Variable Lebesgue Spaces and Hyperbolic Systems

173 Pages·2014·1.652 MB·English

by  David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth (auth.), Sergey Tikhonov (eds.)

Advanced Courses in Mathematics CRM Barcelona
David Cruz-Uribe
Alberto Fiorenza
Michael Ruzhansky
Jens Wirth
Variable Lebesgue
Spaces and
Hyperbolic Systems

Advanced Courses in Mathematics
CRM Barcelona
Centre de Recerca Matemàtica
Managing Editor:
Carles Casacuberta
More information about this series at http://www.springer.com/series/5038
David Cruz-Uribe • Alberto Fiorenza
Michael Ruzhansky • Jens Wirth
Variable Lebesgue Spaces
and Hyperbolic Systems
Editor for this volume:
Sergey Tikhonov (ICREA and CRM Barcelona)
David Cruz-Uribe Alberto Fiorenza
Department of Mathematics Dipartimento di Architettura
Trinity College Università di Napoli Federico II
Hartford, CT, USA Napoli, Italy
Michael Ruzhansky Jens Wirth
Department of Mathematics Fachbereich Mathematik
Imperial College London Universität Stuttgart
London, United Kingdom Stuttgart, Germany

ISBN 978-3-0348-0839-2 ISBN 978-3-0348-0840-8 (eBook)
DOI 10.1007/978-3-0348-0840-8
Springer Basel Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014945352

Mathematics Subject Classification (2010): Primary: 35B45, 42B20, 42B25, Secondary: 35L30, 35L45, 42B35

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Foreword
This book contains expository lecture notes for two of the courses given under
the title Advanced Courses on Approximation Theory and Fourier Analysis in
the Centre de Recerca Matem`atica, Bellaterra, Barcelona, from November 7 to
11, 2011. These courses were among the main activities of a six-month research
program on Approximation Theory and Fourier Analysis.
Five courses were delivered by David Cruz-Uribe (Trinity College) on vari-
able Lebesgue spaces, Feng Dai (University of Alberta) on weighted polynomial
approximation on the sphere, Michael Ruzhansky (Imperial College London) on
the asymptotic behaviour of solutions to hyperbolic partial differential equations,
VladimirTemlyakov(UniversityofSouthCarolina)ongreedyapproximation,and
Yuan Xu (University of Oregon) on approximation theory and harmonic analysis
on the unit sphere.
This book contains expanded versions of the lectures on variable Lebesgue
spacesbyDavidCruz-Uribeandthelecturesonasymptoticsforhyperbolicsystems
by Michael Ruzhansky. The lectures by Vladimir Temlyakov, Feng Dai, and Yuan
Xu will be published in separate volumes of this series.
The lectures by Cruz-Uribe (written jointly with Alberto Fiorenza) offer an
introductiontothetheoryofvariableLebesguespacesLp(·) andcoverawiderange
of topics including boundedness properties of the Hardy–Littlewood maximal op-
erator, convolution operators, and norm inequalities for the Riesz potentials. The
second part consists of the lectures given by Ruzhansky (written jointly with
Jens Wirth) and provides an overview of the asymptotic properties of solutions
to hyperbolic partial differential equations and systems with time-dependent co-
efficients, containing the presentation of very recent results by the authors on the
topic.
I am indebted to the Centre de Recerca Matema`tica and its staff for hosting
the advanced courses, and would like to express gratitude to Joaquim Bruna for
hissupportwiththeorganizationofthecoursesandofthewholeprogram.Iwould
also like to thank the authors for their active participation and cooperation.
Sergey Tikhonov
v

