Table of Contents

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 norm and continuity of 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 ‘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

1.5 Forerunners of the Musielak-Orlicz class: Orlicz spaces, Lp(x) spaces

2 Musielak-Orlicz spaces

2.1 Introduction, De nition and Examples

2.2 Embeddings between Musielak-Orlicz spaces

2.2.1 The <2-condition

2.2.2 Absolute continuity of the norm

2.3 Separability

2.4 Duality of Musielak-Orlicz spaces

2.4.1 Conjugate Musielak-Orlicz functions

2.4.2 Conjugate functions and the dual of L'()

2.5 Density of regular functions

2.6 Uniform convexity of Musielak-Orlicz spaces

2.7 Caratheodory 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

2.9.1 Properties

2.10 Historical notes

3 Sobolev spaces of Musielak-Orlicz type

3.1 Sobolev spaces: de nition and basic properties

3.1.1 Examples

3.2 Separability

3.3 Duality of Sobolev spaces of Musielak-Orlicz type

3.4 Embeddings, compactness, Poincare-type inequalities

4 Applications

4.1 Preparatory results and notation

4.2 Compactness of the Sobolev embedding and the modular setting

4.3 The variable exponent p-Laplacian

4.3.1 Stability of the solutions

4.4 -convergence

4.5 The eigenvalue problem for the p-Laplacian

4.6 More on Eigenvalues