Analysis on Function Spaces of Musielak-Orlicz Type

Osvaldo Mendez is an associate professor at University of Texas at El Paso. His areas of research include Harmonic Analysis, Partial Differential Equations and Theory of Function Spaces. Professor Mendez has authored one book and one edited book.Jan Lang is a professor of mathematics at The Ohio State University. His areas of interest include the Theory of Integral operators, Approximation Theory, Theory of Function spaces and applications to PDEs. He is the author of two books and one edited book.

"The family of Musielak-Orlicz (M-O) spaces mentioned in the title includes not only those of classical Lebesgue and Orlicz type, but also the spaces with variable exponent that have attracted such a great deal of interest in recent years. After a preparatory chapter in which basic facts are established, a detailed study is made of M-O spaces, following which Sobolev spaces based on them are examined. Finally, there is a chapter giving applications, dealing in particularwith the variable exponent p Laplacian.A particular virtue of the book is that the unified approach adopted to deal with very general circumstances is accomplished by keeping the technicalities firmly subordinate to the main ideas. It is a welcome addition to the number of books dealing with related topics and should be of definite interest to many."-Professor David Edmunds, University of Sussex

• 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 convergence1.3.2 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 spaces1.4.5 Further remarks1.5 Forerunners of the Musielak-Orlicz class: Orlicz spaces, Lp(x) spaces 2 Musielak-Orlicz spaces 2.1 Introduction, De nition and Examples2.2 Embeddings between Musielak-Orlicz spaces 2.2.1 The <2-condition2.2.2 Absolute continuity of the norm 2.3 Separability 2.4 Duality of Musielak-Orlicz spaces2.4.1 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 spaces2.8.1 Duality maps on spaces of variable integrability2.9 The Matuszewska-Orlicz index of a Musielak-Orlicz space 2.9.1 Properties2.10 Historical notes3 Sobolev spaces of Musielak-Orlicz type 3.1 Sobolev spaces: de nition and basic properties 3.1.1 Examples3.2 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 notation4.2 Compactness of the Sobolev embedding and the modular setting4.3 The variable exponent p-Laplacian 4.3.1 Stability of the solutions4.4 □□-convergence4.5 The eigenvalue problem for the p-Laplacian 4.6 More on Eigenvalues