Cabrelli / Torrea Recent Developments in Real and Harmonic Analysis

In Honor of Carlos Segovia
1. Auflage 2010
ISBN: 978-0-8176-4588-5
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 196 Seiten

Reihe: Applied and Numerical Harmonic Analysis

A collection of invited chapters dedicated to Carlos Segovia, this unified and self-contained volume examines recent developments in real and harmonic analysis. The work begins with a chronological description of Segovia's mathematical life, highlighting his original ideas and their evolution. Also included are surveys dealing with Carlos' favorite topics, and PDE works written by students and colleagues close to Segovia whose careers were in some way influenced by him. Contributors: H. Aimar, A. Bonami, O. Blasco, L.A. Caffarelli, S. Chanillo, J. Feuto, L. Forzani, C.E. Gutíerrez, E. Harboure, A.L. Karakhanyan, C.E. Kenig, R.A. Macías, J.J. Manfredi, F.J. Martín-Reyes, P. Ortega, R. Scotto, A. de la Torre, J.L. Torrea.

Autoren/Hrsg.

Cabrelli, Carlos

Torrea, Jose Luis

Inhaltsverzeichnis

1;ANHA Series Preface;7
2;Foreword;10
3;Preface;12
4;Publications of Carlos Segovia;14
5;List of Contributors;18
6;Contents;20
7;Carlos Segovia Fern´andez;22
7.1;1 Square functions;23
7.2;2 Spaces of homogeneous type;27
7.3;3 Weighted inequalities;31
7.4;4 One-sided operators;33
7.5;5 Vector-valued Fourier analysis;34
7.6;6 Harmonic analysis associated with generalized Laplacians;37
7.7;References;41
8;Balls as Subspaces of Homogeneous Type: On a Construction due to R. Macías and C. Segovia;46
8.1;1 Introduction;46
8.2;2 Quasi-distance on X and diagonal neighborhoods in X × X;49
8.3;3 Regularization of neighborhoods of .;52
8.4;4 The main result;53
8.5;References;57
9;Some Aspects of Vector-Valued Singular Integrals;58
9.1;1 Introduction and notation;58
9.2;2 Theorems and proofs;66
9.3;3 Commutators;71
9.4;Acknowledgement;75
9.5;References;75
10;Products of Functions in Hardy and Lipschitz or BMO Spaces;78
10.1;1 Introduction;78
10.2;2 Prerequisites on Hardy and Lipschitz spaces;81
10.3;3 Proofs of Theorem 1.1 and Theorem 1.2;84
10.4;4 Generalization to spaces of homogeneous type;89
10.5;Acknowledgements;91
10.6;References;91
11;Harmonic Analysis Related to Hermite Expansions;93
11.1;1 Introduction;93
11.2;2 Hermite polynomials;100
11.2.1;2.1 The Ornstein–Uhlenbeck maximal operator;101
11.2.2;2.2 Riesz transforms;107
11.2.3;2.3 The Littlewood–Paley–Stein functions;108
11.3;3 Hermite functions;110
11.3.1;3.1 The maximal operator for the heat-diffusion semigroup;111
11.3.2;3.3 Littlewood–Paley–Stein g functions;113
11.4;Acknowledgements;113
11.5;References;113
12;Weights for One–Sided Operators;117
12.1;1 Introduction;117
12.2;2 Weighted Hardy inequalities;119
12.3;3 Weights for the one-sided Hardy–Littlewood maximaloperators;122
12.4;4 Some remarks and properties of the one-sided weights;125
12.4.1;4.1 Basic weights;125
12.4.2;4.2 The doubling condition and examples of A+p weights;125
12.4.3;4.3 The reverse Hölder inequality;126
12.4.4;4.4 Sharp functions and BMO;127
12.5;5 Some approximations of the identity;130
12.6;6 One-sided singular integrals;132
12.6.1;6.1 One-sided strongly singular integrals;133
12.6.2;6.2 One-sided Calderón–Zygmund kernels;134
12.6.3;6.3 Further examples of one-sided singular kernels;136
12.7;7 Some applications to ergodic theory;140
12.7.1;7.1 The strong type;141
12.7.2;7.2 The weak type;143
12.7.3;7.3 Back to Dunford–Schwartz;145
12.8;8 The one-sided Hardy–Littlewood maximal operator in dimensions greater than 1;146
12.9;Acknowledgements;149
12.10;References;149
13;Lectures on Gas Flow in Porous Media;153
13.1;1 Introduction;153
13.1.1;1.1 Travelling fronts;157
13.1.2;1.2 Quadratic solution (separation of variables);157
13.1.3;1.3 Fundamental solution;157
13.2;2 Scaling;159
13.3;3 Regularity of the free boundary;164
13.4;4 Differentiability of the free boundary;169
13.4.1;4.1 Blow-up;169
13.4.2;4.2 Classification of the global solutions;171
13.5;5 Remarks;171
13.5.1;5.1 N-dimensional results;171
13.5.2;5.2 Waiting time;172
13.5.3;5.3 Viscosity solutions;172
13.5.4;5.4 Global profiles and regularity;174
13.5.5;5.5 Moving plane method;176
13.6;References;177
14;Sharp Global Bounds for the Hessian onPseudo-Hermitian Manifolds;178
14.1;1 Introduction;178
14.2;2 The main theorem;182
14.3;3 Applications to PDE;186
14.4;Acknowledgements;190
14.5;References;190
15;Recent Progress on the Global Well-Posednessof the KPI Equation;192
15.1;References;196
16;On Monge–Ampère Type Equationsand Applications;198
16.1;1 The Monge–Ampère equation;198
16.1.1;1.1 Basic facts;198
16.1.2;1.2 The Dirichlet problem;200
16.1.3;1.3 Regularity of solutions;201
16.1.4;1.4 Estimates for the linearized Monge–Ampère equation;203
16.2;2 A Monge–Ampère type equation for reflectors;204
16.2.1;2.1 Snell’s law;204
16.2.2;2.2 The reflector problem;204
16.2.3;2.3 Notion of weak solution for the reflector problem;205
16.2.4;2.4 Results;206
16.3;Acknowledgements;208
16.4;References;209
17;Index;210
17.1;Applied and Numerical Harmonic Analysis;213