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Variable Lebesgue Spaces and Hyperbolic Systems

  • David Cruz-Uribe
  • Alberto Fiorenza
  • Michael Ruzhansky
  • Jens Wirth
  • Sergey Tikhonov

Part of the Advanced Courses in Mathematics - CRM Barcelona book series (ACMBIRK)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Introduction to the Variable Lebesgue Spaces

    1. Front Matter
      Pages 1-1
    2. David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth
      Pages 3-9
    3. David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth
      Pages 11-34
    4. David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth
      Pages 35-56
    5. David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth
      Pages 57-82
    6. Back Matter
      Pages 83-90
  3. Asymptotic Behaviour of Solutions to Hyperbolic Equations and Systems

    1. Front Matter
      Pages 91-91
    2. David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth
      Pages 93-98
    3. David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth
      Pages 99-110
    4. David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth
      Pages 111-117
    5. David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth
      Pages 119-141
    6. David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth
      Pages 143-155
    7. David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth
      Pages 157-162
    8. David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth
      Pages 163-164
    9. Back Matter
      Pages 165-169

About this book

Introduction

This book targets graduate students and researchers who want to learn about Lebesgue spaces and solutions to hyperbolic equations. It is divided into two parts.

Part 1 provides an introduction to the theory of variable Lebesgue spaces: Banach function spaces like the classical Lebesgue spaces but with the constant exponent replaced by an exponent function. These spaces arise naturally from the study of partial differential equations and variational integrals with non-standard growth conditions. They have applications to electrorheological fluids in physics and to image reconstruction. After an introduction that sketches history and motivation, the authors develop the function space properties of variable Lebesgue spaces; proofs are modeled on the classical theory. Subsequently, the Hardy-Littlewood maximal operator is discussed. In the last chapter, other operators from harmonic analysis are considered, such as convolution operators and singular integrals. The text is mostly self-contained, with only some more technical proofs and background material omitted.

Part 2 gives an overview of the asymptotic properties of solutions to hyperbolic equations and systems with time-dependent coefficients. First, an overview of known results is given for general scalar hyperbolic equations of higher order with constant coefficients. Then strongly hyperbolic systems with time-dependent coefficients are considered. A feature of the described approach is that oscillations in coefficients are allowed. Propagators for the Cauchy problems are constructed as oscillatory integrals by working in appropriate time-frequency symbol classes. A number of examples is considered and the sharpness of results is discussed. An exemplary treatment of dissipative terms shows how effective lower order terms can change asymptotic properties and thus complements the exposition.

Keywords

Rubio de Francia extrapolation hyperbolic Cauchy problems maximal operators oscillating time-dependent coefficients singular integrals variable Lebesgue spaces

Authors and affiliations

  • David Cruz-Uribe
    • 1
  • Alberto Fiorenza
    • 2
  • Michael Ruzhansky
    • 3
  • Jens Wirth
    • 4
  1. 1.Department of MathematicsTrinity CollegeHartfordUSA
  2. 2.Dipartimento di ArchitetturaUniversità di Napoli Federico IINapoliItaly
  3. 3.Department of MathematicsImperial College LondonLondonUnited Kingdom
  4. 4.Fachbereich Mathematik Institut für Analysis, Dynamik und ModelUniversität StuttgartStuttgartGermany

Editors and affiliations

  • Sergey Tikhonov
    • 1
  1. 1.ICREA and CRMBarcelonaSpain

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-0840-8
  • Copyright Information Springer Basel 2014
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-0348-0839-2
  • Online ISBN 978-3-0348-0840-8
  • Series Print ISSN 2297-0304
  • Series Online ISSN 2297-0312
  • Buy this book on publisher's site