Contents
Introduction to the Variable Lebesgue Spaces
David Cruz-Uribe and Alberto Fiorenza 1
1 Introduction and Motivation 3
1.1 An intuitive introduction . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Organization of this monograph . . . . . . . . . . . . . . . . . . . . 7
1.5 A word on proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Properties of Variable Lebesgue Spaces 11
2.1 Exponent functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 The modular and the norm . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Convergence and completeness . . . . . . . . . . . . . . . . . . . . 18
2.4 Embeddings and dense subsets . . . . . . . . . . . . . . . . . . . . 22
2.5 H¨older’s inequality, the associate norm and duality . . . . . . . . . 25
2.6 The Lebesgue differentiation theorem. . . . . . . . . . . . . . . . . 33
3 The Hardy–Littlewood Maximal Operator 35
3.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 The maximal operator on Lp, 1≤p<∞ . . . . . . . . . . . . . . 36
3.3 The maximal operator on variable Lebesgue spaces . . . . . . . . . 41
3.4 The necessity of the hypotheses in Theorem 3.15 . . . . . . . . . . 46
3.5 Weakening the hypotheses in Theorem 3.15 . . . . . . . . . . . . . 52
3.6 Modular inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Extrapolation in Variable Lebesgue Spaces 57
4.1 Convolution operators and approximate identities . . . . . . . . . . 58
4.2 The failure of Young’s inequality in Lp(·) . . . . . . . . . . . . . . . 59
4.3 Approximate identities on variable Lebesgue spaces . . . . . . . . . 61
4.4 Muckenhoupt weights and weighted norm inequalities . . . . . . . 65
4.5 Rubio de Francia extrapolation . . . . . . . . . . . . . . . . . . . . 68
4.6 Applications of extrapolation . . . . . . . . . . . . . . . . . . . . . 74
vii
viii Contents
Bibliography 83
Asymptotic Behaviour of Solutions to Hyperbolic Equations and
Systems
Michael Ruzhansky and Jens Wirth 91
1 Introduction 93
1 Energy and dispersive estimates. . . . . . . . . . . . . . . . . . . . 93
2 Equations with constant coefficients . . . . . . . . . . . . . . . . . 94
3 Stationary phase estimates . . . . . . . . . . . . . . . . . . . . . . 96
2 Equations with constant coefficients 99
1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . 99
2 Combined estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3 Properties of hyperbolic polynomials . . . . . . . . . . . . . . . . . 104
4 Estimates for oscillatory integrals . . . . . . . . . . . . . . . . . . . 107
3 Some interesting model cases 111
1 Scale invariant weak dissipation . . . . . . . . . . . . . . . . . . . . 111
1.1 Reduction to special functions . . . . . . . . . . . . . . . . 111
1.2 High frequency asymptotics . . . . . . . . . . . . . . . . . . 113
1.3 Low frequency asymptotics . . . . . . . . . . . . . . . . . . 114
1.4 Notions of sharpness . . . . . . . . . . . . . . . . . . . . . . 115
2 Scale invariant mass terms . . . . . . . . . . . . . . . . . . . . . . . 116
4 Time-dependent hyperbolic systems 119
1 Motivating examples . . . . . . . . . . . . . . . . . . . . . . . . . . 119
2 Symbol classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3 Uniformly strictly hyperbolic systems . . . . . . . . . . . . . . . . 122
4 Diagonalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.1 Initial step . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.2 The diagonalisation hierarchy . . . . . . . . . . . . . . . . . 125
4.3 Zone constants and invertibility . . . . . . . . . . . . . . . . 126
5 Solving the diagonalised system . . . . . . . . . . . . . . . . . . . . 127
5.1 Treating the diagonal terms . . . . . . . . . . . . . . . . . . 127
5.2 Generalised energy conservation . . . . . . . . . . . . . . . 128
5.3 Perturbation series arguments . . . . . . . . . . . . . . . . . 129
6 Examples and resulting representations of solutions . . . . . . . . . 131
6.1 Symmetric hyperbolic systems . . . . . . . . . . . . . . . . 132
6.2 Second-order equations . . . . . . . . . . . . . . . . . . . . 133
7 Dispersive estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.1 Contact indices for families of surfaces . . . . . . . . . . . . 135
7.2 Estimates for t-dependent Fourier integrals . . . . . . . . . 136
Contents ix
7.3 Extensions to fully variable setting . . . . . . . . . . . . . . 139
8 An alternative low-regularity approach: asymptotic integration . . 140
5 Effective lower order perturbations 143
1 The diffusion phenomenon . . . . . . . . . . . . . . . . . . . . . . . 143
2 Diagonalisation for small frequencies . . . . . . . . . . . . . . . . . 145
2.1 Initial step . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
2.2 The diagonalisation hierarchy . . . . . . . . . . . . . . . . . 148
3 Asymptotic integration and small frequency expansions . . . . . . 150
4 Lyapunov functionals and parabolic type estimates . . . . . . . . . 151
5 A diffusion phenomenon for partially dissipative hyperbolic systems 153
6 Examples and counter-examples 157
1 Parametric resonance phenomena . . . . . . . . . . . . . . . . . . . 157
2 Construction of coefficients and initial data . . . . . . . . . . . . . 159
7 Related topics 163
Bibliography 165