ∈ C(Ω) such that ks − vkϕ < . Proof. Select s as in the statement. By assumption, there are distinct real numbers s1 , s2 , ..., sk (which can be assumed to be non-zero) and disjoint measurable sets Ω1 , Ω2 , ..., Ωk such that s=

k X

sk 1Ωj .

j=1

Because of the assumption s ∈ Lϕ (Ω), there must exist λ > 0 for which R ϕ(x, λ|s(x)|) dx < ∞. Consequently, on account of the ∆2 condition and the Ω

definition of Ωj , one has: k Z X j=1Ω

j

ϕ(x, |sj |)dx < ∞.

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Analysis on Function Spaces of Musielak-Orlicz Type

Since ϕ is assumed to satisfy condition (2.28) it is clear that for any j : j = 1, 2, ..., k and for γ as in statement (2.28), one has Z µ(Ωj ) ≤ γ −1 ≤ ϕ(x, 1)dx < ∞. Ωj

Moreover, the support of s is contained in the union

k S

Ωj , which the above

j=1

discussion implies has finite measure and obviously |s(x)| ≤ sup{|sj |, j = 1, 2, ..., k}. Lemma 2.5.2 implies now that s can be approximated at will, in norm k · kϕ by a continuous function. Lemmas 2.5.2 and 2.5.3 yield: Lemma 2.5.4. If ϕ is a Musielak-Orlicz function satisfying the ∆2 condition and (2.28), the set C(Ω) is dense in Lϕ (Ω). The next lemma follows as a consequence of the preceding discussion. Lemma 2.5.5. C0∞ (Ω) is dense in C(Ω) ∩ Lϕ (Ω). Proof. Fix u ∈ C(Ω) ∩ Lϕ (Ω) S and let (Kj ) be a non-decreasing sequence of compact subsets of Ω with Kj = Ω. Given  > 0 it follows from the ∆2 condition that for any u ∈ C(Ω) ∩ Lϕ (Ω)   2|u(x)| ϕ x, ∈ L1 (Ω).  The theorem of monotone convergence implies   Z  Z  2|u(x)| 2|u(x)| ϕ x, dx % ϕ x, dx as j → ∞,   Kj

Ω

and it is easy to derive from here that for some j ∈ N,   Z 2|u(x)| ϕ x, dx ≤ 1.  Ω\Kj

Also, there exists a compact subset K of Ω such that   Z 2 ϕ x, |u(x)| dx ≤ 1.  Ω\K

(2.29)

Musielak-Orlicz spaces

127

Write u1 = u1K . Then (2.29) implies ku − u1 kϕ ≤

 . 2

(2.30)

Next, select a mollifier function Ψ ∈ C0∞ (Rn ) supported in the unit ball, that R n is Ψ ≥ 0 in R , Ψ(x)dx = 1; it is well known that under these assumptions, Rn

for any δ > 0 the mollified function (u1 ∗ Ψ)δ (·) = δ

−n



Z Ψ

·−y δ

 u1 (y)dy

Ω

is in C0∞ (Ω) and converges uniformly to u1 on every compact subset of Ω, as δ → 0. In fact, by definition u1 is continuous and hence uniformly continuous on K ; for arbitrary δ0 , select δ so small that |u1 (x − δ) − u1 (x)| < δ0 for any x ∈ K . Then it is easy to derive that |u1 ∗ Ψ(x) − u1 (x)| ≤ δ0 . Therefore, for x ∈ K ,   Z  Z  2|u1 ∗ Ψ(x) − u1 (x)| 2δ0 ϕ x, dx ≤ ϕ x, dx   K

(2.31)

K

and the latter tends to 0 as δ0 → 0. It follows that ku1 ∗ Ψ − u1 kϕ ≤

 . 2

As an immediate consequence of (2.30) and (2.31), one concludes ku − u1 ∗ Ψkϕ ≤ ku − u1 kϕ + ku1 ∗ Ψ − u1 kϕ ≤

  + = . 2 2

The following theorem is an immediate consequence of Lemmas 2.5.1, 2.5.2, 2.5.3 and 2.5.5. Theorem 2.5.6. Let Ω ⊆ Rn be open and ϕ be a locally integrable Musielak Orlicz function on Ω that satisfies the ∆2 condition and (2.28). Then C0∞ (Ω) is dense in Lϕ (Ω).

Example 2.5.1. The ∆2 condition in Theorem 2.5.6 is necessary. Indeed, C0∞ (Ω) need not have the density property. For example, let n = 1, Ω = (1, ∞)

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Analysis on Function Spaces of Musielak-Orlicz Type

and p(x) = x (x ∈ Ω); let f be the constant function on Ω that is everywhere equal to 1. Then f ∈ Lp (Ω) since Z∞

−1

λ−x dx = (λ log λ)

if λ > 1.

1

Let g ∈

C0∞ (Ω),

so that supp g ⊂ [1 + ε, N ] for some ε > 0 and N > 1. Hence   Z∞   kf − gkp ≥ inf λ > 0 : λ−x dx ≤ 1 ,   N

and as the integral is finite only if λ > 1, it follows that kf − gkp ≥ 1. Thus C0∞ (Ω) is not dense in Lp (Ω) . Before closing this section we emphasize the following corollary to Theorem 2.5.6: Corollary 2.5.7. Let Ω ⊆ Rn be open, p ∈ P(Ω) and suppose that p+ < ∞. Then the set of all bounded, measurable functions is dense in Lp (Ω), as is C(Ω) ∩ Lp (Ω); and Lp (Ω) is separable. Moreover, C0∞ (Ω) is dense in Lp (Ω).

2.6

Uniform convexity of Musielak-Orlicz spaces

Recall from Definition 1.2.7 in Section 1.2.3 that a convex modular ρ on a vector space V is said to be uniformly convex if for every  > 0 there exists δ = δ() > 0 such that   u+v (u ∈ V, v ∈ V, ρ(u) = ρ(v) = 1 and ρ(u − v) > ) ⇒ ρ < 1 − δ(). 2 It is only natural to expect a close connection between some form of convexity of the Musielak-Orlicz function ϕ and the uniform convexity of the modular ρϕ . To that effect, the following definition is introduced. Definition 2.6.1. A ϕ be a Musielak-Orlicz function on Ω (Chapter 2) is said to be uniformly convex if there exists w : (0, 1) −→ (0, 1) such that for any r ∈ (0, ∞), α ∈ (0, 1) and β ∈ [0, α] the inequality   1+β ϕ(x, r) + ϕ(x, βr) ϕ x, r ≤ (1 − w(α)) 2 2 holds a.e. in Ω.

Musielak-Orlicz spaces

129

In the interest of completeness we present the proof of the following theorem, following [24, 82]. Theorem 2.6.1. ([24, Theorem 2.4.11], [82, Theorem 11.6]) Let Ω ⊆ Rn be a domain. If the Musielak-Orlicz function ϕ : Ω × [0, ∞) −→ [0, ∞) is uniformly convex and satisfies the ∆2 condition, Then the Luxemburg norm k · kρϕ is uniformly convex. Proof. Fix  : 0 <  < 1 and functions u and v in Lϕ (Ω) such that ρϕ (u) = ρϕ (u) = 1 and ρϕ

u+v 2




. Set n o  M = x ∈ Ω : |u(x) − v(x)| > max{|u(x)|, |v(x)|} . 4

Convexity yields  

Z

ϕ x, Ω\M

u+v 2

 ≤

1  2

 Z

Z ϕ(x, u)dx +

Ω\M

 ϕ(x, v)dx

Ω\M

and it follows from here that     u u u+v γ = ρϕ 1Ω\M + ρϕ 1Ω\M − ρϕ 1Ω\M ≥ 0. 2 2 2 Thus, invoking the uniform convexity of ϕ:     u+v 1 u+v 1 − ρϕ = (ρϕ (u) + ρϕ (v)) − ρϕ 2 2 2   1 u+v = (ρϕ (1M u) + ρϕ (1M v)) − ρϕ 1M +γ 2 2 1 ≥ (ρϕ (1M u) + ρϕ (1M v)) 2 1 − w(1 − ) − (ρϕ (1M u) + ρϕ (1M v)) 2 = w(1 − ) (ρϕ (1M u) + ρϕ (1M v)) . It follows from the very definition of M that



 ρϕ 1Ω\M (u − v) ≤ ρϕ 1Ω\M u + ρϕ 1Ω\M v 4   ≤ (ρϕ (u) + ρϕ (v)) ≤ ; 4 2

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Analysis on Function Spaces of Musielak-Orlicz Type

consequently ρϕ ((u − v)1M ) = ρϕ (u − v) − ρϕ (u − v)1Ω\M   ≥ ρϕ (u − v) − > . 2 2




On account of condition (1.45) it follows that  < ρϕ ((u − v)1M ) 2 1 ≤ (ρϕ (2u1M ) + ρϕ (2v1M )) 2 1 ≤ K (ρϕ (2u1M ) + ρϕ (2v1M )) 2    K u+v ≤ 1 − ρϕ . 2w(1 − ) 2 In conclusion

 ρϕ

u+v 2

 . By virtue of the ∆2 condition necessarily ρϕ (u) = ρϕ (v) = 1 and   u−v ρϕ > 1. ku − vkϕ Invoking the ∆2 condition again it is readily seen that, for some constant C = C() > 1, ρϕ (u − v) is subject to the inequality:     u−v u−v 1 < ρϕ ≤ ρϕ ≤ Cρϕ (u − v); ku − vkϕ  which, in turn, yields, on account of the uniform convexity of ρϕ , the existence of δ = δ(C()) > 0 such that:   u−v ρϕ < 1 − δ, 2 valid for some δ = δ() > 0. We claim that there exists η : 0 < η() < 1 such that

u + v

2 < 1 − η. ϕ For otherwise, as is easy to see, there would exist a sequence (wn ) ∈ Lϕ (Ω) such that kwn kϕ % 1 and ρϕ (wn ) < 1 − δ.

Musielak-Orlicz spaces

131

This quickly leads to a contradiction, as follows: Set xn = kwn1 kϕ wn ; then ρϕ (xn ) = 1 and let C stand for the ∆2 constant of ρϕ . Then       1 1 1 = ρϕ (xn ) = ρϕ 2 − 1 wn + 2 − wn kwn kϕ kwn kϕ     1 1 ≤ ρϕ (wn ). (2.32) − 1 Cρϕ (wn ) + 2 − kwn kϕ kwn kϕ The right-hand side in (2.32) tends to 1 − δ as n → ∞, which concludes the proof. A direct consequence of the above discussion provides a criterion for the uniform convexity of Lp (Ω). Corollary 2.6.2. The variable exponent Lebesgue space Lp (Ω) (Section 1.5) is uniformly convex if 1 < p− = ess inf p ≤ ess inf p = p+ < ∞. Ω

Ω

(2.33)

Proof. It is easy to verify that for constant q : q > 1 the function v : (0, 1) −→ (0, 1) v(s) =

(1 + s)q 1 + sq

is increasing and that strictly less than 2q−1 ; consequently, for r, α and β as in Definition 2.6.1 one has q

(1 + αq ) (1 + β) ≤ (1 + α)q (1 + β q )

(2.34)

and it follows easily from (2.34) that q  1+β q q 2 (1 + α ) r ≤ (1 + α)q (rq + (rβ)q ). 2 Some elementary algebraic manipulations yield that as long as q > 1, for all s : s ∈ (0, 1) one has w(s) = 21−q v(s) < 1. (2.35) Thus if (2.33) holds,    p(x) 1+β 1+β ϕ x, r = r 2 2 (1 + α)p(x) 21−p(x) (1 + αp(x) )−1 p(x) (r + (rβ)p(x) ) 2 (ϕ(x, r) + ϕ(x, βr)) = (1 − (1 − 21−p(x) v(α)) 2 (ϕ(x, r) + ϕ(x, βr)) ≤ (1 − (1 − 21−p− v(α)) . 2 ≤

132

Analysis on Function Spaces of Musielak-Orlicz Type

Thus the norm k · kp is uniformly convex according to Definition 2.6.1 and on account of inequality (2.35).

2.7

Carath´ eodory functions and Nemytskii operators on Musielak-Orlicz spaces

In the study of boundary value problems it is of paramount importance to understand the action of so-called Nemytskii-type operators, that is, operators of the form X 3 u −→ f (x, u(x)), where X is a given function space on a domain Ω and f : Ω × R −→ R is a measurable function. In this section we analyze some fundamental questions related to the situation described above in the particular case when X is a Musielak-Orlicz space. Let Ω ⊆ Rn be an open set and consider a Musielak-Orlicz function and ϕ on Ω, as defined in Definition 2.1.1. Definition 2.7.1. Let Ω ⊆ Rn be a domain. A function w : Ω × [0, ∞) → [0, ∞) such that w(x, ·) is continuous a.e. x ∈ Ω and Lebesgue-measurable for every t ∈ R is said to be a Carath´eodory function. We remark at this point that a Carath´eodory function w naturally defines an operator Tw on M(Ω) by way of the assignment u(·) −→ Tw (u)(·) = w(·, |u(·)|).

(2.36)

The following theorem can be found in [61]; we include its proof for the sake of completeness. Theorem 2.7.1. Let Ω ⊆ Rn be a bounded domain and w : Ω → [0, ∞) be a Carath´eodory function. Let S be the subspace of M(Ω) consisting of sequences of measurable functions, that are convergent in measure. Then S is invariant under the operator Tw defined in (2.36). That is, if (uj ) ⊆ M(Ω) converges in measure, so does (Tw (uj )).

Musielak-Orlicz spaces

133

Proof. Let (un ) ⊆ M(Ω) converge in measure to u ∈ M(Ω). Fix  > 0. For any j ∈ N let   1 Ωj = x ∈ Ω : t − < |u(x)| < t ⇒ |w(x, |u(x)|) − w(x, t)| <  . j It is plain that the sequence (Ωj ) is increasing and from the a.e. left-continuity assumption on w(·, s) it is readily concluded that ∞ [ |Ω| = Ωj . 1

Thus, |Ω| = lim |Ωj |. j→∞

For any δ > 0, then, one can select j0 ∈ N such that |Ωj0 | > |Ω| − 2δ . Next, for each j ∈ N we set   1 Ej = x ∈ Ω : |u(x) − uj (x)| < . j0 Now, since (un ) converges to u in measure one has, for large enough j,   x ∈ Ω : |u(x) − uj (x)| > 1 < δ j0 2 and it is therefore clear that for some N ∈ N, j > N implies δ |Ej | > |Ω| − . 2 By definition, if x ∈ Ωj0 ∩ Ej , then |w(x, |u(x)|) − w(x, |uj (x)|)| < . It follows then that |{x ∈ Ω : |w(x, |u(x)|) − w(x, |uj (x)|)| < }| ≥ |Ωj0 ∩ Ej |. Since for j > N it holds that \ [ Ωj0 = |Ej | + |Ωj0 | − Ej Ωj 0 Ej δ ≥ |Ω| − , 2 it is clear that for j > N one has: |{x ∈ Ω : |w(x, |u(x)|) − w(x, |uj (x)|)| ≥ }| > The theorem follows immediately from the last statement.

δ . 2

134

Analysis on Function Spaces of Musielak-Orlicz Type

For the next discussion we fix an Musielak-Orlicz function (Definition 2.1.1) ϕ : Ω × [0, ∞) −→ [0, ∞) on the domain Ω ⊆ Rn . We assume that both, ϕ and its conjugate ϕ∗ (see (2.24) in Section 2.4.1) satisfy the ∆2 condition and impose on ϕ the additional condition that it be an N -function, that is (see [2, 24]): ϕ(x, t) = 0 a.e. t→0 t

x ∈ Ω.

lim

(2.37)

Theorem 2.7.2. Let ϕ be an N -function (i.e., it satisfies (2.37)) and assume that ∂ϕ ϕt (x, t) = (x, t) ∂t exists a.e. x ∈ Ω. Define the operator Tϕt as Tϕt : M(Ω) −→ M(Ω) Tϕt (u) = ϕt (x, |u(x)|) =

∂ϕ (x, |u(x)|). ∂t

Then, from the assumption ∗

Tϕt (Lϕ (Ω)) ⊆ Lϕ (Ω)

(2.38)

it follows that the operator ∗

Tϕt : Lϕ (Ω) −→ Lϕ (Ω) is continuous and bounded. Proof. Assume (un ) ⊆ Lϕ (Ω) converges to u ∈ Lϕ (Ω). On Ω × [0, ∞) define w(x, t) = ϕt (x, |u(x) + t|) − ϕt (x, |u(x)|); then on account of the assumption on ϕt , w is a Carath´eodory function and w(x, 0) = 0. If ∗ Tw : Lϕ (Ω) −→ Lϕ (Ω) ∗

is continuous at 0, then Tw (un − u) −→ 0 in Lϕ (Ω) as n → ∞. If ϕ∗ satisfies the ∆2 condition, the latter is equivalent to Z ρϕ∗ (Tw (un − u)) = ϕ∗ (x, |ϕt (x, |un (x)|) − ϕt (x, |u(x)|)|) → 0 as n → ∞, Ω

that is

∗

Tϕt (un ) −→ Tϕt (u) in Lϕ (Ω) as n → ∞.

Musielak-Orlicz spaces

135

Therefore, it is enough to show that Tϕt is continuous at 0 under the assumption that ϕt (x, 0) = 0 a.e. in Ω. Assume that Tϕt is not continuous at 0; let r > 0 and let (un ) be a sequence that converges to 0 in Lϕ (Ω), for which kTϕt (un )kϕ∗ ≥ r for any n ∈ N. Since norm convergence implies modular convergence, one can, without loss of generality assume that max {ρϕ (un ), kun kϕ } < and hence that

P∞ R n=1

1 , 2n

ϕ(x, |un (x)|) dx < ∞. Due to the validity of the ∆2

Ω

∗

condition for ϕ , norm convergence and modular convergence are equivalent ∗ on Lϕ (Ω). It follows that there exists (r) > 0 such that ρϕ∗ (Tϕt (un )) ≥ (r) for any n ∈ N. We next claim the existence of a sequence of real numbers (k ), a sequence (Ωk ) of subsets of Ω and a subsequence (unk ) of (un ) satisfying the following conditions: (i) k+1 < 12 k , (ii) |Ωk | ≤ k , (iii)

R Ωk

ϕ∗ (x, |Tϕt (unk )|) dx > 23 (r).

