Analysis on Function Spaces of Musielak-Orlicz Type

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 Carath?eodory 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