目次

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