(iv) If E ⊆ Ω is measurable and |E| < 2k+1 , then Z (r) ϕ∗ (x, |Tϕt (unk )|) dx < . 3 E

Set Ω1 = Ω, 1 = |Ω|, n1 = 1. We assume that k , nk and Ωk are given, then, ∗ by assumption, ϕt (·, |unk (·)|) ∈ Lϕ (Ω) and on account of the ∆2 condition one has Z ϕ∗ (x, |ϕt (x, |unk (x)|)|) dx < ∞. Ω

Since the measure Z A −→ µ(A) =

ϕ∗ (x, |ϕt (x, |unk (x)|)|) dx

A

defined on the Borel σ algebra B of subsets of Ω is absolutely continuous with respect to the Lebesgue measure, one can find k+1 such that any X ∈ B R with |X| < 2k+1 satisfies ϕ∗ (x, |ϕt (x, |unk (x)|)|) dx < (r) 3 . The assumption X

k ≤ 2k+1 would contradict (iii). We now proceed to the construction of

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Analysis on Function Spaces of Musielak-Orlicz Type

Ωk+1 and nk+1 . On account of Theorem 2.7.1, the strong convergence of (un ) in Lϕ (Ω) implies the convergence in measure of both (Tϕt (un )) and (Tϕ∗ (Tϕt (un ))). Consequently, there exists nk+1 ∈ N such that   x ∈ Ω : Tϕ∗ (Tϕt (un )) > (r) < k+1 < k < k . k+1 3|Ω| 2 Define

 Ωk+1 =

(r) x ∈ Ω : Tϕ∗ (Tϕt (unk+1 )) > 3|Ω|

 .

Next,  Z

 ϕ∗ (x, |Tϕt (unk+1 )|) dx = 

Ωk+1

 Z

Z

 ∗  ϕ (x, |Tϕt (unk+1 )|) dx

−

Ω

Ω\Ωk+1

> (r) −

(r) 2(r) = . 3 3

By construction [ ∞ X ∞ Ω ≤ j < 2k+1 . j j=k+1 j=k+1 Set v(x) =

∞ S

 un (x) if x ∈ Ωk \ k

(2.39)

Ωj

j=k+1



0 otherwise.

ϕ

It is clear that v ∈ L (Ω). Next, observe that: Z

∗

ϕ (x, Tϕt (v(x))) dx ≥

∞ X

Z

ϕ∗ (x, Tϕt (unk (x))) dx

k=1 Ωk \∪∞ j=k+1 Ωj

Ω

 Z ∞ X  =  − k=1

≥

Ωk

∞  X k=1

 Z

 ∗  ϕ (x, Tϕt (unk (x))) dx

∪∞ j=k+1 Ωj

 2 1 (r) − (r) 3 3

= ∞, which follows by (2.39) and condition (iv). This contradicts assumption (2.38). Hence, Tϕt is continuous at 0.

Musielak-Orlicz spaces

2.8

137

Further properties of variable exponent spaces

Our next objective is to pause and exploit the general theory hitherto established to derive further properties in the particular case of the variable exponent Lebesgue spaces, such as uniform convexity. We follow the line of reasoning given in [76] and begin with a lemma that characterizes spaces with absolutely continuous norms. We fix a σ-finite measure space (Ω, A, µ). Recall that P(Ω) stands for the vector space of all real-valued, Borel-measurable functions defined on Ω. Lemma 2.8.1. Let p ∈ P(Ω). Then the following statements are equivalent: (i) Lp (Ω) has absolutely continuous norm (see Definition 1.2.3); (ii) p+ < ∞; (iii) Lp (Ω) is separable. Proof. First suppose that p+ = ∞. If |Ω∞ | = 0, set: Ωk = {x ∈ Ω : k ≤ p(x) < k + 1} (k ∈ N) and let {nk }k∈N be a sequence of natural numbers such that |Ωnk | > 0 for each k ∈ N. Let ck > 0 be such that Z p(x) ck dx = 1 (k ∈ N) Ωnk

and put u=

∞ X

ck 1Ωnk , Ej =

k=1

∞ [

Ωn k ;

k=j

thus Ej → ∅. Since  

 ∞ Z    X p(x) ck kukp = inf λ > 0 : dx ≤ 1   λ k=1Ωnk ( ) ∞   nk X 1 ≤ inf λ > 1 : dx ≤ 1 ≤ 2, λ k=1

it follows that u ∈ Lp (Ω). As   Z  p(x)  

c j

u1Ej ≥ inf λ > 1 : dx ≤ 1 = 1, p   λ Ωnj

u does not have absolutely continuous norm. If |Ω∞ | > 0, take any A ⊂ Ω∞

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Analysis on Function Spaces of Musielak-Orlicz Type

with positive measure and observe that 1A evidently does not have absolutely continuous norm. Thus (i) implies (ii). For the reverse implication, assume that p+ < ∞, let u ∈ Lp (Ω) have norm 1, take

ε > 0, let {Ek } be a sequence of sets with |Ek | & 0, let j ∈ N be such that u1Ej p ≥ 1 − ε, set φ = u1Ω\Ej and put ψ = u1Ej . By Proposition 1.5.2, p p Z Z φ ψ dx = dx = 1. kφkp kψkp Ω

Ω

Thus

Z

p

kφkp+ ≤

p

p

Ω

and

Z

Z

|φ| dx, kψkp+ ≤

p

|ψ| dx Ω

Z

p

|φ| dx + Ω

p

|ψ| dx ≤ 1. Ω

Hence p kψkp+

Z ≤

p

Z

|ψ| dx ≤ 1 − Ω

p

p

|φ| dx ≤ 1 − kφkp+ ≤ 1 − (1 − ε)p+ ,

Ω 1/p

so that kψkp ≤ (1 − (1 − ε)p+ ) + and the absolute continuity follows. Thus (i) and (ii) are equivalent. For the equivalence of (i) and (iii) we refer to [11, Corollary 1.5.6]. We are now ready to tackle the following theorem: Theorem 2.8.2. Let p ∈ P(Ω). The following statements are equivalent: (i) Lp (Ω) is reflexive; 0 (ii) Lp (Ω) and Lp (·) (Ω) have absolutely continuous norms; (iii) Lp (Ω) is uniformly convex; (iv) 1 < p− ≤ p+ < ∞. Proof. Since Lp (Ω) is a Banach function space, the equivalence of (i) and (ii) follows immediately from Theorem 1.2.19. In view of Lemma 2.8.1, (ii) implies (iv); and that (iii) implies (i) is obvious. All that is left is to show that (iv) implies (iii). Suppose that (iv) holds, let ε ∈ (0, 1) and let u, v have unit Lp (Ω) norm. Set s = (u + v)/2, t = (u − v)/2 and Γ = {x ∈ Ω : p− ≤ p(x) ≤ p+ } , so that |Ω\Γ| = 0; put S = {x ∈ Γ : |t(x)| < ε |s(x)|} , T = {x ∈ Γ : |t(x)| ≥ ε |s(x)|} .

Musielak-Orlicz spaces

139

Then Z

Z

p

|t| dx ≤ S

p

εp |s| dx ≤

S

≤ε

Z

p

εp |s| dx

(2.40)

Ω

p−

Z

p

|s| dx ≤ εp− .

Ω t

The function λ 7−→ λ is strictly convex on R whenever 1 < t < ∞; hence   t t t |λ| < |λ + 1| + |λ − 1| /2 (λ ∈ R). (2.41) q

q

q

Thus the function f : (q, λ) 7−→ (|λ + 1| + |λ − 1| ) /2 − |λ| (q ∈ (1, ∞), λ ∈ R) is continuous and strictly positive on (1, ∞)×R. It follows that there exists α > 0 such that f (q, λ) ≥ α whenever q ∈ [p− , p+ ] and λ ∈ [−1/ε, 1/ε]. We therefore see that  1 p(x) p(x) p(x) |λ + 1| + |λ − 1| − |λ| ≥α 2 for all x ∈ Γ and λ ∈ [−1/ε, 1/ε], so that  1 p(x) p(x) p(x) p(x) |s(x) + t(x)| + |s(x) − t(x)| ≥ |s(x)| + α |t(x)| (x ∈ T ), 2 while by (2.41),  1 p(x) p(x) p(x) |s(x) + t(x)| + |s(x) − t(x)| ≥ |s(x)| (x ∈ S). 2 Thus Z 1 p p 1= (|s + t| + |s − t| ) dx 2 Ω Z Z p p ≥ |s| dx + α |t| dx, Ω

and so

Z

T

p

p−

|t| dx < ε T

Z if

p

|s| dx > 1 − αεp− .

Ω

Thus if δ := αεp− and k(u + v)/2k > 1 − δ, then

R

p

|s| dx > 1 − δ, from which

Ω

with the aid of (2.40) it follows that Z p |t| dx < 2εp− , Ω

so that ku − vkp = k2tkp ≤ 2(2εp− )1/p+ , which establishes the uniform convexity of Lp (Ω).

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Analysis on Function Spaces of Musielak-Orlicz Type

Recall from Section 1.2.2, if X be a Banach function space, three subspaces of X are to be distinguished, namely: (i) the set of all elements of X with absolutely continuous norm (denote Xa ); (ii) the closure of the set of all bounded functions supported in sets of finite measure (denote Xb ); (iii) the space of all functions u with a continuous norm, by which we mean that for every x ∈ Ω we have

lim u1B(x,ε)∩Ω p = 0 ε→0

(denote Xb ). From Lemma 1.2.17, we have that Xa is a closed subspace of X and Xa ⊂ Xb ; it has been shown in [68] that Xc is a closed subspace of X and Xa ⊂ Xc . Now take X = Lp (Ω). Then in view of Lemma 2.8.1 it follows that if p+ < ∞, then Xa = Xb = Xc = X.

(2.42)

The position when p+ may be infinite is more complicated, as we now illustrate with the next two theorems from [35]. Theorem 2.8.3. If Ω is bounded and p ∈ P(Ω), then Xa = Xc . Proof. It suffices to show that Xc ⊂ Xa . First suppose that |Ω∞ | = 0. Assume that u ∈ / Xa . Then there exist α > 0 and a decreasing sequence of sets {Gk }k∈N such that |Gk | → 0 and, for every k ∈ N,   Z   p ku1Gk kp = inf λ > 0 : (u/λ) dx ≤ 1 ≥ α.   Gk

With β := α/2 this implies that Z p (f /β) dx > 1 (k ∈ N). Gk

If there existed m such that Z

p

(u/β) dx ≤ K < ∞, Gm

then as the Gk are decreasing, with |Gk | → 0, by dominated convergence we would have Z p lim (u/β) dx = 0, k→∞ Gk

a contradiction. Thus

Z

p

(u/β) dx = ∞ Gk

Musielak-Orlicz spaces

141

for every k ∈ N. Since Ω is bounded, there is a closed cube Q with side length `(Q) such that Ω ⊂ Q. By subdivision of Q we obtain a decreasing sequence (Qk ) of closed cubes with l(Qk ) = 2−k l(Q) and Z p (u/β) dx = ∞ Qk ∞ T

for every k. Let x ∈

Qk and ε > 0. Then Qk ⊂ B(x, ε) for large enough k

k=1

and so Z

Z

p

p

(u/β) dx ≥

(u/β) dx Ω∩Qk

Ω∩B(x,ε)

Z

p

(u1Qk /β) dx = ∞.

= Ω

Hence u1Ω∩B(x,ε) p > β, and so u ∈ / Xc . This establishes the theorem when |Ω∞ | = 0. Finally, we dispense with

this assumption and take any u ∈ Xc . The norms kukp and u1Ω\Ω∞ p + ess supx∈Ω∞ |u(x)| are equivalent. Since u ∈ Xc , u(x) = 0 on Ω∞ . Now apply the case of the theorem already proved to the set Ω\Ω∞ and the function p1 := p1Ω1 to obtain u ∈ Xa . Theorem 2.8.4. Let p ∈ P(Ω) and suppose that Ω is bounded; for each k ∈ N put Λk = {x ∈ Ω : k ≤ p(x) < k + 1} . (i) If p ∈ L∞ (Ω\Ω∞ ), then Xb = X; (ii) if |Ω∞ | = 0 and p+ = ∞, then Xb 6= X; (iii) if |Ω∞ | = 0 and p+ = ∞, then Xa = Xb if and only if ∞ X

Ak |Λk | < ∞ for all A > 1.

(2.43)

k=1

Proof. (i) This is immediate from (2.42) and the fact that L∞ (Ω∞ ) = Xb (Ω∞ ). R p (ii) For each k ∈ N let ck be chosen so that ck dx = 2−k if |Λk | > 0, ck = 0 Ωk

otherwise; put dk = max(ck , 2k) if ck > 0, dk = 0 otherwise. There are sets 0 Λk ⊂ Λk such that Z dpk dx = 2−k if |Λk | > 0. 0

Λk

142

Analysis on Function Spaces of Musielak-Orlicz Type

Let u =

∞ P k=1

dk 1Λ0 . Since u(x) = 0 on Ω∞ , k

Z

up dx =

ρp (u) =

∞ Z X k=1 0 Λk

Ω\Ω∞

hence kf kp ≤ 1. Now define

 uk (x) =

dpk dx ≤

∞ X

2−k = 1;

k=1

u(x), 0 ≤ u(x) ≤ k, k, u(x) > k,

and let v be any measurable function on Ω with |v| ≤ k. Since |u(x) − v(x)| ≥ |u − uk (x)| , we have ku − vkp ≥ ku − uk kp ; moreover, u − uk = 0 on Ω∞ . Thus Z

p

(4 |u − uk |) dx ≥

∞ X

j

4

j=k

Ω\Ω∞

≥

∞ X

R

(dj − k)p dx

0

Λj

4j

j=k

But

Z

Z

(dj /2)p dx ≥

Z ∞ X 4j dj p dx. 2j+1 j=k

0

Λj

0

Λj

dj p dx = 2−j for infinitely many j 0 s. Hence the last sum is infinite and

0

Λj

ku − vkp > 1/4. Since v is an arbitrary bounded function, the proof of (ii) is complete. (iii) Suppose that (2.43) holds and let u be a bounded function on Ω with |u| ≤ K. Let {Gm }m∈N be a decreasing sequence of measurable subsets of Ω with |Gm | → 0, and let λ ∈ (0, K). Then for every m,  p Z p Z  p Z ∞ X f K K dx ≤ dx = dx λ λ λ Gm

≤

Gm ∞  X k=1

k=1G ∩Λ m k

K λ

k+1 |Gm ∩ Λk | .

Since K/λ > 1, (2.43) implies that there exists k0 such that  k+1 ∞ X K 1 |Λk | ≤ ; λ 2 k=k0 +1

as |Gm | → 0, there exists m0 such that if m ≥ m0 , then k+1 ∞  X K 1 |Gm ∩ Λk | ≤ . λ 2 k=1

Musielak-Orlicz spaces

143

Thus, Z p u dx ≤ 1 if m ≥ n0 . λ Gm

It follows that for any small λ > 0 we have ku1Gm kp ≤ λ if m ≥ m0 , so that limm→∞ ku1Gm kp = 0. Hence u ∈ Xa . Now assume that u ∈ Xb . Then there is a sequence of bounded functions um with ku − um kp → 0. By the last part of the proof, each um ∈ Xa ; as Xa is closed, u ∈ Xa . Now suppose that there exists A > 1 such that ∞ P Ak |Λk | = ∞. Let v be the function that is identically 1 on Ω, and put k=1

Gm =

∞ S

Λk . Then

k=m

Z 

u1Gm 1/A

p dx =

∞ Z X

Ap 1Gm dx ≥

k=1Λ k

Ω

∞ X

Am |Λk | = ∞.

k=m

Hence ku1Gm kp > 1/A for any m. As the Gm are decreasing and |Gm | → 0, it follows that u ∈ / Xa . Since plainly u ∈ Xb the proof is complete. We underline that under the conditions (iii) of the preceding theorem another characterization of the equality of Xa and Xb can be given in terms of the non-increasing rearrangement p∗ of the function p. In fact, since Z |Ω| Z p∗ (t) A dt = Ap(x) dx, 0

Ω

it is immediate that Xa = Xb if and only if Z |Ω| ∗ Ap (t) dt < ∞ for all A > 1. 0

By way of illustration, consider the case in which n = 1, Ω = (0, 1/e) and p∗ (x) = xα for some α < 0. Then Z

1/e

A

p∗ (x)

0

Z dx =

1/e

A 0

xα

Z dx = 0

1/e

X∞ (xα log A)k dx k=0 k!

=∞ for all A > 1. Thus Xa = Xb . On the other hand, if n = 1, Ω = (0, 1/e) and p∗ (x) = (log x−1 )α for some α ≥ 0, then Z 1/e Z 1/e Z ∞
∗ Ap (x) dx = exp (log x−1 )α log A dx = exp (y α log A − y) dy 0

0

1 : thus Xa = Xb when α ∈ (0, 1). However, if α ∈ [1, ∞), the choice A = e gives Z

1/e

Ap

0

∗

(x)

Z

∞

dx =

exp (y α log A − y) dy = ∞,

1

and so Xa $ Xb . Our discussion so far has been focused on those aspects of the theory of spaces with variable exponent that emphasize the similarities between the properties of such spaces and those of their classical counterparts. Remarkable though these are, important differences remain, even when the exponent p is non-pathological. The most prominent of these relates to the matter of p-mean continuity. Recall that if p is a constant, p ∈ [1, ∞), and Ω is a non-empty open bounded subset of Rn , then every function u ∈ Lp (Ω), extended by 0 outside Ω, is p-mean continuous in the sense that given any ε > 0, there exists δ > 0 such that  1/p Z p(x)  |u(x + h) − u(x)| dx < ε if |h| < δ. Ω

By analogy with this, when p is any element of P(Ω), we say that a function u ∈ Lp (Ω) is p-mean continuous if ρp (u − τh u) < ε whenever |h| < δ : here τh f := u(·+h). It turns out that elements of Lp (Ω) are not, in general, p-mean continuous: more precisely, in [60] the following theorem is given: Theorem 2.8.5. Let p ∈ P (Ω) and suppose that Ω contains a ball B(x0 , r) on which p is continuous and non-constant. Then there is a function u ∈ Lp (Ω) which is not p-mean continuous. Proof. Let z ∈ B(x0 , r) be a point at which p does not have a local extremum. There are sequences (xk ), (yk ) of points of B(x0 , r), each converging to z, with p(xk ) < p(z) < p(yk ) for all k ∈ N. Since p is continuous in B(x0 , r), for each k ∈ N there exists rk > 0 such that p(x)
p(z) if if x ∈ B(yk , rk ). For each k ∈ N put qk = 12 (p(z) + p(xk )) , let uk be a function on supp uk ⊂ B(xk , rk ), uk ∈ Lqk (B(xk , rk )) \ Lp (B(xk , rk )) and and define u=

∞ X k=1

2−k uk .

