Recent Developments in Real and Harmonic Analysis (eBook)

ANHA Series Preface 7
Foreword 10
Preface 12
Publications of Carlos Segovia 14
List of Contributors 18
Contents 20
Carlos Segovia Fern´andez 22
1 Square functions 23
2 Spaces of homogeneous type 27
3 Weighted inequalities 31
4 One-sided operators 33
5 Vector-valued Fourier analysis 34
6 Harmonic analysis associated with generalized Laplacians 37
References 41
Balls as Subspaces of Homogeneous Type: On a Construction due to R. Macías and C. Segovia 46
1 Introduction 46
2 Quasi-distance on X and diagonal neighborhoods in X × X 49
3 Regularization of neighborhoods of . 52
4 The main result 53
References 57
Some Aspects of Vector-Valued Singular Integrals 58
1 Introduction and notation 58
2 Theorems and proofs 66
3 Commutators 71
Acknowledgement 75
References 75
Products of Functions in Hardy and Lipschitz or BMO Spaces 78
1 Introduction 78
2 Prerequisites on Hardy and Lipschitz spaces 81
3 Proofs of Theorem 1.1 and Theorem 1.2 84
4 Generalization to spaces of homogeneous type 89
Acknowledgements 91
References 91
Harmonic Analysis Related to Hermite Expansions 93
1 Introduction 93
2 Hermite polynomials 100
2.1 The Ornstein–Uhlenbeck maximal operator 101
2.2 Riesz transforms 107
2.3 The Littlewood–Paley–Stein functions 108
3 Hermite functions 110
3.1 The maximal operator for the heat-diffusion semigroup 111
3.3 Littlewood–Paley–Stein g functions 113
Acknowledgements 113
References 113
Weights for One–Sided Operators 117
1 Introduction 117
2 Weighted Hardy inequalities 119
3 Weights for the one-sided Hardy–Littlewood maximaloperators 122
4 Some remarks and properties of the one-sided weights 125
4.1 Basic weights 125
4.2 The doubling condition and examples of A+p weights 125
4.3 The reverse Hölder inequality 126
4.4 Sharp functions and BMO 127
5 Some approximations of the identity 130
6 One-sided singular integrals 132
6.1 One-sided strongly singular integrals 133
6.2 One-sided Calderón–Zygmund kernels 134
6.3 Further examples of one-sided singular kernels 136
7 Some applications to ergodic theory 140
7.1 The strong type 141
7.2 The weak type 143
7.3 Back to Dunford–Schwartz 145
8 The one-sided Hardy–Littlewood maximal operator in dimensions greater than 1 146
Acknowledgements 149
References 149
Lectures on Gas Flow in Porous Media 153
1 Introduction 153
1.1 Travelling fronts 157
1.2 Quadratic solution (separation of variables) 157
1.3 Fundamental solution 157
2 Scaling 159
3 Regularity of the free boundary 164
4 Differentiability of the free boundary 169
4.1 Blow-up 169
4.2 Classification of the global solutions 171
5 Remarks 171
5.1 N-dimensional results 171
5.2 Waiting time 172
5.3 Viscosity solutions 172
5.4 Global profiles and regularity 174
5.5 Moving plane method 176
References 177
Sharp Global Bounds for the Hessian onPseudo-Hermitian Manifolds 178
1 Introduction 178
2 The main theorem 182
3 Applications to PDE 186
Acknowledgements 190
References 190
Recent Progress on the Global Well-Posednessof the KPI Equation 192
References 196
On Monge–Ampère Type Equationsand Applications 198
1 The Monge–Ampère equation 198
1.1 Basic facts 198
1.2 The Dirichlet problem 200
1.3 Regularity of solutions 201
1.4 Estimates for the linearized Monge–Ampère equation 203
2 A Monge–Ampère type equation for reflectors 204
2.1 Snell’s law 204
2.2 The reflector problem 204
2.3 Notion of weak solution for the reflector problem 205
2.4 Results 206
Acknowledgements 208
References 209
Index 210
Applied and Numerical Harmonic Analysis 213