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