(2.45) Ω such that kukqk = 1,

Musielak-Orlicz spaces

145

From (2.44) and Theorem 2.2.2 we have kukp ≤

∞ X

2−k kuk kp ≤

k=1

∞ X

2−k kuk kqk (|B(xk , rk )| + 1)

k=1

≤ 1 + sup |B(xk , rk )| < ∞. k

Now put hk = yk − xk (k ∈ N) : from (2.45) and Theorem 2.2.2 we obtain

−1 kτhk ukp ≥ 1B(yk ,rk ) τhk u p ≥ (1 + |B(yk , rk )|) 1B(yk ,rk ) τhk u p(z)

−1 = (1 + |B(yk , rk )|) 1B(yk ,rk ) u p(z) = ∞. It follows that τhk u − u ∈ / Lp (Ω). Thus u is not p-mean continuous. Despite this result, if p+ < ∞ and u ∈ Lp (Ω) is bounded, with compact support, then there is a compact set K such that for all h ∈ Rn with |h| < 1, supp τh u ⊂ K. Thus by Theorem 2.2.2, lim kτh u − ukp ≤ (1 + |K|) lim kτh u − ukp+ = 0.

|h|→0

|h|→0

The lack of satisfactory behavior under translations is perhaps the biggest technical difficulty faced when dealing with spaces with variable exponent. Naturally it manifests itself in connection with convolutions. For example, if p is a constant, p ∈ [1, ∞], then a special case of Young’s inequality asserts that k u ∗ vkp,Rn ≤ kukp,Rn kvk1,Rn for all u ∈ Lp (Rn ) and v ∈ L1 (Rn ); here u ∗ v denotes the convolution of u and v. However (see [24]) if p ∈ P(Ω) and 1 < p− ≤ p+ < ∞, then there exists c > 0 such that k u ∗ vkp,Rn ≤ c kukp,Rn kvk1,Rn for all u ∈ Lp (Rn ) and all v ∈ L1 (Rn ) if and only if p is constant. A more general form of Young’s inequality for constant exponents asserts that k u ∗ gkr,Ω ≤ c kukp,Ω kvkq,Ω , when Ω ⊂ Rn and p, q, r are constants lying in the interval [1, ∞] and satisfying the equality 1 1 1 +1= + . r p q Further insight into the severe restrictions that have to be imposed in order to obtain an analogue of the classical results is provided by the following result (see [23]), which also illustrates the usefulness of Theorem 2.8.5.

146

Analysis on Function Spaces of Musielak-Orlicz Type

Theorem 2.8.6. Let p, r ∈ P(Ω) ∩ C(Ω) be such that 1 < p− ≤ p+ < ∞ and 1 < r− ≤ r+ < ∞; suppose that Ω is bounded and contains a ball on which r is not constant. Then k u ∗ vkr,Ω . kukp,Ω kvk1,Ω (u ∈ Lp (Ω), v ∈ L1 1(Ω)) if and only if p− ≥ r+ . Proof. First suppose that p− ≥ r+ . Then from the classical form of Young’s inequality and Theorem 2.2.2, k u ∗ vkr,Ω . k u ∗ vkr+ ,Ω . kukr+ ,Ω kvk1,Ω . kukp,Ω kvk1,Ω . For the converse, assume that k u ∗ vkr,Ω . kukp,Ω kvk1,Ω and suppose that p− < r+ . It is sufficient to deal with the case in which Ω is a ball on which r is not constant and p(x) < r(x). Theorem 2.2.2 guarantees that Lr (Ω) ,→ Lp (Ω); it follows from Theorem 2.8.5 that there is a function u ∈ Lp (Ω) which is not r-mean continuous. Hence there exists h ∈ Rn such that, in the notation used in Theorem 2.8.5, uh ∈ / Lr (Ω). Now let φ be a Friedrichs mollifier and for each ε > 0 define φε as usual by φε (x) = ε−n φ((x − h) /ε), x ∈ Rn . Then as ε → 0, u ∗ φε → uh in L1 (Ω); and by the continuity of the convolution operator, k u ∗ φε kr,Ω . kukp,Ω kφε k1,Ω = kukp,Ω . As Lr (Ω) is reflexive, there is a sequence (εk ) , with εk → 0, such that ( u ∗ φεk ) converges weakly in Lr (Ω); evidently the limit is uh . But uh ∈ / Lr (Ω) and we have a contradiction.

Yet another difficulty arises in connection with the Hardy-Littlewood maximal operator M, defined for all functions u that are locally integrable on Ω, by Z −1

(M u)(x) = sup |B|

|u(y)| dy

(x ∈ Ω),

B∩Ω

where the supremum is taken over all balls B that contain x and for which |B ∩ Ω| > 0. It is well known that if p is a constant lying in the interval (1, ∞), then M maps Lp (Ω) boundedly into itself: there is a positive constant c = c(p, Ω) such that for all u ∈ Lp (Ω), kM ukp ≤ c kukp .

(2.46)

However, (2.46) does not hold for all p ∈ P(Ω). A sufficient condition for its validity, when Ω is open and bounded, is that for some constant C > 0, |p(x) − p(y)| ≤ −

C for all x, y ∈ Ω with 0 < |x − y| < 1/2. log |x − y|

Musielak-Orlicz spaces A detailed discussion of this is given in [24].

147 As Theorem 2.8.2 settles the

question of uniform convexity for the variable exponent Lebesgue’s spaces, we now turn to the modular uniform convexity. We recall the Definition of U CC2 (see Definition 1.3.9). Theorem 2.8.7. Let Ω ⊆ Rn be open and ∈ P(Ω). If p− > 1 then the modular ρp : Lp (Ω) −→ [0, ∞) Z ρp (u) =

|u(x)|p(x) dx

Ω

satisfies the U CC2 condition. Proof. The proof follows along the same lines as that of Theorem 1.4.32: We include the details for the sake of completeness. Let Ω ⊆ Rn be a domain and p ∈ P(Ω). We retain the terminology from Section 1.5; in addition, for a subset A ⊆ Ω and u ∈ M(Ω), we write Z ρA (u) = |u|p(x) dx. p A

Let r > 0,  > 0 and consider u, v ∈ Lp (Ω) such that   u−v ρp (u) ≤ r , ρp (v) ≤ r , ρp ≥ r. 2 On account of the convexity of ρp we have  ≤ 1: indeed,   u−v r ≤ ρp ≤ r. 2 Let Ω1 := {x ∈ Ω : p(x) ≥ 2}. Then either Z u(x) − v(x) p(x) r dx ≥ 2 2

(2.47)

Ω1

or Z

u(x) − v(x) p(x) r dx ≥ . 2 2

Ω\Ω1

In case (2.47) holds, one has, by virtue of inequality (1.52): Z Z u(x) − v(x) p(x) u(x) + v(x) p(x) dx + dx 2 2 Ω1 Ω1   Z Z 1 ≤ |u(x)|p(x) dx + |v(x)|p(x) dx . 2 Ω1

Ω1

(2.48)

148

Analysis on Function Spaces of Musielak-Orlicz Type

It is thus concluded that   Z Z Z u(x) + v(x) p(x) 1 r p(x) p(x) dx ≤  |u(x)| dx + |v(x)| dx − . 2 2 2 Ω1

Ω1

Ω1

Thus,  ρp

u+v 2



Z Z u(x) + v(x) p(x) = dx + 2 Ω1

u(x) + v(x) p(x) dx 2

Ω\Ω1

  Z Z 1 r ≤  |u(x)|p(x) dx + |v(x)|p(x) dx − 2 2 Ω1 Ω1   Z Z 1  +  |u(x)|p(x) dx + |v(x)|p(x) dx 2 Ω\Ω1

Ω\Ω1

1 r (ρp (u) + ρp (v)) − 2 2  ≤r 1− . 2 =

In case (2.48) holds, we define n o  Ω2 := x ∈ Ω \ Ω1 : |u(x) − v(x)| ≤ (|u(x)| + |v(x)|) . 4 It follows that   Z Z Z u(x) − v(x) p(x)  dx ≤  |u(x)|p(x) dx + |v(x)|p(x)  2 8 Ω2

Ω2

Ω2

 r ≤ (ρp (u) + ρp (u)) ≤ . 8 8 The validity of (2.48) implies in particular that Z Ω\(Ω1 ∪Ω2 )

Z u(x) − v(x) p(x) = 2

Z u(x) − v(x) p(x) − 2

Ω\Ω1

r r r ≥ − = . 2 4 4

Ω2

u(x) − v(x) p(x) 2

Musielak-Orlicz spaces

149

It follows from (1.53) that if x ∈ Ω \ (Ω1 ∪ Ω2 ), one has p(x) u(x) + v(x) p(x)  u(x) − v(x) + (p − 1) − 2 8 2 p(x) u(x) + v(x) p(x)(p( x) − 1)   2−p(x) u(x) − v(x) p(x) ≤ + 2 2 4 2 p(x) 2−p(x) u(x) + v(x) u(x) − v(x) p(x) p(x)(p(x) − 1) u(x) − v(x) ≤ + |u(x)| + |v(x)| 2 2 2 1 ≤ (|u(x)|p(x) + |v(x)|p(x) ). 2 Integrating the last inequality over Ω \ (Ω1 ∪ Ω2 ) it follows that u(x) + v(x) p(x) dx 2

Z Ω\(Ω1 ∪Ω2 )

u(x) − v(x) p(x) dx 2

Z

  + inf p(x) − 1 Ω 8

Ω\(Ω1 ∪Ω2 )

 ≤

1  2

 Z

|u(x)|p(x) dx +

Ω\(Ω1 ∪Ω2 )

Z

 |v(x)|p(x) dx .

Ω\(Ω1 ∪Ω2 )

We arrive thus at Z u(x) + v(x) p(x) dx 2 Ω\(Ω1 ∪Ω2 )

 ≤

1  2

 Z

Ω\(Ω1 ∪Ω2 )

|u(x)|p(x) dx +

Z

Ω\(Ω1 ∪Ω2 )

2  |v(x)|p(x) dx − (p− − 1) r. 32

150

Analysis on Function Spaces of Musielak-Orlicz Type

In all   u+v ρp = 2

Z

u(x) + v(x) p(x) dx + 2

Ω1 ∪Ω2

Z

u(x) + v(x) p(x) dx 2

Ω\(Ω1 ∪Ω2 )



 Z

Z

1 |u(x)|p(x) dx + |v(x)|p(x) dx 2 Ω1 ∪Ω2 Ω1 ∪Ω2   Z Z 1  +  |u(x)|p(x) dx + |v(x)|p(x) dx 2 Ω\(Ω1 ∪Ω2 ) ≤

Ω\Ω1 ∪Ω2

2

− (p− − 1)

 r 32

≤ r − (p− − 1)

  2 2 r = r 1 − (p− − 1) . 32 32

We conclude that for any r > 0,  > 0 and arbitrary u, v ∈ D(r, ) as specified in Definition 1.21, it holds that     1 u+v  2 ≥ min , (p− − 1) > 0, 1 − ρp r 2 2 32 and Lp (Ω) is U U C2.

2.8.1

Duality maps on spaces of variable integrability

Finally we turn to the determination of duality maps on Lp (Ω). When p is a constant in the interval (1, ∞), it is a familiar fact that the Gˆateaux derivative of k·kp is given by   1−p p−1 grad kukp (x) = kukp |u(x)| sgn u(x) (u 6= 0, x ∈ Ω), from which the duality map J with gauge function µ may be calculated by means of the formula   Jf = µ kukp grad kukp . Now suppose that p ∈ P(Ω) and 1 < p− ≤ p+ < ∞. By Theorems 1.2.25 and 2.8.2, Lp (Ω) is uniformly convex and uniformly smooth, and so its norm is even Fr´echet-differentiable. Theorem 2.8.8. Let Ω be a bounded open subset of Rn and let p ∈ P(Ω) be such that 1 < p− ≤ p+ < ∞. Then for every u ∈ Lp (Ω)\{0}, −p(z)

(grad kukp )(z) =

p(z)−1

p(z) kukp |u(z)| sgn u(z) . R −p−1 p p kukp |u| dx Ω

(2.49)

Musielak-Orlicz spaces

151

Proof. Put A(x) =

|u(x)| kukp

!p(x)−1

Z

−p−1

p kukp

sgn u(x), B =

p

|u| dx.

Ω −1

Thus the right-hand side of (2.49) equals p(x) kukp A(x)/B. Note that   p− p− p+ = φp u/ kukp ≤ B ≤ . kukp kukp kukp Moreover, Z φp0 (A) = Ω

|u| kukp

!p dx = 1. 0

Hence the right-hand side of (2.49) represents an element of Lp (Ω) and so can be identified with an element of the dual of Lp (Ω), the value of which at u is kukp . The result follows from this observation.

2.9

The Matuszewska-Orlicz index of a Musielak-Orlicz space

We begin this section with a few elementary facts concerning subadditive functions. For an exhaustive treatment of the topic we refer the reader to [78]. See also [11]. Lemma 2.9.1. Let g : R −→ (0, ∞) be measurable, increasing and subadditive (i.e., g(s + t) ≤ g(t) + g(s) for all (s, t) ∈ R2 ) and assume g(0) = 0. Then 0 < lim

t→∞

g(t) g(t) = inf < ∞. t>0 t t

Proof. For one thing it is clear that L = inf

t>0

g(t) ≤ 1. t

Given a fixed δ > 0, let t be chosen so that L≤

g(t) < L + δ, t

(2.50)

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Analysis on Function Spaces of Musielak-Orlicz Type

then for some natural number k it holds (1 + k −1 )

g(t) < L + δ. t

For any real number s with s ≥ kt, let n be chosen such that s ∈ [n, n + 1) t (obviously, n ≥ k). From the conditions on g it follows that g(s) ≤ (n + 1)g(t). In all g(s) (n + 1)g(t) ≤ ≤ s nt

    1 g(t) 1 g(t) 1+ ≤ 1+ < L + δ. n t k t

The latter shows both the existence of the limit and the equality in (2.50). Definition 2.9.1. For a Musielak-Orlicz function ϕ and each x ∈ Ω, set Mϕ (x, t) = lim sup u→∞

ϕ(x, tu) . ϕ(x, u)

(2.51)

The Matuszewska index of ϕ (or Matuszewska-Orlicz index) is defined to be mϕ (x) = lim

t→∞

ln Mϕ (x, t) ln Mϕ (x, t) = inf . t>1 ln t ln t

In particular, for t, s > 1, lim sup u→∞

ϕ(x, tsu) ϕ(x, tsu) ϕ(x, tu) ≤ lim sup lim sup ; ϕ(x, u) u→∞ ϕ(x, tu) u→∞ ϕ(x, u)

this yields Mϕ (x, ts) ≤ Mϕ (x, ts)Mϕ (x, ts). We remark that for fixed x ∈ Ω, M (ϕ) : [0, ∞) −→ [0, ∞) M (ϕ) (x, s) = Mϕ (x, es ) is subadditive and that M (ϕ) (x, t) Mϕ (x, es ) = lim t→∞ s→∞ ln t s Mϕ (x, es ) Mϕ (x, t) = inf = inf . s>0 t>1 s ln t lim

(2.52)

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153

Example 2.9.1. It is easy to verify that if ϕ(x, t) = tp(x) , then mϕ (x) = p(x). Example 2.9.2. If ϕ(x, t) = tp(x) (log(1 + t))q(x) , then mϕ (x) = p(x). The Matuszewska-Orlicz index of an Orlicz function was first introduced in [79]. We also refer the interested reader to [53, 78, 91, 92, 102] for a more detailed account and further ramifications of this topic. Definition 2.9.2. Let ϕ1 and ϕ2 be Musielak-Orlicz functions defined on Ω ⊂ Rn . The symbol ϕ1 ∼ ϕ2 will be denoted to indicate that there exist positive constants A, B, C, D, T such that for any t ≥ T the inequalities Aϕ1 (x, Bt) ≤ ϕ2 (x, t) ≤ Cϕ1 (x, Dt) hold uniformly in Ω. Proposition 2.9.2. Let ϕi , i = 1, 2 be Musielak-Orlicz functions on Ω ⊂ Rn with ϕ1 ∼ ϕ2 . Then mϕ1 = mϕ2 . Proof. The proof is elementary: For any t > 0, u > 0 it follows from Definition 2.9.2 that uniformly in Ω one has: ϕ2 (x, tu) ϕ1 (x, Dtu) ϕ2 (x, B −1 tu) ≤ CA−1 ≤ CA−1 . ϕ2 (x, u) ϕ1 (x, Bu) ϕ2 (x, BD−1 u) The claim follows from the last string of inequalities and Definition 2.9.1. It is well known (and the proof is elementary, see [78]) that the finiteness of the Matuszewska-Orlicz index of an Orlicz function ϕ is equivalent to the validity of the ∆2 condition for ϕ. The following example shows that this is not so in the general case of a Musielak-Orlicz function ϕ on a domain Ω. Example 2.9.3. Let Ω stand for the Euclidean unit ball in R6 and for n ∈ N, let Bn be the ball of radius 2−n−2 centered at xn = (2−n , 0, 0, 0, 0, 0), denote the ball concentric with Bn of radius 2−n−3 with Bn− and set Bn+ = Bn \ Bn− . For each n ∈ N choose zn = 23(n+3) and fix sn > zn such that s3n − zn2 = 3s2n (sn − zn )

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Analysis on Function Spaces of Musielak-Orlicz Type

and consider the function:  2  t φzn (t) = 3s2n (t − zn ) + zn2  3 t

if t ∈ (0, zn ) if t ∈ (zn , sn ) if t ∈ (sn , ∞).

Next, we define ϕ : Ω × [0, ∞) → [0, ∞) as ϕ(x, t) =

 t3

if x ∈ Ω \

φ (t) zn

if x ∈ Bn+ .

S j∈N

Bj+

It is a matter of course to verify that ϕ is an Musielak-Orlicz function on Ω; in addition, we claim that ϕ does not fulfill the ∆2 condition. Indeed, for any x ∈ Bn+   ϕ(x, zn ) = ϕ(x, 23(n+3) ) = φ23(n+3) 23(n+3) = 26(n+3) , whereas ϕ(x, 2zn ) = ϕ(x, 23n+10 ) = φ23(n+3) 23n+10




≥ 3s2n (23n+10 − 23(n+3) ) + 26(n+3) ≥ 23(n+3) (3s2n + 23(n+3) ) = ϕ(x, zn )

3s2n + 23(n+3) . 23n+9

(2.53)

Since sn > 23n+9 and, by definition sn → ∞ as n → ∞, inequality (2.53) shows that ϕ fails the ∆2 condition, as claimed. Yet, it is easy to show that the Matuszewska index of ϕ is equal to 3 in Ω and that ϕ(x, t) −→ t2 as x −→ (0, 0, 0, 0, 0, 0), on Bn+ ϕ(x, t) −→ t3 as x −→ (0, 0, 0, 0, 0, 0) on Ω \

[

Bj+ .

j∈N

A simple calculation reveals that in the preceding example the limit in Definition 2.9.1 is not uniform in Ω. Lemma 2.9.3. Let Ω ⊂ Rn be a bounded domain and ϕ an MO function as described above. If the convergence to the limits (2.51) and (2.52) is uniform in Ω, then ϕ satisfies the ∆2 condition. Proof. Fix δ > 0, then, owing to the uniformity of (2.52), one has for some T0 > 1 and any t ≥ T0 : tm(x)−δ < M (x, t) < tm(x)+δ

Musielak-Orlicz spaces

155

uniformly in Ω. By definition of M (x, t) and by virtue of the uniformity assumption of the infimum (2.51), there exists a positive number N for which, uniformly for t ≥ T0 and x ∈ Ω, it holds that sup s≥N

ϕ(x, st) < tm(x)+δ . ϕ(x, s)

In particular, for all s ≥ N : supΩ m(x)+δ

ϕ(x, sT0 ) ≤ T0

ϕ(x, s);

by virtue of Lemma 2.2.11 it is clear that ϕ satisfied the ∆2 condition.

