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Variable Lebesgue Spaces

Foundations and Harmonic Analysis

  • David V. Cruz-Uribe
  • Alberto Fiorenza

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Table of contents

  1. Front Matter
    Pages i-ix
  2. David V. Cruz-Uribe, Alberto Fiorenza
    Pages 1-12
  3. David V. Cruz-Uribe, Alberto Fiorenza
    Pages 13-77
  4. David V. Cruz-Uribe, Alberto Fiorenza
    Pages 79-128
  5. David V. Cruz-Uribe, Alberto Fiorenza
    Pages 129-189
  6. David V. Cruz-Uribe, Alberto Fiorenza
    Pages 191-237
  7. David V. Cruz-Uribe, Alberto Fiorenza
    Pages 239-270
  8. Back Matter
    Pages 271-312

About this book

Introduction

This book provides an accessible introduction to the theory of variable Lebesgue spaces. These spaces generalize the classical Lebesgue spaces by replacing the constant exponent p with a variable exponent p(x). They were introduced in the early 1930s but have become the focus of renewed interest since the early 1990s because of their connection with the calculus of variations and partial differential equations with nonstandard growth conditions, and for their applications to problems in physics and image processing.

The book begins with the development of the basic function space properties. It avoids a more abstract, functional analysis approach, instead emphasizing an hands-on approach that makes clear the similarities and differences between the variable and classical Lebesgue spaces. The subsequent chapters are devoted to harmonic analysis on variable Lebesgue spaces. The theory of the Hardy-Littlewood maximal operator is completely developed, and the connections between variable Lebesgue spaces and the weighted norm inequalities are introduced. The other important operators in harmonic analysis - singular integrals, Riesz potentials, and approximate identities - are treated using a powerful generalization of the Rubio de Francia theory of extrapolation from the theory of weighted norm inequalities. The final chapter applies the results from previous chapters to prove basic results about variable Sobolev spaces.​

Keywords

Banach Function Spaces Extrapolation Theory Hardy-Littlewood Maximal Operator Harmonic Analysis Variable Lebesgue Spaces

Authors and affiliations

  • David V. Cruz-Uribe
    • 1
  • Alberto Fiorenza
    • 2
  1. 1., Department of MathematicsTrinity CollegeHartfordUSA
  2. 2.Federico II, Dipartimento di ArchitetturaUniversità di NapoliNapoliItaly

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-0548-3
  • Copyright Information Springer Basel 2013
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-0348-0547-6
  • Online ISBN 978-3-0348-0548-3
  • Series Print ISSN 2296-5009
  • Series Online ISSN 2296-5017
  • Buy this book on publisher's site
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