1st Edition

Analysis on Function Spaces of Musielak-Orlicz Type

By Osvaldo Mendez, Jan Lang Copyright 2019
    276 Pages
    by Chapman & Hall

    Analysis on Function Spaces of Musielak-Orlicz Type provides a state-of-the-art survey on the theory of function spaces of Musielak-Orlicz type. The book also offers readers a step-by-step introduction to the theory of Musielak–Orlicz spaces, and introduces associated function spaces, extending up to the current research on the topic

    Musielak-Orlicz spaces came under renewed interest when applications to electrorheological hydrodynamics forced the particular case of the variable exponent Lebesgue spaces on to center stage. Since then, research efforts have typically been oriented towards carrying over the results of classical analysis into the framework of variable exponent function spaces. In recent years it has been suggested that many of the fundamental results in the realm of variable exponent Lebesgue spaces depend only on the intrinsic structure of the Musielak-Orlicz function, thus opening the door for a unified theory which encompasses that of Lebesgue function spaces with variable exponent.


    • Gives a self-contained, concise account of the basic theory, in such a way that even early-stage graduate students will find it useful
    • Contains numerous applications
    • Facilitates the unified treatment of seemingly different theoretical and applied problems
    • Includes a number of open problems in the area

    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


    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 particular
    with the variable exponent p Laplacian.

    A particular virtue of the book is that the unifi…ed 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 defi…nite interest to many."

    -Professor David Edmunds, University of Sussex