2.10

Historical notes

“The theorem commonly known as Parseval’s Theorem, which in its latest form as extended by Fatou, asserts that if f (x) and g(x) are two functions whose squares are summable and whose Fourier constants are an , bn and αn , βn , then the series ∞ X 1 a0 α0 + (an αn + bn βn ) 2 n=1 converges absolutely and has for its sum Z 1 π f (x)g(x) dx, π −π must be regarded as one of the most important results in the whole of the theory of Fourier series.” Such is the opening paragraph in Young’s work “On a class of parametric integrals and their application to the theory of Fourier Series”, [104]. The author then proceeds to generalize the above result to the case when f ∈ Lp ((−π, π)), g ∈ Lq ((−π, π)) with p and q H¨older conjugate exponents and “the series in question is summed in the C´esaro way.” Far from being our intention to discuss the details of this work, we call the reader’s attention to [104, Paragraph 7]. The following observation (whose proof is elementary) is found there: If p > 0, sp = r, then for U > 0, V > 0 1

(1 + p)U V ≤ pU 1+ p + V 1+p .

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Analysis on Function Spaces of Musielak-Orlicz Type

About a year later, in [105] the author recalls this result by observing that it can be easily generalized in te following fashion: If U : R −→ R is increasing and positive with inverse V , then Z u Z uv − aU (a) ≤ U (t) dt + a

v

V (t) dt.

U (a)

As an obvious application, the author points out almost in passing what constitutes perhaps the first public appearence of what decades later came to be known as H¨ older’s inequality for Orlicz spaces: If u, v are functions of x and u ln u and ev are integrable, then so is uv. In this latter work, Young explicitly directs his attention to the class of functions u for which U (u) is integrable, of which class the set of functions in Lp is a particular case. Moreover, he suggests that generalizations of the known integration results are possible for functions in this wider class. A further line of research in the second decade of the twentieth century inevitably led back to the idea introduced by Young. In 1915, de la Vall´ee Poussin [25] arrived to the class of “functions a function of which is integrable” while dealing with interchanging limits and integrals. More specifically (see [25, Theorem VI]) let E ⊆ R be a Borel-measurable set and a sequence of non-negative functions (fj ) ⊂ L1 (E) such that lim fn (x) = f (x)

n→∞

for every x ∈ E. Then f ∈ L1 (E) and Z Z lim fn dx = f dx n→∞

E

E

provided there exists a non-decreasing function ψ : [0, ∞) → [0, ∞) , lim ψ(x) = ∞ ξ→∞

for which sup n

 Z 

E

  fn ψ(fn )dx < ∞. 

The latter leads to what came to be known as the de la Vall´ee Poussin-Lemma. A family of measurable random variables F = (Xα )α ⊆ L1 (Ω) on a probability space (Ω, A, µ) is said to be uniformly integrable iff      Z  lim sup |g| dµ = 0. a→∞ g∈F     |g|>a

Musielak-Orlicz spaces

157

Theorem 2.10.1. In the terminology of the preceding paragraph, F is uniformly integrable if and only if there exists a convex, even function ϕ : R −→ R with ϕ(0) = 0, limx→∞

ϕ(x) x

= ∞ and Z sup ϕ(|g|) dµ < ∞. g∈F Ω

The interested reader might refer to [57] for the proof of this theorem and further details. These early versions of what would later be called Orlicz spaces reappear in 1926, in an article by W. Young [106]. In this work, Young specifically mentions that this enhanced class of functions includes as particular cases the family of measurable functions u for which up is integrable for some exponent p : 1 ≤ p < ∞, as well as the class of those functions u for which u ln ln ... ln u is integrable. In the second half of 1930 the field received new impetus with the publication of the work by Birnbaum and Orlicz [13]. By writing p

M (u) = |u|p , N (v) = |v| p−1 , the authors point out that the classical H¨older’s inequality can be viewed as the assertion that if f and g are measurable on the interval (0, 1) and both 1

Z

M (f )dx < ∞

(2.54)

N (g)dx < ∞

(2.55)

0

and Z

1

0

then Z

1

f gdx < ∞

(2.56)

0

and that conversely, if (2.56) holds for all f satisfying (2.54), then g must a fortiori fulfill (2.55). Next, the authors go on to challenge the necessity of confining themselves to the specific forms (2.54) and (2.55) for M and N , and open the question of what other functions M and N would be admissible to guarantee the validity of the above conditions. Birnbaum and Orlicz devote the rest of this 67-page long article to settle the issue. In Theorem 6 in Chapter I and Theorem 3 in Chapter II, necessary and sufficient conditions are given for the existence of a conjugate function to a suitable function M (we refer the reader to the actual article for the specifics) and Chapter 3 is devoted to the carrying over of further properties of conjugate functions, properties

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Analysis on Function Spaces of Musielak-Orlicz Type

known at the time to hold for the Lp setting. Though much of the modern terminology is still not included in their work, in the process, they unleashed a number of ideas that would impregnate the field to the very day the present book is being written. At this juncture the field was already mature for further development. In 1932, almost in tandem with [13], W. Orlicz published a research article on “se¨ quences with variable exponents,” as he himself calls them (Uber konjugierte Exponentenfolgen, [85]). To begin his analysis he adduces the fact that ”in the Theory of Orthogonal expansion one encounters series which satisfy ∞ X

|bν |β < ∞

ν=1

but in which the exponents are not all equal.” Following this line of thought he defines the `p spaces (for p > 1) and proves the corresponding version of H¨ older’s inequality for conjugate exponents. As an application of other theorems in this work, he corrects the following statement made by S. Banach in [8]: Theorem 2.10.2. Let (n )n be a sequence of positive real numbers converging to 0. Then, there exists a continuous function u ∈ C([−π, π]) with Fourier expansion ∞ a0 X + (an cos nx + bn sin nx) 2 1 such that

∞ X

|an |2−n + |bn |2−n




1

diverges. Theorem 30 (Satz 30 ) in [85] shows that this is in general not the case. In fact, Theorem b in [8] sates that if 0 < λn −→ ∞ as n −→ ∞, then one can find an integrable function u with Fourier coefficients (an )n and (bn )n for which the series ∞ X

|an |λn + |bn |λn




n=1

diverges. Based on a number of results on series with variable exponent, Orlicz goes further and generalizes this theorem by observing that the particular choice of the trigonometric system is immaterial: any orthonormal system of uniformly bounded functions would do and that the condition on (λn ) can be weakened to require that ∞ X k λn n=1

Musielak-Orlicz spaces

159

converges for every real number k : 0 < k < 1 (Theorem 8). The second section in [85] should inevitably attract the attention of the specialist interested in the history of Musielak-Orlicz spaces. In what seems to be the first reference to variable exponent Lebesgue spaces, Orlicz defines “f (x) being integrable with α(x) > 1” (straight translation from the German original) for two measurable functions f and α by requiring Z

1

|f (x)|α(x) dx < ∞.

0

“A variety of results about integrable functions with constant α,” in Orlicz’ own words, “can be generalized to certain classes of functions that are integrable with a function α.” Putting his words into action he sets out to prove several theorems that are now part of the standard theory of Lebesgue spaces with variable exponent. [85, Theorem 4] is essentially the completeness of Lα(·) , [85, Theorem 5] relates to the weak-compactness of the unit ball in Lα(·) and [85, Theorem 6] is the generalized H¨older’s inequality. Finally, in 1932, Orlicz formally defines the spaces LM in the splendorous elegance of its modern form [86]. The opening paragraph of this work might add to the mathematical understanding of the history of the emergence of Orlicz spaces in Analysis. As Orlicz puts it: The problem concerning the generalization of the concept of conjugate functions poses the question of the extent to which theorems whose validity is known for the class S α (functions whose αth power is integrable) can be carried over to other classes of functions. Thus, the question of whether a given functional class can be regarded as a B-space1 of similar character to that of S α plays an important role in Functional Analysis. A series of Orlicz’ works throughout the 1930s [85, 87, 88, 89, 90] reflects the attention given to problems arising in the Theory of Fourier series at the time. The last article in the sequence is of particular interest from the point of view of the history of Orlicz spaces. In [90] the author sets out to characterize the sequences of real numbers that are actually Fourier coefficients of functions in some class. As Orlicz himself points out [90], the consideration of functions in the class L2 made this characterization extremely difficult. On the spot he invokes the spaces in [85] and proceeds to prove several theorems aiming at characterizing the collection of sequences of real numbers that are Fourier coefficients of functions in (among other spaces) LM . During the 1950 the mathematical importance of these “LM ” spaces as well as their natural role as generalizations of the Lebesgue spaces that had been introduced by Riesz [93, 94], seem to have been realized and accordingly, considerable amount of research was devoted to them: gradually these spaces start to emerge not only as examples or auxiliary elements, as had generally been hitherto the case. Mazur, Musielak and Orlicz, specifically were at this point interested in these spaces per se. Are the LM -spaces linear? What 1 Orlicz

is using the expression“B-space” for what is known today as a Banach space.

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Analysis on Function Spaces of Musielak-Orlicz Type

conditions on M would make LM an F -space? It is only natural that these questions were first posed and answered in the discrete situation [80] and almost immediately thereafter in the general measure-theoretic realm [81]. It is worthwhile noticing, however that the classes of functions introduced in the research mentioned in the preceding paragraph did not include the variableexponent Lebesgue spaces that had been mentioned in [85]. There seems to be general agreement in the field about the fact that the first abstract generalization of the variable-exponent Lebesgue spaces introduced in [85] appeared in 1950 and is due to Nakano [83], who included such generalization in a rather arcane work on part of which he claims to have been working during the Second World War. It is plausible that Nakano had introduced these spaces in an unpublished work he mentions among the bibliographic references in [83]. It looks in retrospective that the introduction of some Sobolev version of Orlicz spaces was imminent. Inevitably, this finally happened in the mid sixties. According to Adams [2], Dankert [22] and Donaldson obtained the first Sobolev-type embedding theorems for Sobolev-Orlicz spaces. In 1971 this line of work was continued by Donaldson and Trudinger [30]. The systematic study of Sobolev spaces based on the Nakano class had to wait a little longer and seems to have been initiated by Hudzik in the mid seventies [45, 44, 46, 47, 48, 50, 51]. In 1983 Musielak published his celebrated work [82] that was bound to become the standard reference for Musielak-Orlicz spaces (the name with which the Nakano class came to be known ever since) to this day.

Chapter 3 Sobolev spaces of Musielak-Orlicz type

3.1

Sobolev spaces: definition and basic properties . . . . . . . . . . . . . . . . . 3.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Duality of Sobolev spaces of Musielak-Orlicz type . . . . . . . . . . . . . . 3.4 Embeddings, compactness, Poincar´e-type inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

161 164 165 166 166

Sobolev spaces: definition and basic properties

In this chapter we retain the notation introduced in Chapter 1 but unless otherwise indicated, Ω ⊂ Rn will stand for an open set and we will consider the measure space (Ω, B, µ) where B stands for the Borel σ-algebra of subsets of Ω, µ denotes the restriction of the Lebesgue measure to the Borel σ-algebra of Ω and ϕ : Ω × [0, ∞) −→ [0, ∞) denotes an M O-function on Ω. Recall (Definition 2.1.2) that the M O-function ϕ is said to be locally integrable if for any t > 0 and any W ∈ A with µ(W) < ∞, Z ϕ(x, t) dx < ∞. W

Lemma 3.1.1. Let ϕ be an M O function on the domain Ω ⊆ Rn . If γ = ess inf ϕ(x, 1) > 0, x∈Ω

(3.1)

then every function in the M O space Lϕ (Ω) is locally integrable in Ω. Proof. It follows from the uniformity condition (3.1) and convexity that for any s ≥ 1 and a.e. x ∈ Ω, 1 1 0 < γ ≤ ϕ(x, s ) ≤ ϕ(x, s). s s 161

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Analysis on Function Spaces of Musielak-Orlicz Type

We underline the fact, due to convexity, that for a fixed x ∈ Ω and any measurable v ∈ Lϕ (Ω) for which |v(x)| < 1 it holds that: ϕ (x, |v(x)|) ≤ |v(x)|ϕ (x, |v(x)|) . Therefore, for any compact Ω0 ⊂ Ω and any u ∈ Lϕ (Ω):   Z Z Z |u(x)|   |u(x)| dx =  + dx  kukϕ kukϕ Ω0

{x:|u(x)|/kukϕ ≥1}

Z ≤



{x:|u(x)|/kukϕ n, W01,p (Ω) embeds compactly in C(Ω) (and hence in Lq (Ω) for any q ∈ (1, ∞)). Finally, W01,n (Ω) is compactly embedded in Lq (Ω) for any q ∈ [1, ∞). Moreover, for any p ∈ (1, ∞), the embedding W01,p (Ω) ,→ Lp (Ω)

(3.2)

is compact. In what follows, we aim at a Musielak-Orlicz version of the embedding (3.2). Specifically, we undertake the task of finding conditions on a MusielakOrlicz function ϕ that guarantee the compactness of the embedding W01,ϕ (Ω) ,→ Lϕ (Ω).

(3.3)

In the particular case of Orlicz spaces, that is when ϕ is independent of x related results were obtained by [22, 30]. The variable exponent Sobolev embedding theorem, ϕ(x, t) = tp(x) was anounced in [60, Theorem 3.10] (see also [66] for a detailed proof.) Example 3.4.1. We define the Musielak-Orlicz function ϕ : (0, 1) −→ [0, ∞) as ϕ(x, t) =

( 10n+1 10n+9 1 t if x ∈ [ n+1 , 10n(n+1) ] ∪ [ 10n(n+1) , n1 ] 10n+1 10n+9 tn if x ∈ [ 10n(n+1) , 10n(n+1) ]

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Analysis on Function Spaces of Musielak-Orlicz Type

and consider the sequence (un ) defined by  10n+1 10n+9  1 for x ∈ [ 10n(n+1) , 10n(n+1) ]     1 1 10n+1 un (x) = 10n(n + 1) x − n+1 if x ∈ [ n+1 , 10n(n+1) ] 
 −10n(n + 1) x − 1 if x ∈ [ 10n+9 , 1 ] n

10n(n+1) n

Example 3.4.2. Here we set ϕ : Ω = (0, 1) −→ [0, ∞) defined by ϕ(x, t) =

( 1 t2 if x ∈ [ n+1 + 1 2 n5 t

if x ∈

1 1 1 10n(n+1) , n − 10n(n+1) ] 1 10n+1 10n+9 [ n+1 , 10n(n+1) ] ∪ [ 10n(n+1) , n1 ].

In this case we define the sequence  10n+1 10n+9  n for x ∈ [ 10n(n+1) , 10n(n+1) ]     1 1 10n+1 2 un (x) = 10n (n + 1) x − n+1 if x ∈ [ n+1 , 10n(n+1) ] 
 −10n2 (n + 1) x − 1 if x ∈ [ 10n+9 , 1 ] n

10n(n+1) n

In both examples above, no subsequence of (un ) is a Cauchy sequence in Lϕ ((0, 1)), which implies that no subsequence can converge in the latter space. Yet, it is readily verified that both sequences are bounded in W01,ϕ ((0, 1)). We highlight the fact that in Example 3.4.1, the Matuszewska-Orlicz index (see Section 2.9) of ϕ is unbounded on Ω, whereas in Example 3.4.2, the Matuszewska-Orlicz index is identically equal to 2 on Ω. As we shall see, these observations are by no means incidental, for the behavior of the MatuszewskaOrlicz index in fact determines the validity of the Sobolev Embedding (3.3). In preparation for the main result in this section we prove: Lemma 3.4.3. If ϕ is an M O function for which the limits Mϕ (x, t) = lim sup u→∞

and

ϕ(x, tu) ϕ(x, u)

(3.4)

ln Mϕ (x, t) ln Mϕ (x, t) = inf (3.5) t>1 ln t ln t are uniform on Ω, then, there are constants C1 > 1, C2 > 1 and S0 > 1 for which ϕ(x, C1 s) ≤ C2 ϕ(x, s) f or s ≥ S0 . mϕ (x) = lim

t→∞

Sobolev spaces of Musielak-Orlicz type

169

Proof. A straightforward calculation shows that if δ > 0 then there exists a constant C1 > 1 for which t ≥ C1 implies M (x, t) < tm(x)+δ ; the assumed uniformity of the limit yields the existence of S0 > 1 for which sup s≥S0

ϕ(x, ts) 1 [ sup m(x)+δ] < tm(x)+δ + C1x∈Ω ϕ(x, s) 2

(3.6)

whenever s ≥ S0 , t ≥ C1 ; in particular, setting t = C1 in (3.6) one easily sees that for s ≥ S0 it holds that ϕ(x, C1 s) ≤

sup m(x)+δ] 3 [x∈Ω C1 ϕ(x, s), 2

whence the lemma follows immediately. Theorem 3.4.4. Let Ω ⊂ Rn be a bounded domain and ϕ : Ω × [0, ∞) −→ R a locally integrable Musielak-Orlicz function. Assume (i) ess inf ϕ(x, 1) > 0. x∈Ω

(ii) The limits (3.4) and (3.5) are uniform on Ω. (iii) The Matuszewska-Orlicz index mϕ is the restriction to Ω of a continuous function m ˜ defined on the closure of Ω. (iv) 1 < m− := inf mϕ . Ω

(v) There exists a function β : (0, ∞) −→ (0, ∞) such that the inequality ϕ(x, t) ≤ β(t) holds uniformly in Ω.

