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Lebesgue and Sobolev Spaces with Variable Exponents

  • Book
  • © 2011

Overview

Authors:
  1. Lars Diening

    1. Institute of Mathematics, LMU Munich, Munich, Germany

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  2. Petteri Harjulehto

    1. Dept. of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

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  3. Peter Hästö

    1. Dept. of Mathematical Sciences, University of Oulu, Oulu, Finland

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  4. Michael Ruzicka

    1. Mathematisches Institut, Universität Freiburg, Freiburg, Germany

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  • First book in the area of research
  • Self-contained presentation suitable for both graduate students and researchers
  • Comprehensive book with extensive index and nomenclature
  • Applications for the developed theory is presented in the book
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2017)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-ix
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  2. Introduction

    • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
    Pages 1-17

  3. Lebesgue Spaces

    1. Front Matter

      Pages 19-19
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  4. Lebesgue spaces

    1. A Framework for Function Spaces

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 21-68

    2. Variable Exponent Lebesgue Spaces

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 69-97

    3. The Maximal Operator

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 99-141

    4. The Generalized Muckenhoupt Condition*

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 143-197

    5. Classical Operators

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 199-212

    6. Transfer Techniques

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 213-244

  5. Sobolev Spaces

    1. Front Matter

      Pages 245-245
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  6. Sobolev spaces

    1. Introduction to Sobolev Spaces

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 247-288

    2. Density of Regular Functions

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 289-314

    3. Capacities

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 315-338

    4. Fine Properties of Sobolev Functions

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 339-366

    5. Other Spaces of Differentiable Functions

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 367-398

  7. Applications to Partial Differential Equations

    1. Front Matter

      Pages 399-399
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  8. Applications to partial differential equations

    1. Dirichlet Energy Integral and Laplace Equation

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 401-436

    2. PDEs and Fluid Dynamics

      • Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička
      Pages 437-481

  9. Back Matter

    Pages 483-509
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Keywords

About this book

The field of variable exponent function spaces has witnessed an explosive growth in recent years. The standard reference article for basic properties is already 20 years old. Thus this self-contained monograph collecting all the basic properties of variable exponent Lebesgue and Sobolev spaces is timely and provides a much-needed accessible reference work utilizing consistent notation and terminology. Many results are also provided with new and improved proofs. The book also presents a number of applications to PDE and fluid dynamics.

Reviews

From the reviews:

“The authors provide a comprehensive survey of the state of the art concerning Lebesgue and Sobolev spaces with variable exponents. … The book is also having a rich bibliography of 399 entries, a long list of symbols and an index. It will certainly become a standard reference in this field and stimulate further work in this direction.” (H. G. Feichtinger, Monatshefte für Mathematik, Vol. 165 (1), January, 2012)

“The book is devoted to Lebesgue and Soboley spaces with variable exponents. … The present book consists of the introduction and three parts. … The majority of the results presented in the monograph were obtained by the authors and their collaborators. … the books is a useful source of unified information on Lebesgue and Soboley spaces with variable exponents.” (Alexei Yu. Karlovich, Zentralblatt MATH, Vol. 1222, 2011)

“This book consists of three parts of different lengths and intentions, sub-divided into several chapters. There is a nice figure at the very beginning of the monograph explaining the dependencies among the chapters, together with some recommendations on which parts should be used for first reading or when teaching the subject in a graduate course. … the presentation can thus also be considered as a textbook and extremely useful reference for graduate students and researchers working in related fields … .” (Dorothee D. Haroske, Mathematical Reviews, January, 2013)

Authors and Affiliations

  • Institute of Mathematics, LMU Munich, Munich, Germany

    Lars Diening

  • Dept. of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

    Petteri Harjulehto

  • Dept. of Mathematical Sciences, University of Oulu, Oulu, Finland

    Peter Hästö

  • Mathematisches Institut, Universität Freiburg, Freiburg, Germany

    Michael Ruzicka

Bibliographic Information

  • Book Title: Lebesgue and Sobolev Spaces with Variable Exponents

  • Authors: Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Ruzicka

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-642-18363-8

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2011

  • Softcover ISBN: 978-3-642-18362-1Published: 31 March 2011

  • eBook ISBN: 978-3-642-18363-8Published: 29 March 2011

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: IX, 509

  • Number of Illustrations: 10 b/w illustrations

  • Topics: Analysis, Functional Analysis, Partial Differential Equations

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