(3.7)

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Analysis on Function Spaces of Musielak-Orlicz Type

Then the embedding W01,ϕ (Ω) ,→ Lϕ (Ω)

(3.8)

is compact. Proof. Assume first m− < n. We remark the technical point that if γ satisfies the inequalities m− (n − m− ) 0 . n − m− + γ 2 n − m−

Set µ :=

1 m− . 2 n − m−

Let r > 0 be small enough to guarantee the inequality n(n − r) > n + 2r r

(3.9)

and set   < min{µ, 5r} , 0 < γ < min

 m− (n − m− ) m− − 1 , , 30 2n − m− 2

 .

(3.10)

Then, obviously w(x) := m(x) − γ < m(x) < m(x) − γ +

  = w(x) + . 20 20

The uniformity conditions (3.5) thus yield the existence of γ satisfying the second inequality in (3.10) and of a constant T0 > 1 such that t ≥ T0 implies that the following inequality holds uniformly for all t ≥ T0 , x ∈ Ω: tm(x)−γ = tw(x) < M (x, t) < tm(x)−γ+/20 = tw(x)+/20 . The uniformity of the infimum (3.5) with respect to t and x yields a positive number S0 (without loss of generality it can be assumed S0 > 1) such that, for all (x, t) ∈ Ω × [0, ∞), one has, for t ≥ T0 > 1 and any δ such that γ 1. By assumption and by virtue of Tietze’s extension theorem, w is the restriction to Ω of a continuous function p : Rn −→ [w− , w+ ]. Let p1 = w− . For k > 1 if pk−1 < n, set pk =

npk−1  − . n − pk−1 5

Clearly, if pj−1 < n, pj > p1 + (j − 1) 45  and pj−1 < pj . Let J be the first subindex for which pJ > n − 2r , where r is as in (3.9). Let I = [w− , w+ ] and    w− + 1 np1  Ω1 := p−1 , − ∩I 2 n − p1 10 and for 1 < k ≤ J − 1 set −1

Ωk := p



npk  pk , − n − pk 10



 ∩I ;

furthermore, define ΩJ and ΩJ+1 as ΩJ := p−1 ((n − r, n + r) ∩ I)    r ΩJ+1 := p−1 n + , ∞ ∩ I 2 and let (χk )1≤k≤J+1 be a partition of unity subordinated to the cover (Ωk )k of Ω. A straightforward argument shows that if v ∈ C0∞ (Ω) (which can be considered extended by 0 to Rn ) then, for each k : 1 ≤ k ≤ J + 1, vχk ∈

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Analysis on Function Spaces of Musielak-Orlicz Type

C0∞ (Ω ∩ Ωk ). It follows from this observation that if v ∈ W01,w (Ω), then for each fixed k, vχk ∈ W01,w (Ω ∩ Ωk ) . Fix a sequence (uj ) bounded in W01,ϕ (Ω); then inequalities (3.12) in concert with a simple calculation imply that (uj ) is bounded in W01,w (Ω). We contend that (uj χk )j is bounded in W01,pk (Ω), for any subindex k : 1 ≤ k ≤ J − 1. Denote the indicator function of any set A by IA . Then by construction wk := wIΩ∩Ωk ≥ pk IΩ∩Ωk so the embedding 1,pk IΩ∩Ωk

W01,wk (Ω) ,→ W0

(Ω)

(3.13)

is continuous, that is, for some positive constant C kuj χk k

1,pk IΩ∩Ω

W0

k

(Ω)

≤ Ckuj χk kW 1,wk (Ω) . 0

On the other hand, if Fkj stands for any uj ∇χk Z Z |Fkj | pk = 1= kFkj kpk Ω

of the functions uj χk , (∇uj ) χk or |Fkj | kFkj kp

Ω

and with the same token Z 1=

|Fkj | kFkj kw

k

Ω

w k Z =

k

pk IΩ∩Ω k

|Fkj | kFkj kw

Ω

k

w

(3.14)

the two preceding observations and (3.13) yield kFkj kpk = kFkj kpk IΩ∩Ωk ≤ CkFkj kwk = kFkj kw , whence it follows the contention. Hence, by vitue of Theorem 3.4.2 there is no loss of generality in assuming npk −  that (uj χk )j converges in L n−pk 20 (Ω). For simplicity, let qk be the rightendpoint of p(Ωk ) for 1 ≤ k ≤ J. Next, if 1 ≤ j ≤ J, set       dj := qj + IΩj + w+ + IΩ\Ωj . 20 20 Then dj ≥ w +

 20

for all x ∈ Ω and one has the continuous embedding 

Ldj (Ω) ,→ Lw+ 20 (Ω).

(3.15)

For any function u ∈ W01,ϕ (Ω) and 1 ≤ k ≤ J − 1: Z Z Z Z npk  −  dk d k IΩ k qk + 20 |uχk | = |uχk | = |uχk | = |uχk | n−pk 20 . Ω

Ω

Ω

Ω

(3.16)

Sobolev spaces of Musielak-Orlicz type

173

The preceding string of inequalities yields the following observation: npk

−



If (uj χk )j is a Cauchy sequence in L n−pk 20 (Ω), 1 ≤ k ≤ J − 1 then it is  convergent in Ldk (Ω) and by virtue of (3.15), (uj χk )j converges in Lw+ 20 (Ω). We claim that the latter observation yields the convergence of (uj χk )j in Lϕ (Ω) for 1 ≤ k ≤ J − 1. Indeed, there is no loss of generality by assuming that (uj χk )j converges pointwise a.e; on the other hand: Z ϕ(x, |uj (x) − ui (x)|χk (x)) dx =

(3.17)

Ω

Z ϕ(x, |uj (x) − ui (x)|χk (x)) dx+ {x:|uj (x)−ui (x)|χk (x)≤S0 }

Z ϕ(x, |uj (x) − ui (x)|χk (x)) dx. {x:|uj (x)−ui (x)|χk (x)>S0 }

Since for any fixed x ∈ Ω ϕ(x, ·) is nondecreasing, the integrand in the first term above satisfies the inequality ϕ(x, |uj (x) − ui (x)|χk (x)) ≤ ϕ(x, S0 ). The assumption of local integrability on ϕ and a straightforward application of Lebesgue’s dominated convergence yield Z lim ϕ(x, |uj (x) − ui (x)|χk (x)) dx = 0. i,j→∞ {x:|uj (x)−ui (x)|χk (x)≤S0 }

Since S0 > 1, the second integral in (3.17) is dominated by Z  |uj (x) − ui (x)|w(x)+ 20 χk (x) dx. {x:|uj (x)−ui (x)|χk (x)>S0 }

In all, (ρϕ ((uj − ui )χk )j ) −→ 0 as i, j −→ ∞. Next, we observe that for C1 as in the statement of Lemma 3.4.3 one has: ρϕ (C1 (ui − uj )χk ) = Z ϕ(x, C1 |uj − ui |χk )) dx {x:|uj (x)−ui (x)|χk (x)≤S0 }

Z +

ϕ(x, C1 |uj − ui |χk )) dx;

{x:|uj (x)−ui (x)|χk (x)>S0 }

a straightforward application of Lebesgue’s dominated convergence theorem

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Analysis on Function Spaces of Musielak-Orlicz Type

on the first integral and the consideration of Lemma 3.4.3 in the second one easily yield ρϕ (C1 (ui − uj )χk ) −→ 0 as i, j −→ ∞. It follows automatically by induction that for any l ∈ N one has ρϕ (C1l (ui − uj )χk ) −→ 0 as i, j −→ ∞, and it is concluded from here that the sequence (uj χk )j is Cauchy in Lϕ (Ω), as claimed. The remaining intervals in the covering are handled similarly: define wJ := wIΩ∩ΩJ ≥ (n − r)IΩ∩ΩJ the embedding W01,wJ (Ω) ,→ W01,n−r (Ω) is bounded. Retaining the notation of the above discussion, the sequence (uj χJ )j is bounded in W01,n−r (Ω); without loss of generality it can be considn(n−r) ered convergent in L r (Ω), which by the choice of r in (3.9) is continuously embedded in Ln+2r (Ω). Setting    h = (n + 2r)IΩJ + w+ + IΩ\ΩJ 20 

it is clear that Lh (Ω) is continuously embedded in Lw+ 20 (Ω). It follows immediately that (uj χJ )j is Cauchy in the latter space. Theorem 2.2.1 ensures now that (uj χJ )j is convergent in Lϕ (Ω). Finally, via the continuous embeddings 1,(n+ r2 )IΩJ+1 +w− IΩ\ΩJ+1

W01,ϕ (Ω) ,→ W01,w (Ω) ,→ W0

(Ω)

the boundedness of (uj χJ+1 )j in W01,ϕ (Ω) yields its boundedness in 1,n+ r2 (Ω) and by way of Theorem 3.4.2 it is readily concluded that (uj χJ+1 )j W0 can be considered convergent in C(Ω), hence convergent in Lϕ (Ω). In all, for m− < n any bounded sequence (uj )j ⊂ W01,ϕ (Ω) has a subsequence that converges in Lϕ (Ω). Since the case n ≤ m− is handled along the same lines, we only sketch the proof for this instance. For r chosen as in (3.9) we observe that there exists T0 > 1 such that uniformly on Ω and for all t ≥ T0 : r

tm(x)− 4 < M (x, t) < tm(x)+r .

Sobolev spaces of Musielak-Orlicz type

175

As before, one can conclude that given the conditions on the index, there are positive constants c1 > 1, c2 > 1 and T > 1 for which r

c1 tn− 4 ≤ ϕ(x, t) < tm(x)+r

(3.18)

uniformly in Ω, for all t ≥ T . Consider a partition of unity (χ1 , χ2 ) subordinated to the cover of Ω that consists of the open sets    r Ω1 = p−1 ((n − r, n + r) ∩ I) , Ω2 = p−1 n + , ∞ ∩ I . 2 1,n− r4

If (uj )j is a bounded sequence in W01,ϕ (Ω) (hence in W0  r q = n− IΩ1 + m− IΩ\Ω1 4

(Ω)) one can set

and along the same lines as (3.13)−(3.14) conclude that (uj χ1 )j is bounded 1,n− r4 in W0 (Ω). Via Theorem 3.4.2 and the choice (3.9) it follows that (uj χ1 )j has a subsequence that converges in Ln+2r (Ω). If t := (n + 2r)IΩ∩Ω1 + (m+ + 2r) IΩ\Ω∩Ω1 , then the obvious equality Z Z n+2r t |(ui − uj )χ1 | = |(ui − uj )χ1 | Ω

Ω

implies that the subsequence also converges in Lt (Ω) and since m + r < t in Ω it converges also in Lm+r (Ω), hence in Lϕ (Ω) via the right-hand inequality in (3.18). Still denoting this subsequence by (uj χ1 )j , it is easy to see, that (uj χ2 )j 1,n+ r2 is bounded in W0 (Ω); therefore from Theorem 3.4.2 it is clear that it has a subsequence (still denoted by (uj χ2 )j ) that converges in Lm+2r (Ω). The right-hand inequality in (3.11) yields the convergence of (uj χ2 )j in Lϕ (Ω). A straightforward computation reveals that the above conclusion implies the compactness of the embedding (3.8) in all cases. It is apparent from the proof of the preceding theorem functions in W01,ϕ (Ω) belong to a higher order integrability space than just Lϕ (Ω): We state this important fact as a separate corollary: Corollary 3.4.5. For an M O function ϕ on Ω that fulfills the conditions of Theorem 3.4.4, the embedding 

W01,ϕ (Ω) ,→ Lm(x)+ 20 (Ω) $ Lϕ (Ω) is compact. The following corollary generalizes the Poincar´e Inequality to the setting of MO spaces.

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Analysis on Function Spaces of Musielak-Orlicz Type

Corollary 3.4.6 (Poincar´e-type Inequality). For ϕ satisfying the conditions of Theorem 3.4.4, there exists a positive constant C depending only on n, Ω, ϕ such that for any u ∈ W01,ϕ (Ω) kukϕ ≤ Ck|∇u|kϕ . Proof. If not, it would be an elementary matter to construct a sequence (vk ) ⊂ W01,ϕ (Ω) with kvk k1,ϕ = 1 ≥ kvk kϕ ≥ kk|∇vk |kϕ for k ∈ N. Clearly, |∇vk | −→ 0 in Lϕ (Ω)

(3.19)

as k −→ ∞ and the compactness of the Sobolev embedding yields the existence of v ∈ Lϕ (Ω) for which vk −→ v in Lϕ (Ω). Necessarily then, kvk − vj k1,ϕ = kvk − vj kϕ + k∇ (vk − vj ) kϕ −→ 0 as k, j −→ ∞; it follows that (vk )k converges in W01,ϕ (Ω) and it is obvious that the limit must be v. On the other hand, (3.19) forces ∇v = 0 and hence v = 0, which is a contradiction. Remark 3.4.7. A quick examination of the counterexamples presented before Theorem 3.4.4 reveals that in Example 3.4.1, the Matuszewska index p (being unbounded on (0, 1)) is not the restriction to Ω = (0, 1) of any continuous function on R, whereas in Example 3.4.2 estimate (3.7) is violated. For its connection to classical analysis and its ubiquity in current applications, we single out the following particular case, first stated in [60] (see also, [66]): Corollary 3.4.8. Let Ω ⊂ Rn be a domain and p ∈ C(Ω) with 1 < p− ≤ p+ < ∞. Then the embedding W01,p (Ω) ,→ Lp (Ω) is compact and there exists a positive constant C depending only on p and Ω such that k|∇u|kϕ ≤ Ckukϕ . Proof. The proof follows immediately from the fact that under the required conditions on the exponent p, the M O function ϕ(x, t) = tp(x) satisfies the necessary hypothesis of Theorem 3.4.4.

Sobolev spaces of Musielak-Orlicz type

177

Example 3.4.3. In addition, the embedding W01,ϕ (Ω) ,→ Lϕ (Ω) is compact for the following M O functions (1 < p− = inf p(x) ≤ p+ = sup p(x) < ∞) : x∈Ω

(i) ϕ(x, t) = tp(x) (log (1 + t)) −n

(ii) ϕ(x, t) = tp(x) (1 + |x|)

(iii) ϕ(x, t) =

tp(x) p(x) ,

x∈Ω q(x)

for p, q ∈ C(Ω), 1 < p− ≤ p+ < ∞

, p ∈ C(Ω).

p ∈ C(Ω).

The above statements can be easily proved by computing the MatuszewskaOrlicz indexes of each particular case and verifying the hypotheses of Theorem 3.4.4. In what follows we present an example, based in Example 2.9.3, to the effect that the uniform convergence in the limits (2.51) and (2.52) can not be weakened. Example 3.4.4. We keep the notation of 2.9.3. In particular, we consider Ω to be the Euclidean unit ball in R6 and for n ∈ N we set Bn to be the ball of radius 2−n−2 centered at xn = (2−n , 0, 0, 0, 0, 0), denote the ball concentric with Bn of radius 2−n−3 with Bn− and write: Bn+ = Bn \ Bn− . For each h > 0, we define the function vh : R6 −→ R as follows:

  1 vh (x) = 2 −   0

|x| h

if |x| ≤ h if |x| ∈ (h, 2h) if |x| ∈ (2h, ∞).

Finally, we consider the sequence (un ) is defined then as 1

un (x) = 22(n+3) (w6 )− 3 v2−(n+3) (x − xn ), where w6 is the Lebesgue measure of the unit sphere in R6 .

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Analysis on Function Spaces of Musielak-Orlicz Type

For each n ∈ N fix zn = 23(n+3) ≈ |∇un (x)| that  2  t φzn (t) = 3s2n (t − zn ) + zn2  3 t

and recall from Example 2.9.3 if t ∈ (0, zn ) if t ∈ (zn , sn ) if t ∈ (sn , ∞),

where for each n ∈ N we have chosen zn = 23(n+3) and sn > zn is chosen in such a way that s3n − zn2 = 3s2n (sn − zn ). Then the M O function ϕ is given by: ( t3 ϕ(x, t) = φzn (t)

S if x ∈ Ω \ n Bn+ if x ∈ Bn+ .

We claim that the sequence (un ) is bounded in W01,ϕ (Ω). To this end observe that   Z Z   ρϕ (un ) =  +  ϕ (x, un (x)) dx − Bn

+ Bn

Z =

2(n+3)

ϕ(x, 2

−1 w6 3 )dx

Z +

− Bn


 −1 ϕ x, 22(n+3) w6 3 2 − 2n+3 |x − xn | dx

+ Bn



3 1

−3

5 |Bn | 22(n+3) w6

Z +

  −1 ϕ x, 22(n+3) w6 3 dx

+ Bn



2 1

−3

≤ C + |Bn | 22(n+3) w6 ≤ C1 + C2

where C1 is a positive constant independent of n and C2 → 0 as n → ∞. −1/3

Notice that for n large enough, 23(n+3)w6 > sn . A simple calculation shows that while ∇un (x) = 0 on Ω \ Bn+ , for x ∈ Bn+ one has: −1/3

|∇un (x)| ≈ 23(n+3) w6

.

Sobolev spaces of Musielak-Orlicz type

179

One derives from here that:   Z Z   ρϕ (|∇un |) =  +  ϕ (x, un (x)) dx − Bn

Z =

+ Bn



− 13


 2 − 2n+3 |x − xn | dx

ϕ x, 22(n+3) w6

+ Bn

 3 −1/3 ≈ |Bn | 22(n+3) w6 ≤C for a positive constant C, independent of n. We have shown that there are positive constants k1 , k2 such that for all n ∈ N, one has: k1 ≤ kun kϕ ≤ k2 k1 ≤ k|∇un |kϕ ≤ k2 , from which it follows that the sequence (un ) is bounded in W01,ϕ (Ω). On the other hand, each un is continuous, any two different functions in the sequence (un ) have disjoint supports and sup un → ∞ as n → ∞. Hence, no subsequence of (un ) converges in Lϕ (Ω).

Ω

Chapter 4 Applications

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Preparatory results and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compactness of the Sobolev embedding and the modular setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The variable exponent p-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Stability of the solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Γ-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The eigenvalue problem for the p-Laplacian . . . . . . . . . . . . . . . . . . . . . Modular eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence properties of the eigenvalues and eigenfunctions . . Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 186 190 198 206 210 219 228 243

In this Chapter we present some applications of the theoretical results developed in Chapters 1-3. In the interest of facilitating the reading, some definitions already given in previous sections will be repeated in the sequel.

4.1

Preparatory results and notation

A recollection of a few standard results from nonlinear functional analysis will facilitate the treatment of the variable exponent p-Laplacian, the operator to be studied in this chapter. We continue to use the notation from Chapter 1. Our presentation follows [96, 65, 66]. Let X, Y be real Banach spaces: all Banach spaces in this chapter will be assumed to be real. An operator T : X −→ Y is said to be bounded if for any bounded set A ⊂ X, the set T (A) is bounded. Recall from Chapter 1 that in the particular case when Y = X ∗ , the operator T is said to be monotone if the inequality hT (x) − T (y), x − yi ≥ 0 holds for any x ∈ X, y ∈ X. T : X → Y is called hemicontinuous if for any fixed x, y ∈ X the real valued 181

182

Analysis on Function Spaces of Musielak-Orlicz Type

function s 7−→ hT (x + sy), yi : R → R is continuous. Another class of operators is of particular importance in the sequel: Definition 4.1.1. An operator T : X −→ X ∗ is said to be of type M if, for any weakly-convergent sequence (xn ) ⊂ X with xn * x, the conditions T (xn ) * f and lim suphT (xn ), xn i ≤ hf, xi, n→∞

imply T (x) = f . Theorem 4.1.1. Let X be a reflexive Banach space and T : X −→ X ∗ be hemicontinuous and monotone. Then T is of type M . Proof. For fixed y ∈ X, (xn ), x and f as in Definition 4.1.1, the assumed monotonicity of T implies that 0 ≤ hT (xn ) − T (y), xn − yi for all n; hence hT (y), x − yi ≤ hf, x − yi . In particular, for any z ∈ X,     1 T x − z , z ≤ hf, zi ; n

(4.1)

inequality (4.1) conjunction with the hemicontinuity property of T , immediately yields hT (x), zi ≤ hf, zi for all z ∈ X. The last inequality implies T (x) = f as claimed. The next lemma is of rather technical nature; its necessity will be revealed shortly: Lemma 4.1.2. If F : Rn −→ Rn is continuous and there exists ε > 0 such that F (u) · u ≥ 0 for each u such that kuk = ε, then F has a zero.

Applications

183

Proof. Denote the unit ball in Rn by Bn . Assuming F (x) 6= 0 for each x ∈ Bn , one can define the function ψ : Bn −→ Bn by ψ(x) = −

F (εx) . kF (εx)k

This function is easily seen to be continuous. Let x0 be a fixed point of ψ whose existence is guaranteed by Brouwer’s theorem, and notice that x0 6= 0. Then F (εx0 ) · x0 − = kx0 k2 , kF (εx0 )k which is a contradiction. From the immediately preceding results, there follows the surjectivity criterion given in the next theorem: Theorem 4.1.3. Consider a separable and reflexive Banach space X and let T : X −→ X ∗ be of type M and bounded. If for some f ∈ X ∗ there exists ε > 0 such that for every x ∈ X with kxk > ε one has the inequality: hT (x), xi > hf, xi,

(4.2)

then f belongs to the range of T . Proof. Let {xk : k ∈ N} denote a basis for the linear span of a countable, dense subset Y ⊂ X. For each natural number n denote the linear span of the set {x1 , x2 , ..., xn } by Yn , that is Yn = h{x1 , x2 , ..., xn }i and consider the canonical isomorphism in : Rn −→ Yn , in (a1 , ..., an ) =

n X

ak xk .

1

We first claim that for each n ∈ N there exists un ∈ Yn satisfying hxk , T (un )i = f (xk ) for k = 1, 2, ..., n.

(4.3)

This is a direct consequence of the assumption on f and of Lemma 4.1.2, for the function Fn : Rn −→ Rn , Fn (z) = i∗n ◦ T ◦ in (z) − i∗n ◦ f

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Analysis on Function Spaces of Musielak-Orlicz Type

satisfies the conditions of Lemma 4.1.2 and the zero un thereby given satisfies the equality (4.3). Consequently, for each n, hT (un ), un i = f (un ), which by virtue of condition (4.2) implies that kun k ≤ ε. Since T is bounded, the reflexivity of X yields the existence of a subsequence (vn ) of (un ) such that vn * v ∈ X , T (vn ) * f. Since T is of type M , we conclude that T (v) = f . In what follows we recall the definition of singular numbers for operators on Banach spaces. Specifically, consider a reflexive Banach space X with norm k · kX , a Banach space Y with norm k · kY and a compact linear operator T : X −→ Y. Simple functional-analytic arguments reveal that the operator norm kT k = kT kB(X,Y ) := sup 06=x∈X

kT (x)kY kxkX

is attained in X. More precisely: Lemma 4.1.4. Given Banach spaces X and Y with X reflexive and a compact linear operator T : X → Y, there exists x0 ∈ X such that kT k = kT (x0 )kY = sup 06=x∈X

kT (x)kY . kxkX

Proof. By definition of kT k there exists a sequence (xn ) ⊂ BX with

1 for each n ∈ N; n from the reflexivity of X, (xn ) can be assumed to be weakly convergent to x0 ∈ BX ; obviously, then, T (xn ) → T (x0 ). kT (xn )kY ≥ kT k −

Invoking the compactness of T , one can extract a subsequence (yk ) of (xn ) such that (T (yk )) converges strongly in Y ; the sequence (T (yk )), hence, must necessarily converge strongly to T (x0 ) in Y . As is apparent from the choice of the sequences, it holds for each natural number n that kT k < kT (xn )kY +

1 , n

Applications

185

whence, from the continuity of the norm one has kT k ≤ kT (x0 )kY . The lemma follows at once from the latter inequality. The following section reveals the connection between the eigenvalues of a given compact operator and the maximum problem suggested by the previous lemma. Recall that a Banach space is called smooth if its norm is Gˆateauxdifferentiable at every non-zero point. We refer the reader to [12], where this connection was first observed, for further comments. Theorem 4.1.5. Let X, Y be smooth Banach spaces; assume that X is reflexive; let DX and DY stand for the gradients of the respective norms. If T : X −→ Y is a linear, compact operator (whose transpose is denoted by T ∗ ) and x0 ∈ X\{0} is maximal in the sense that kT k =

kT (x0 )kY , kx0 kX

then (T ∗ DY T )(x0 ) = kT kDX (x0 ).

(4.4)

Proof. The claim follows by direct differentiation: For fixed h ∈ X\{0} and −1 t ∈ (−kx0 kX khk−1 X , kx0 kX khkX ), set F (t) =

kT (x0 + th)kY . kx0 + thkX

By assumption, F has an absolute maximum at t = 0, and the smoothness hypothesis guarantees the differentiability of F ; moreover, as shown with a simple calculation,

0 = F 0 (0) = hDY (T (x0 )), T (h)i which immediately yields (4.4).

1 kT (x0 )kY − hDX (x0 ), hi , kx0 kX kx0 k2X

186

4.2

Analysis on Function Spaces of Musielak-Orlicz Type

Compactness of the Sobolev embedding and the modular setting

This section is devoted to obtaining some further applications of the material covered in Chapters 1 and 3. As a matter of course, we retain the notation introduced in the sequel. In particular, Ω ⊂ Rn will stand for a bounded domain and ϕ will denote an M O function on Ω (as in Definition 2.1.1). We also recall the notation for the modular engendered by ϕ: Z ρϕ (u) = ϕ(x, |u(x)|) dx. Ω

Most of the results presented in this section heavily depend on the fact that if the limits (2.51) and (2.52) are uniform, then the M O function ϕ satisfies the ∆2 condition (Lemma 2.9.3). The following technical theorem will be used in the sequel: Theorem 4.2.1. Let ϕ be an M O function on Ω; assume that ϕ satisfies the conditions of Theorem 3.4.4; in particular, ϕ satisfies the ∆2 condition 2.2.2, i.e., for some K > 0, S0 > 0 it holds that ϕ(x, 2s) ≤ Kϕ(x, s) for all s ≥ S0 , x ∈ Ω.

(4.5)

sup {ρϕ (u) : ρϕ (|∇u|) ≤ r} < ∞.

(4.6)

Then, Proof. It follows from (4.5) that for arbitrary v ∈ Lϕ (Ω) Z ρϕ (2v) = ϕ(x, 2|v(x)|)dx Ω



 Z

 =

Z +

|v| 0. Then there exists a function ur ∈ W01,ϕ (Ω) that is maximal in the following sense: ρϕ (ur ) = Sr = sup {ρϕ (u) : ρϕ (|∇u|) ≤ r} Proof. Pick a sequence (un ) in Br with Sr −

1 < ρϕ (un ). n

The sequence (un ) is bounded in W01,ϕ (Ω); indeed, Theorem 4.2.1 guarantees that the numerical sequence (ρϕ (un )) is bounded and it follows from here that either kun kϕ ≤ 1 or 1 = ρϕ (un /kun kϕ ) ≤ kun k−1 ϕ ρϕ (un ),

(4.11)

so that the boundedness of the sequence (kun kϕ ) follows from that of the sequence (ρϕ (un )) (4.6). Likewise, since by definition, (ρϕ (|∇un |)) is bounded, either k|∇un |kϕ ≤ 1 or 1 = ρϕ (|∇un |/k|∇un |kϕ ) ≤ k|∇un |k−1 ϕ ρϕ (|∇un |).

(4.12)

Inequalities (4.11) and (4.12) together with the discussions preceding them show the claimed boundedness of (un ).

Applications

189

On account of the theorem of Banach-Alaoglu and the reflexivity of W01,ϕ (Ω) (Theorem 3.3.1) there is no loss of generality by assuming that (un ) converges weakly in W01,ϕ (Ω); let un

W01,ϕ (Ω)

*

u ∈ W01,ϕ (Ω).

(4.13)

In particular, in the light of Theorem 4.2.2, statement (4.13) ensures that u ∈ Br . The compactness of the Sobolev embedding (Theorem 3.4.4) guarantees the strong convergence Lϕ (Ω)

un −→ u. In view of Sobolev’s embedding theorem (Theorem 3.4.4), one must have a subsequence (still denoted by un ) that is strongly convergent to u in the Lϕ (Ω) norm. Owing to Corollary 2.1.7 one can assume that un −→ u µ-a.e. in Ω. In fact, u is the sought-for maximal function. To see this, we notice that a.e. in Ω, ϕ(x, |un (x)|) −→ ϕ(x, |u(x)|) and that on account of convexity, for any n ∈ N: ϕ(x, |un (x)|) ≤

1 1 ϕ(x, 2|un (x) − u(x)|) + ϕ(x, 2|u(x)|). 2 2

(4.14)

Select n large enough so that 2ku − un kϕ < 1; for such n (4.14) yields:   |un (x) − u(x)| 1 ϕ(x, |un (x)|) ≤ kun − ukϕ ϕ x, + ϕ(x, 2|u(x)|). (4.15) ku − un kϕ 2 Denote the left-hand side and the right-hand side of (4.15) by vn and wn respectively. Then the following conditions hold: (i) vn (x) → v(x) = ϕ(x, |u(x)|) ∈ L1 (Ω) a.e. in Ω (ii) wn (x) → w(x) = 12 ϕ(x, 2|u(x)|) ∈ L1 (Ω) a.e. in Ω (iii) vn , wn ∈ L1 (Ω) for any n ∈ N (iv)

R Ω

wn dx →

R Ω

1 2 ϕ(x, 2|u|)dx

= 12 ρϕ (2u).

Since w − v ≥ 0 a.e in Ω, Fatou’s Lemma leads to: Z Z Z (w − v)dx ≤ w dx + lim inf (−vn ) dx n

Ω

Ω

Ω

Z

Z w dx − lim sup

=

vn dx

n Ω

Ω

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Analysis on Function Spaces of Musielak-Orlicz Type

and Z

Z (w + v)dx ≤

Ω

Z w dx + lim inf

vn dx.

n

Ω

Ω

The two last statements yield Z Z lim ϕ(x, |un (x)|)dx = ϕ(x, |u(x)|)dx n→∞

Ω

Ω

or, equivalently ρϕ (un ) −→ ρϕ (u) as n → ∞.

(4.16)

By construction ρϕ (un ) −→ Sr ; (4.16) is therefore the desired result.

4.3

The variable exponent p-Laplacian

At this stage we take a detour from the general theory to focus on the particular case of the Musielak-Orlicz function given by ϕ(x, t) = tp(x) . We devote this section to present a self contained discussion on the variable exponent p(x)-Laplacian. We base our presentation on [38]. The main result here is the proof of Theorem 4.3.4, originally due to Fan and Zhang [38]. We start with the following lemma: Lemma 4.3.1. For n ∈ N, x, y ∈ Rn and constant p,
1 (|x|p−2 − |y|p−2 )(|x|2 − |y|2 ) + (|x|p−2 + |y|p−2 )|x − y|2 2 = (|x|p−2 x − |y|p−2 y) · (x − y). Proof. This follows by straightforward calculation: (|x|2 − |y|2 )(|x|p−2 − |y|p−2 ) + |x − y|2 (|x|p−2 + |y|p−2 )
= (x − y) (x + y)(|x|p−2 − |y|p−2 ) + (x − y)(|x|p−2 + |y|p−2 ) = 2(x − y)((|x|p−2 x − |y|p−2 y)).

Let Ω ⊆ Rn be a domain and consider a Borel-measurable function p : Ω −→ [1, ∞). For u ∈ W01,p (Ω), the Dirichlet p-Laplacian ∆p (u) is defined by
∆p (u) = div |∇u|p−2 ∇u .

(4.17)

Applications

191

In weak form, ∆p is the (non-linear) operator  ∗ ∆p : W01,p (Ω) −→ W01,p (Ω) such that for all u, h ∈ W01,p (Ω), Z h∆p u, hi = −

|∇u|p−2 ∇u · ∇h dx.

Ω

A few remarks are in order. The functional F : W01,p (Ω) −→ [0, ∞)

(4.18)

defined by Z F (w) =

|∇w|p dx p

Ω

is convex, weakly lower-semicontinuous, Fr´echet-differentiable and in fact (see also [19]), F 0 = −∆p . (4.19) For the sake of completeness, a thorough proof of (4.19) will be given in Section 4.6 in a more general setting (Lemma 4.6.1 and Remark 4.6.2). When p is constant, it is obvious that the functional (4.18) is a multiple of the pth power of the Sobolev norm, namely F (w) =

1 k∇wkp . p

For variable p the consideration of the derivative of the norm as opposed to the derivative of the modular (which is essentially the definition we have adopted for the p-Laplacian) leads to a different differential operator, as will be discussed later (Section 4.5). Lemma 4.3.2. Let Ω ⊂ Rn be a bounded, Lipschitz domain and p : Ω → R a Borel-measurable function satisfying 1 < p− ≤ p(x) ≤ p+ < ∞ a.e. in Ω. Under these conditions, the operator  ∗ ∆p : W01,p (Ω) −→ W01,p (Ω) is bounded, hemicontinuous and monotone. In addition, ∆p is of type M . Proof. Let S ⊂ W01,p (Ω) be bounded, with sup{k∇ukp : u ∈ S} ≤ C.

192

Analysis on Function Spaces of Musielak-Orlicz Type

For u ∈ S and w in the unit ball of W01,p (Ω), set Z h∆p u, wi = |∇u|p−2 ∇u · ∇w dx.

(4.20)

Ω

Taking absolute values in equality (4.20) and invoking the variable exponent form of H¨ older’s inequality it is clear that   n o 1 1 ∗ sup k∆p uk(W 1,p (Ω)) : u ∈ S ≤ C 1 + − , 0 p− p+ which shows that ∆p is bounded. For the proof of hemicontinuity, fix t ∈ R. It is clear that for |s| < |t| +

1 and 1 < p ≤ 2 2

it holds that |∇(u + sv)(x)|p−1 ≤ |∇u|p−1 + |s|p−1 |∇v(x)|p−1 ,

(4.21)

whereas for p > 2 the corresponding inequality reads: |∇(u + sv)|p−1 ≤ 2p−1 (|∇u|p−1 + |s|p−1 |∇v|p−1 ).

(4.22)

On the other hand, by definition: Z h∆p (u + sv), vi = |∇(u + sv)|p−2 ∇(u + sv) · ∇v dx.

(4.23)

Ω

In view of (4.21) and (4.22) the integrand in (4.23) is bounded above by |∇u|p−1 |∇v| + |s|p−1 |∇v|p 1{1 0. Likewise, for 1 < p ≤ 2 (with the obvious provision that x 6= 0 and y 6= 0) we have (p − 1)|x − y|2 (1 + |x| + |y|)p−2 ≤ (|x|p−2 x − |y|p−2 y)(x − y),

(4.25)

which follows from (|x|p−2 x − |y|p−2 y)(x − y) Z 1
d = (x − y) (|y + t(x − y)|p−2 (y + t(x − y)) dt 0 dt Z 1 = |x − y|2 |y + t(x − y)|p−2 0 1

Z

|y + t(x − y)|p−4 ((y + t(x − y))(x − y))

+ (p − 2)

2

0

≤ (p − 1)|x − y|

2

1

Z

|y + t(x − y)|p−2 ≥ (p − 1)|x − y|2 (1 + |x| + |y|)p−2 .

0

For fixed u and v in W01,p (Ω), it follows straight from the definition of ∆p that: Z h∆p (u) − ∆p (v), u − vi = (|∇u|p−2 ∇u − |∇v|p−2 ∇v) · ∇(u − v)dx. (4.26) Ω

The desired conclusion is obtained by splitting the integral in the right-hand side of (4.26) over {x : 1 < p(x) ≤ 2} and {x : p(x) ≥ 2} and applying the inequalities (4.24) and (4.25). Lemma 4.3.3. [38, Theorem 3.1] Let Ω ⊂ Rn be a bounded, Lipschitz domain and p : Ω → R a Borel-measurable function satisfying 1 < p− ≤ p(x) ≤ p+ < ∞ a.e. in Ω. Then the operator  ∗ ∆p : W01,p (Ω) −→ W01,p (Ω) given by Z h∆p u, vi = − Ω

is a homeomorphism.

|∇u|p−2 ∇u∇v dx

194

Analysis on Function Spaces of Musielak-Orlicz Type

Proof. The surjectivity of ∆p is derived from Theorem 4.1.3, injectivity follows from Poincar´e inequality and inequalities (4.24) and (4.25). The continuity of ∆−1 p ensues from a functional-analytic argument coupled with inequalities  ∗ (4.24) and (4.25). We proceed to the details of the proof. Fix f ∈ W01,p (Ω) . For u ∈ W01,p (Ω) with   1/(p− −1) k|∇u|kp > max 1, kf k W 1,p , ∗ ( 0 (Ω)) one can write |∇u|p 1 p p dx ≤ k|∇u|kp k|∇u|kp−

Z 1= Ω

Z

|∇u|p dx;

Ω

thus for such u, Z h∆p u, ui =

|∇u|p dx ≥ k|∇u|kpp− = k|∇u|kpp− −1 k|∇u|k

Ω

> kf k(W 1,p (Ω))∗ k|∇u|kp . 0

As a direct consequence of Theorem 4.1.3 it follows that f is in the range of ∆p , hence that the latter is surjective. The injectivity of the map can be obtained as a consequence of inequalities (4.24) and (4.25) in conjunction with Poincar´e inequality. Indeed, let u ∈ W01,p (Ω) and v ∈ W01,p (Ω) satisfy ∆p (u) = ∆p (v). In this case, we estimate ρp (∇u − ∇v) as follows: Z |∇u − ∇v|p dx = Ω

Z

p

Z

|∇u − ∇v| dx + {p>2}

|∇u − ∇v|p dx.

(4.27)

{1 1 and (1 + )) > 1, then, that for i sufficiently large one has:  pi pi Z  Z  |vk | |vk | −1 dx ≤ (1 + ) dx ≤ 1. (4.65) (1 + )kvk kq kvk kq Ω

Ω

Analogously, (4.49) shows that for i sufficiently large it holds:  pi pi Z  Z  |vk | |vk | 1 < (1 − )−1 dx ≤ dx. kvk kq (1 − )kvk kq Ω

(4.66)

Ω

It follows immediately by definition of the Luxemburg norm that (4.65) implies that for large enough i : kvk kpi ≤ (1 + )kvk kq . Since the function λ → (λ−1 |vk |)pi is non-increasing, as an immediate conR

Ω

sequence of the definition of the Luxemburg norm and inequality (4.66) one derives, for i as above: (1 − )kvk kq ≤ kvk kpi . The last two statements show (4.63) and the exact same reasoning yields (4.64). On account of Proposition 1.5.7, the sequence (ui ) ⊂ Lp1 (Ω) can be considered to be convergent to u a.e. in Ω. By assumption, q− > n; it follows that for i large enough one has W01,pi (Ω) ,→ C(Ω). Then there is a positive constant C, independent of i such that kui k∞ ≤ C, whence it follows that for any λ > 0 Z Z |λui |pi dx → |λu|q dx as i → ∞. Ω

Ω

As in the proof of (4.63) and (4.64), the preceding statement leads to kui kpi → kukq as i → ∞. Consequently, −1 λi = kui k−1 as i → ∞. pi → kukq

Next, (4.63), (4.64) and the extremal character of ui yield, for any  > 0: kvk kq /k|∇vk |kq ≤ kvk kpi /k|∇vk |kpi +  ≤ kui kpi +  i sufficiently large and it follows from the last inequalities by letting i → ∞ that kvk kq /k|∇vk |kq ≤ kukq . (4.67)

Applications

215

Letting κ → ∞ in inequality (4.67) it is concluded that kvkq /k|∇v|kq ≤ kukq .

(4.68)

In particular, for v = u, (4.68) shows that k|∇u|kq ≥ 1 and since k|∇u|kq ≥ 1 one has that, in fact, k|∇u|kq = 1. The claim of the theorem follows by setting u = uq . An analogous result follows for a non-increasing sequence. Specifically, let Ω ⊂ Rn be a regular bounded domain and (pi ) ⊂ C(Ω) be a non-increasing sequence of functions converging uniformly in Ω to its infimum p ∈ C(Ω); assume that 1 < p− ≤ p+ < ∞. 1,pj

As in the setting of Theorem 4.5.1, for each j ∈ N let uj ∈ W0 for an extremal function for the Sobolev embedding 1,pj

(Ω) stand

(Ω) ,→ Lpj (Ω);

W0

we normalize by requiring that k|∇uj |kpj = 1 and notice that according to Theorem 4.1.5, uj is a solution of the eigenvalue problem (4.57) with p = pj with first eigenvalue λ = λj . Let λp be the first eigenvalue of the problem (4.57) for p = inf pj , i.e., j

λp =

inf

06=u∈W01,p (Ω)

k|∇u|kp /kukp .

Theorem 4.5.2. In the above setting, there exists an extremal function up for the embedding W01,p (Ω) ,→ Lp (Ω) and a subsequence of (pj ) (still denoted by (pj )) such that uj

W01,p (Ω)

*

Lp (Ω)

up and uj → up .

Moreover, λj → λp as j → ∞. Proof. The proof will be split into three mutually exclusive cases, namely, n ≤ p− , n ∈ (p− , p+ ] and p+ < n. We observe first that for p+ < n the inequality p(x) +

1 p2− p(x)2 np(x) < p(x) + = 2 n − p− n − p(x) n − p(x)

holds for every x ∈ Ω. The embedding W01,p (Ω)

p+ 12

,→ L

p2 − n−p−

(Ω)

216

Analysis on Function Spaces of Musielak-Orlicz Type

is compact (see [60, Theorem 3.9]). Hence no generality is lost by considering p+ 12

the sequence (ui ) strongly convergent in L

p2 − n−p−

(Ω) to some u ∈ BW 1,p (Ω) 0

W01,p (Ω)

W01,p (Ω))

(the unit ball in and that ui * u. By assumption, there exists i0 such that i ≥ i0 implies pi − p
0, one has, for sufficiently large k, i ∈ N, by virtue of the extremal character of ui : k|∇v|kp k|∇vk |kp k|∇vk |kpi ≥ − /2 ≥ − kvkp kvk kp kvk kpi k|∇ui |kpi 1 ≥ −= − . kui kpi kui kpi This inequality together with (4.69) yields for i large enough: k|∇v|kp 1 ≥ + kvkp kui − ukpi + kukpi Letting i → ∞ it is concluded 1 k|∇v|kp = 1,p inf . kukp W0 (Ω)3v6=0 kvkp In particular, by setting v = u in the above inequality it is obtained that k|∇u|kp ≥ 1 i.e. k|∇u|kp = 1. The claimed extremality of u has been proved. This settles the case p+ < n. Suppose next that n ∈ (p− , p+ ]. In the light of Theorem 3.4.4, one can infer the existence of a function u ∈ W01,p (Ω) and of a subsequence (still denoted by (ui )) of the original sequence such that ui The proof relies on the following lemma:

W01,p (Ω)

*

Lp (Ω)

u and that ui → u.

Applications

217

Lemma 4.5.3. In the above setting, u ∈ Li0 (Ω) for some i0 ∈ N. Based on this result (which for clarity we will prove later) we conclude that for i ≥ i0 one has kui kpi ≤ kui − ukpi + kukpi and kukpi ≤ kui − ukpi + kui kpi Since kui − ukpi → 0 and kukpi → kukp as i → ∞ it follows that kui kpi → kukp as i → ∞. The same argument used in the case p+ < n yields now the extremal character of u. We now proceed to the Proof of Lemma 4.5.3. To this effect we choose positive numbers , δ subject to the conditions:   p2− 1 n(n − r)  < min , , p− − r > 1 and n + r < . (4.70) n − 1 n − p− r Furthermore, define p1 = p− − δ, and inductively, if pk < n define pk =

npk−1  − . n − pk−1 5

Clearly, if pj−1 < n, pj > p1 + (j − 1) 45  and pj−1 < pj . Let J be the first subindex for which pJ > n − 2r , where r is as in (4.70). Let I = [w− , p+ ] and   w− + 1 np1  −1 Ω1 = p , − 2 n − p1 10 and for 1 < k ≤ J − 1 set Ωk = p

−1



npk  pk , − n − pk 10



 ∩I ;

ΩJ = p−1 ((n − r, n + r))    r ΩJ+1 = p−1 n + , ∞ x . 2 It can be readily verified that (Ωk )k is an open cover of Ω; let (χk )1≤k≤J+2

218

Analysis on Function Spaces of Musielak-Orlicz Type

be a partition of unity subordinated to the cover (Ωk )k . For 1 ≤ k ≤ J − 1 the sequence (ui χk )i ∈ W01,p (Ω) ,→ W01,pk (Ω) can npk

without loss of generality be considered to be convergent in L n−pk this is due to the compactness of the embedding npk

W01,pk (Ω) ,→,→ L n−pk

 − 20

 − 20

(Ω);

(Ω).

Thus, npk

ui χk

L n−pk

−  20

−→

(Ω)

npk

vk ∈ L n−pk

 − 20

(Ω).

p

(4.71) p

L (Ω)

L (Ω)

It is easy to see that from ui −→ u it follows that ui χk −→ uχk . Moreover,   Z Z Z   p |vk |p dx =  +  |vk χk | dx Ω

{|vk | 0 for t > 0 and a.e. x ∈ Ω. ∂t

(4.88)

Then there exists at least one (weak) solution (λ0 , u0 ) to the problem   ∂ ∂ −1 div |∇u(·)| ∇u ϕ(·, |∇u(·)|) = λ|u(·)|−1 u ϕ(·, |u(·)|). ∂t ∂t To facilitate the proof of Theorem 4.6.5 we digress on a Lagrangemultiplier-type discussion. Let F and G be the functionals introduced in Lemma 4.6.1 and Remark 4.6.2. For any r > 0, we recall that Theorem 4.2.3 yields the existence of at least ur ∈ W01,ϕ (Ω) such that F (ur ) = ρϕ (ur ) = Sr = sup {ρϕ (u) : ρϕ (|∇u|) = G(u) ≤ r} .

Applications

225

Lemma 4.6.6. In the notation of Theorem 4.2.3, Lemma 4.6.1 and Remark 4.6.2 it follows that hG0 (ur ), ur i > 0.

(4.89)

Proof. Because of Remark 4.6.2, it is clear that Z ∂ϕ hG0 (ur ), ur i = (x, |∇ur (x)|)|∇ur (x)|dx; ∂s Ω

since G(ur ) =

R

ϕ(x, |∇ur (x)|)dx = r > 0 one must have ϕ(x, |∇ur (x)|) > 0

Ω

a.e. x ∈ Ω. Thus, |∇ur (x)| > 0 a.e. ∈ Ω. The claim now follows on account of assumption (4.88). Lemma 4.6.7. Under the assumptions of Lemma 4.6.6, it follows that W01,ϕ (Ω) = ker G0 (ur ) ⊕ h{ur }i.

(4.90)

Proof. The proof is elementary: Any v ∈ W01,ϕ (Ω) can be written as v = v − (hG0 (ur ), vi/hG0 (ur ), ur i) ur + (hG0 (ur ), vi/hG0 (ur ), ur i) ur and a straightforward calculation shows this decomposition to be unique. Next, set ω : ker G0 (ur ) ⊕ R −→ [0, ∞) ω(h, t) = G ((1 + t)ur + h) − r. By definition, it is immediate that w(0, 0) = 0. We claim that w is differentiable in both variables, that ∂w ∂t (0, 0) > 0 and that ∂w ∂h (0, 0) = 0. The differentiability of w is clear from the differentiability of G; from Remark 4.6.2 and from Lemma 4.6.6, it follows that ∂w (0, 0) = hG0 (ur ), ur i > 0. ∂t The last assertion follows by direct computation, namely, for any h ∈ ker G0 (i.e., hG0 (ur ), hi = 0) one has: w(h, 0) − w(0, 0) G(ur + h) − G(ur ) = w(h, 0)/khk1,ϕ = khk1,ϕ khk1,ϕ G(ur + h) − G(ur ) − hG0 (ur ), hi = khk1,ϕ −→ 0 as khk1,ϕ → 0.

226

Analysis on Function Spaces of Musielak-Orlicz Type

Lemma 4.6.8. For w as above, the function ∂w ∗ : ker G0 (ur ) ⊕ R −→ (ker G0 (ur )) ∂h is continuous. Proof. By definition, one has, for η ∈ ker G0 (ur ): 0 ∂w (h0 , t0 )(η) = G (h0 + (1 + t0 )ur )(η) ∂h Z ∂ϕ ∇u0 (x)∇η(x) (x, |∇u0 (x)|) dx. ∂s |∇u0 (x)|

Ω

The continuity claim follows immediately from Theorem 2.7.2, and repeated applications of H¨ older’s inequality and Lebesgue’s dominated convergence theorem, as in Lemma 4.6.3. In all, the implicit function theorem now applies to w. Accordingly, there exist a neighborhood of zero U in W01,ϕ (Ω),  > 0 and a differentiable function ψ : U ∩ ker G0 (ur ) −→ (−, ) such that for any (a, b) ∈ U ∩ ker G0 (ur ) × (−, ) one has w(a, b) = 0 ⇐⇒ b = ψ(a). Pick h ∈ ker G0 , let δ > 0 be small enough so that th ∈ U ∩ ker G0 (ur ) for |t| < δ and set b : (−δ, δ) −→ W01,ϕ (Ω) b(t) = ur + th + ψ(th). Then b(0) = ur , b is differentiable at 0 (since so is ψ) and b0 (0) = h, since ψ 0 (0) = 0. Then the function ξ : (−δ, δ) −→ [0.∞) ξ(t) = F (ur + th + ψ(th)) is differentiable and attains a maximum at t = 0. It follows then that ξ 0 (0) = F 0 (ur )(h) = 0 and from here, by assumption on h, that ker G0 ⊆ ker F 0 .

Applications

227

Since both, ker G0 and ker F 0 have codimension one and neither functional is identically zero, it is concluded that ker G0 (ur ) ⊆ ker F 0 (ur ), and thus that there must exist a constant λ ∈ R such that G0 (ur ) = λF 0 (ur ).

(4.91)

Proof of Theorem 4.6.5: The proof follows from Theorem 4.2.3, Lemma 4.6.1 and (4.91). On account of Theorem 3.4.4, 4.6.1 and corollary 4.6.4, we have the following corollary: Corollary 4.6.9. Let Ω ⊂ Rn be a bounded domain and p ∈ C(Ω). Then there exists at least a solution (λ, u) ∈ (0, ∞) × W01,p (Ω) of the modular eigenvalue problem −∆p u = λ|u|p−2 u. (4.92) Proof. By virtue of Theorem 4.6.5, for each r > 0 there exists an eigenfunction ur satisfying the exremality condition of Theorem 4.2.3 and a corresponding eigenvalue λr for which (4.92) holds. It is well-known that when the exponent p is constant there exists a unique positive eigenfunction u0 ∈ W01,p (Ω) corresponding to the first eigenvalue of the p-Laplacian (up to multiplication by constants). This deep result was proved first by Anane [4]. An elegant proof was given by Kawohl and Lindqvist [56]. Let u0 ∈ W01,p (Ω) be the extremal function resulting from the application of Lemma 4.1.5; that is, k|∇u0 |kp k|∇u|kp = inf1,p . ku0 kp 06=u∈W0 (Ω) kukp 1

u0 For any r > 0, set vr = r p k|∇u . It follows then easily that for each r > 0 0 |kp 1

and any u ∈ W01,p (Ω) with k|∇u|kp = r p one has k|∇u|kpp k|∇u0 |kpp rk|∇u0 |kpp r r = ≥ = = . kukp kukp ku0 kpp rku0 kpp kvr kpp In other words, vr is an extremal function as in Theorem 4.2.3. In all, the eigenvalues and eigenfunctions given by Corollary 4.6.9 coincide with the pth power of the bona-fide eigenvalues if p is constant on Ω.

228

Analysis on Function Spaces of Musielak-Orlicz Type

4.7

Convergence properties of the eigenvalues and eigenfunctions

In this Section we undertake a stability analysis of the eigenvalues and eigenfunctions of the variable exponent p-Laplacian under perturbations of the variable exponent. Throughout this section Ω ⊂ Rn will stand for a bounded domain and all variable exponents will be considered to be in C(Ω). As usual, it will be assumed that all variable exponents p appearing in this Section satisfy the inequalities 1 < p− = inf p ≤ sup p = p+ < ∞. Ω

Ω

Our presentation is based on our work [66]. We recall the following version of Lemma 2.2.5, which can be proved along the exact same lines [33, 34]: Lemma 4.7.1. If f : Ω → R is Borel-measurable, ε > 0 and p, q ∈ C(Ω) are subject to p < q < p + ε, then Z Z p −ε |f | dx ≤ ε|Ω| + ε |f |q dx. Ω

Ω

The following Lemma, of a technical nature, will be used in the sequel: Lemma 4.7.2. Let (pj ) ⊂ C(Ω) be a non-increasing, uniformly convergent sequence in Ω to its infimum p; assume 1 < p− ≤ p+ < ∞ and that for some constant θ > 0 and r ∈ C(Ω) p(x) + θ < r(x) in Ω. If a sequence (ψj ) ⊂ Lr (Ω) converges to ψ in Lr (Ω), then Z Z |ψj |pj |ψ|p lim sup ≤ . pj p j Ω

(4.93)

Ω

Proof. Fix the subindex I ∈ N so that pI < p + θ < r uniformly in Ω; in consequence kψj − ψkpI −→ 0 as j −→ ∞. In particular the convergence in the pI -norm implies that the numerical sequence Z |ψj |pI Ω

Applications

229

is bounded. Consequently Z Z |ψj |pI −→ |ψ|pI as j −→ ∞. Ω

Ω

On account of Lemma 4.7.1, for any integer j ≥ I one has the estimate Z Z |ψj |pj 1 I −pj k∞ ≤ kpI − pj k−kp (|ψj |pI − |ψ|pI ) p /p ∞ I pj pj j Ω Ω Z |ψ|pI I −pj k∞ + kpI − pj k−kp + kpI − pj k∞ |Ω|. (4.94) ∞ p /p pj I j Ω

It is obvious that the right-hand side of (4.94) converges as j −→ ∞ to Z |ψ|pI I −pk∞ kpI − pk−kp + kpI − pk∞ |Ω|. (4.95) ∞ ppI /p Ω

Moreover, the inequality |ψ|pI ≤ |ψ|pI χ{x:|ψ(x)| 0 let Vp,r stands for the set of all maximal functions resulting from the application of Theorem 4.2.3. Theorem 4.7.3. Under the conditions of Lemma 4.7.2, one has the following conclusions: lim λpj ,1 exists (4.96) j−→∞

and (p− /p+ )2 lim λpj ,1 ≤ λp,1 ≤ lim λpj ,1 . j−→∞

j−→∞

(4.97)

230

Analysis on Function Spaces of Musielak-Orlicz Type

Moreover, there exists u ∈ Vp,1 and a subsequence of (uj ) (still denoted by (uj )) such that as j −→ ∞ uj * u in W01,p (Ω)

(4.98)

uj → u in Lp (Ω).

(4.99)

and Proof. By assumption, for each j ∈ N, uj ∈ Vpj ,1 . In particular: Z Z |uj |pj |v|pj = R max . pj pj |∇v|pj /pj ≤1 Ω

(4.100)

Ω

Ω

According to Theorem 4.2.3 one must necessarily have: Z |∇uj |pj = 1; pj

(4.101)

Ω

in particular, because of Lemma 4.7.1 each uj is subject to the bound Z j −pk∞ |∇uj |p ≤ (p1 + )kpj − pk−kp + kpj − pk∞ |Ω|, ∞ Ω

where p1 + = sup p1 . Ω

Inequalities 1.5.4 imply that the sequence (uj ) is bounded in W01,p (Ω), hence it can be assumed to be weakly convergent to u ∈ W01,p (Ω); by virtue of the compactness of the Sobolev embedding (Theorem 3.4.4) there is no loss of generality if uj is taken to be strongly convergent to u, in Lp (Ω). On account of Corollary 2.2.3, (uj ) can be considered to be pointwise almost everywhere convergent to u in Ω. For arbitrary δ > 0, pick N large enough to guarantee that j ≥ N imply j −pk∞ kpj − pk−kp + kpj − pk∞ |Ω|+ < 1 + δ ∞

and that

Then Z

1 1 ppj /p − pj < δ uniformly in Ω, for j ≥ N.

|∇uj |p p

Ω

 j −pk∞  ≤ kpj − pk−kp ∞

Z

Ω

+ kpj − pk∞ |Ω| ≤ (1 + δ)(p1 + δ + 1).

|∇uj |pj



1 ppj /p

1 − pj



Z dx + Ω

 |∇uj |pj  dx pj

Applications

231

Thus, since (uj )j≥N converges weakly to u in W01,p (Ω), the weak lowersemicontinuity of the functional F : W01,p (Ω) −→ [0, ∞) Z |∇w|p F (w) = dx p Ω

guarantees that Z

|∇u|p dx ≤ (1 + δ)(p1 + δ + 1) p

Ω

and the arbitrariness of δ implies that Z |∇u|p dx ≤ 1 p Ω

Next, it will be shown that u is maximal in the sense of Theorem 4.2.3. Fix ω ∈ W01,p (Ω) such that Z |∇ω|p dx = 1. p Ω

pick a sequence of smooth functions (ωk ) ⊂ C0∞ (Ω) converging to ω in W01,p (Ω) and such that ∇ωk → ∇ω and ωk → ω a.e in Ω. For arbitrary δ > 0 let k be large enough to satisfy Z |∇ωk |p 1 − δ/2 < dx < 1 + δ/2. p Ω

Because of the particular choice of the sequence (pj ), one has: Z

|∇ωk |p j −pk∞ dx ≤ kpj − pk−kp ∞ p

Ω

Z

|∇ωk |pj dx + kpj − pk∞ |Ω|. pj

(4.102)

Ω

Lebesgue’s dominated convergence theorem yields, for each fixed k, that Z Z |∇ωk |pj |∇ωk |p dx −→ dx as j −→ ∞. pj p Ω

Ω

Next, observe that Z Z |∇ωk |p |∇ω|p dx −→ dx = 1 as k −→ ∞. p p Ω

Ω

232

Analysis on Function Spaces of Musielak-Orlicz Type

Thus, for k fixed, j can be chosen so large that Z |∇ωk |pj dx ≤ 1 + δ. pj Ω

The last inequality implies that for fixed k ∈ N there exists J ∈ N such that j ≥ J implies
pj Z ∇ wk /(1 + δ)1/pj− dx ≤ 1, pj Ω

whence the maximality of uj gives for j ≥ J: Z Z |wk |pj |uj |pj dx ≤ dx. pj pj pj− (1 + δ) p j Ω Ω

(4.103)

We split the remaining part of the proof the cases p+ < n, p− ≤ n ≤ p+ and p− > n. For p+ < n, M ∈ N can be chosen sufficiently large that kpM − pk∞
1} |u|r + 1, whence Lebesgue’s theorem yields Z Z |u|pj dx −→ |u|p dx. Ω

Ω

Lemma 4.7.1 yields the estimate   j −rk∞ ku − uj kpj ≤ kpj − rk−kp + kp − rk |Ω| ku − uj kr . j ∞ ∞

(4.107)

On the other hand, for 0 < t < 1 it holds that: Z Z pj |uj | dx − |u|p dx Ω

Ω

Z

t1−p+ |uj − u|pj dx + (1 − t)1−p+

≤ Ω

Z

|u|pj dx −

Ω

Z

|u|p dx

Ω

 Z

t1−p+ |uj − u|pj dx + (1 − t)1−p+ 

= Ω

 Z

|u|pj dx −

Ω

+ (1 − t)1−p+


−1

Z

Z

|u|p dx

Ω

|u|p dx,

Ω

likewise: Z Z |u|p dx − |uj |pj dx Ω

Ω

Z ≤

t

1−p+

p

1−p+

|u − uj | dx + (1 − t)

Ω

Z ≤

Z

Z

p

|uj | dx − Ω

|uj |pj dx

Ω

t1−p+ |uj − u|p dx

Ω

   Z j k∞ + (1 − t)1−p+  kp − pj k−kp−p −1 |uj |pj dx + kp − pj k∞ |Ω| ∞ Ω 1−p+

+ (1 − t)

−1




Z

p

|u| dx. Ω

234

Analysis on Function Spaces of Musielak-Orlicz Type

Letting j tend to infinity in the last two inequalities, recalling the fact that (uj )pj ≤r converges to u in Lr (Ω) and invoking (4.107), one obtains (4.106). As to the case n ∈ (p− , p+ ], we point out that in the proof of Theorem 3.4.4 a finite collection of subdomains of Ω was obtained, namely Ω1 , Ω2 , ...ΩM such that

M [

Ωk = Ω

1

and sequences of positive real numbers s1 , ...sM and θ1 , ...θM with p(x) + θk ≤ sk in Ωk . Consider a partition of unity ϕk , 1 ≤ k ≤ M subordinated to the open covering just described. By the assumption of uniform convergence, for sufficiently large I, pI + θ < sk uniformly in Ω. As in the proof of Theorem 3.4.4, it follows the inclusion (uj ϕk ) ⊂ Lsk (Ω) and thus, for each k = 1, 2, ...M , the sequence (uj ϕk ) can be assumed to be convergent in Lsk (Ω). Denote the limit of the sequence (uj ϕk ) in Lsk (Ω) by v ∈ Lsk (Ω). In particular, for each k and for j ≥ I, it holds that uj ϕk ∈ LpI (Ω). Moreover: uj −→ u in Lp (Ω), and by reason of the inclusion Lsk (Ω) ⊆ Lp (Ω) one has kuj ϕk − vkp → 0 as j −→ ∞, whence the inequality kv − ϕk ukp ≤ kuj ϕk − vkp + k(uj − u)ϕk kp implies that as j −→ ∞, uj ϕk −→ uϕk in Lsk (Ω) ,→ LpI (Ω).

(4.108)

Applications

235

Consequently, u=

M X

uϕk ∈ LpI (Ω).

k=1

According to (4.108), ku − uj kpI ≤

M X

k(u − uj )ϕk kpI → 0 as j → ∞;

(4.109)

k=1

from which it immediately follows that for j ≥ I, ku − uj kpj −→ 0 as j −→ ∞, whence

Z

|u − uj |pj dx −→ 0 as j −→ ∞.

Ω

Since |u(x)|pi (x) −→ |u(x)|p(x) as i −→ ∞ a.e in Ω, the pointwise inequality M pi M M X X X pi |u| = uϕk ≤ M pi −1 |uϕk |pi ≤ C |uϕk |pi k=1

k=1

k=1

and the pointwise estimate (which is valid in Ω for each k = 1, 2...M and for any natural number j ≥ I): |uϕk |pj ≤ χ{x:|uϕk (x)|>1} |u|pj + χ{x:|uϕk (x)|1} |u|sk + 1, imply easily via a straightforward application of Lebesgue’s theorem that Z Z |u|pj dx −→ |u|p dx as j −→ ∞. (4.110) Ω

Ω

On the other hand, one has, for any t ∈ (0, 1), and j ≥ I, Z Z pj |uj | dx − |u|p dx Ω

Ω

Z ≤

t1−p+ |uj − u|pj dx + (1 − t)1−p+

Ω

Z =

Z

|u|pj dx −

Ω

Z

|u|p dx

Ω

  Z Z t1−p+ |uj − u|pj dx + (1 − t)1−p+  |u|pj dx − |u|p dx

Ω

Ω 1−p+

+ (1 − t)


−1

Z

p

|u| dx, Ω

Ω

236

Analysis on Function Spaces of Musielak-Orlicz Type

and Z Z p |u| dx − |uj |pj dx Ω

Ω

Z ≤

t

1−p+

|u − uj |p dx + (1 − t)1−p+

Ω

Z ≤

Z

|uj |p dx −

Ω

Z

|uj |pj dx

Ω

t1−p+ |uj − u|p dx

Ω

   Z j k∞ + (1 − t)1−p+  kp − pj k−kp−p −1 |uj |pj dx + kp − pj k∞ |Ω| ∞ Ω

+ (1 − t)1−p+ − 1




Z

|u|p dx,

Ω

whence it transpires from the prior discussion that Z Z |uj |pj dx −→ |u|p dx as j −→ ∞. Ω

(4.111)

Ω

Given the assumptions on p and on the sequence (pj ), (4.111) implies that Z Z |uj |pj |u|p dx −→ dx as j −→ ∞. (4.112) pj p Ω

Ω

Letting j tend to infinity in (4.103) we obtain : Z Z |uj |pj |u|p lim sup dx ≤ dx. pj p j Ω

Ω

In conjunction with inequality (4.103) the preceding statement yields the maximality of u in the required sense. If p− ≥ n all Sobolev spaces involved in the above proof are contained in L∞ (Ω); the sequence (uj ), in particular, fulfills the hypothesis of Lemma 4.7.2 for any real number r > 1. The maximality of u follows immediately from this observation and inequality (4.103). Next, by reason of (4.112), for sufficiently large j one has R |∇uj |pj dx p1 p− Ω R ≤ R dx ≤ R p + |u|p dx + 1 |uj |pj dx |u| dx + 1 Ω

Ω

Ω

where pj + = sup pj . There is therefore no loss of generality in assuming the Ω

Applications

237

sequence |∇uj |pj dx R |uj |pj dx

R

Ω

Ω

to be convergent, to say L. Clearly, from (4.112): Z Z pj |∇uj | dx −→ L |u|p dx. Ω

(4.113)

Ω

From the weakly-lower semi-continuity of the functional G : W01,p (Ω) −→ [0, ∞) Z G(w) = |∇w|p dx Ω

it is easily concluded that Z Z |∇u|p dx ≤ lim inf |∇uj |p dx j−→∞

Ω

Ω





jk ≤ lim inf kpj − pk−kp−p ∞

Z

j−→∞

|∇uj |pj dx + kp − pj k∞ |Ω| ;

Ω

from which, utilizing (4.112) it is readily concluded that R |∇u|p dx Ω R ≤ L. |u|p dx

(4.114)

Ω

On the other hand, for arbitrary δ > 0 and large enough j, R R R |∇uj |pj dx |∇uj |pj /pj dx |∇u|p /p dx |∇u|p dx Ω Ω Ω Ω R ≤ pj + R p = pj + R p ≤ pj + /p− R p . |uj |pj dx |u| dx − δ |u| dx − δ |u| dx − δ R

Ω

Ω

Ω

Ω

In conjuction with (4.114), the preceding inequality yields (4.97). In fact, the preceding convergence results can be substantially improved if p(x) ≥ 2 in Ω. Specifically: Theorem 4.7.4. For p− ≥ 2, and u, (ui ) as in Theorem 4.7.3, we have: Z lim |∇u − ∇uj |p dx = 0. j→∞

Ω

238

Analysis on Function Spaces of Musielak-Orlicz Type

Proof. Notice that since u ∈ W01,p (Ω), one has Z Z |∇u|pk |∇u|p lim dx = dx = 1. k−→∞ pk p Ω

(4.115)

Ω

Also, the functional G : W01,p (Ω) → [0, ∞) Z |∇v|p G(v) = dx p Ω

is weakly lower- semicontinuous, whence Z Z |∇u|p |∇uj |p 1= dx ≤ lim inf dx j→∞ p p Ω

and Z

(4.116)

Ω

|∇u|p dx ≤ lim inf j p

Ω

Z ∇(u + uj ) p 1 p dx. 2

(4.117)

Ω

From the equality (valid for all natural numbers j): Z |∇uj |pj dx = 1 pj Ω

and using the embedding estimate (4.7.1) one has, for j ∈ N and ε = kpj −pk∞ : Z Z |∇uj |p |∇uj |pj −ε dx ≤ ε dx + ε|Ω| (4.118) p ppj /p Ω Ω   Z Z 1 |∇uj |pj 1 −ε −ε ≤ε |∇uj |pj − dx + ε dx + ε|Ω| pj pj ppj /p Ω Ω   Z 1 1 = ε−ε |∇uj |pj − dx + ε−ε + ε|Ω|, pj ppj /p Ω

which implies Z lim sup j→∞

|∇uj |p dx ≤ 1. p

(4.119)

Ω

Inequalities (4.116) and (4.119) yield Z |∇uj |p lim dx = 1. j→∞ p Ω

(4.120)

Applications

239

Moreover, Z

|∇uj |pj dx ≤ pj+ ≤ p+

(4.121)

Ω

for all positive integers j and 1 ppj /p

−

1 −→ 0 as j → ∞. pj

(4.122)

Consequently, for each j ∈ N: Z Z Z p p ∇(u + uj ) p 1 1 |∇u| 1 |∇uj | dx ≤ dx + dx p 2 2 p 2 p Ω Ω Ω   Z Z p 1 |∇u| 1  −ε |∇uj |pj ≤ dx + ε dx + ε|Ω| 2 p 2 ppj /p Ω Ω   Z pj 1 1  −ε |∇uj | = + ε dx + ε|Ω| . (4.123) 2 2 ppj /p Ω

It follows that the expression to the right of the equal sign in (4.123) remains bounded as j → ∞ and hence that Z ∇(u + uj ) p 1 Q = lim inf p dx < ∞. j 2 Ω

Fix δ > 0 and select k so large that for j ≥ k one has kp − pj k∞ |Ω| < δ and j k∞ kp − pj k−kp−p < 1 + δ. ∞

Pick I large enough so that i ≥ I implies Z ∇(u + uj ) p 1 dx. Q − δ < inf p j≥i 2 Ω

Then for j ≥ I;

240

Analysis on Function Spaces of Musielak-Orlicz Type

Z 1= Ω

Z ∇(u + uj ) p 1 dx p 2 Ω   Z p 1 |∇uj | ≤ δ + 1 + dx 2 p Ω   Z pj 1+δ  |∇uj | 1+ ≤δ+ dx 2 ppj /p Ω     Z Z pj 1+δ  1 1 |∇uj | ≤δ+ 1 + |∇uj |pj dx − + dx 2 pj pj ppj /p Ω Ω     Z 1+δ  1 1 =δ+ 1 + |∇uj |pj − dx + 1 . 2 pj ppj /p

|∇u|p dx ≤ δ + p

Ω

By virtue of (4.121) and (4.122) it is apparent that the last term above tends to 1 as j → ∞ and δ −→ 0, whence Z ∇(u + uj ) p 1 dx −→ 1 as j −→ ∞. (4.124) p 2 Ω

For any real number t : t ≥ 2 the inequality a + b t a − b t
1 t t + 2 2 ≤ 2 |a| + |b| holds for any complex numbers a and b (see [[2], Lemma 2.27]). In particular, for each x ∈ Ω p Z Z ∇(u + uj ) p dx + ∇(u − uj ) dx ≤ 2p1/p 2p1/p Ω Ω   p Z Z ∇uj p 1  ∇u  p1/p dx + p1/p dx 2 Ω

Ω

whence (4.120) and (4.124) imply Z Z |∇(u − uj )|p dx ≤ 2p+ pp+ /p− Ω

∇(u − uj ) p 2p1/p dx −→ 0 as j −→ ∞.

Ω

(4.125) This completes the proof of Theorem 4.7.4.

Applications

241

Next, we single out the one dimensional case, for which the convergence statement is perhaps the best one can hope for: Theorem 4.7.5. For n = 1, Ω = (a, b), a < b. Let (pj ) ⊂ C([a, b]) be a non-increasing sequence of functions, uniformly convergent to p = inf pj with 1 < p− ≤ p+ < ∞. For each j ∈ N let λj and uj be, respectively, the first eigenvalue and a first eigenfunction of the problem  0 0 0 |u |pj −2 u = γ|u|pj −2 u; (4.126) analogously, let λ be the first modular eigenvalue of the one-dimensional pLaplacian and u be its associated eigenfunction in the sense of Corollary 4.6.9. Then there exists a subsequence (uk ) of (uj ) and a (first) eigenfunction u of the p-Laplacian such that uk −→ u in W01,p (Ω).

(4.127)

Proof. For p− ≥ 2, Theorem 4.7.5 follows directly from Theorem 4.7.4. We hence focus on the case p− < 2. To that end we notice the elementary fact that there exists a positive constant c such that as long as 1 < s < S < ∞ and (q, r) ∈ (s, S) × [−M, M ] 1 (|r + 1|q + |r − 1|q ) − |r|q ≥ c. 2

(4.128)

Fix δ > 0 and set 0

0

A = {x ∈ (a, b) : |(uj − u) | > δ|(uj + u) |}. Then if x ∈ A, r = vj =

0

1/p uj /pj j

v(x)+vj (x) v(x)−vj (x) ,

0

in (4.128) yields, pointwise in Ω: v + v j pj v − vj pj 1 pj pj , (|v| + |vj | ) − ≥ c 2 2 2

whence it follows by integrating on A that   Z Z Z Z v + vj pj 1 pj pj dx ≥ c |v| dx + |vj | dx − 2 2 Ω

Ω

Ω

A

Ω

Z Ω

|v|pj dx −→ 1 as j −→ ∞.

,

(4.129)

v − vj pj 2 dx. (4.130)

As seen in Theorem 4.7.4 Z v + vj pj 2 dx −→ 1 as j −→ ∞, whereas

1/pj

q = pj (x), the substitution v = u /pj

242

Analysis on Function Spaces of Musielak-Orlicz Type

Letting j tend to infinity in (4.130) it is clear that Z v − vj pj 2 dx −→ 0 as j → ∞

(4.131)

A

which through an easy calculation yields : Z |u0 − u0j |p dx −→ 0 as j −→ ∞.

(4.132)

A

On the other hand Z Z v − vj pj dx ≤ δ p− 2

v + v j pj p− 2 dx ≤ δ .

(4.133)

Ω

Ω\A

In all, Z

|u0 − u0j |p dx −→ 0 as j −→ ∞.

Ω

This last statement completes the proof. A result analogous to that in Theorem 4.7.3 holds for a non-decreasing sequence (pi ). Specifically: Theorem 4.7.6. Let (pi ) be a non-decreasing sequence of functions in C(Ω) with lim pi = q ∈ C(Ω). i→∞

Assume furthermore that inf Ω q = q− > n. For each i ∈ N let ui ∈ W01,pi (Ω) be an eigenfunction corresponding to the least eigenvalue λpi ,1 of the pi Laplacian. Then there exists a subsequence, still denoted by (ui ), that converges weakly to u ∈ Vq,1 (Ω). Moreover Z |∇u|q dx = 1. q Ω and the limit lim λpi ,1

i−→∞

R |∇ui |pi dx = lim RΩ =L