Weights, Extrapolation and the Theory of Rubio de Francia (eBook)

Contents 6
Preface 10
Preliminaries 12
Part I One-Weight Extrapolation 16
Chapter 1 Introduction to Norm Inequalities and Extrapolation 17
1.1 Weighted norm inequalities 18
1.2 The theory of extrapolation 25
1.3 The organization of this book 28
Chapter 2 The Essential Theorem 31
2.1 The new proof 32
2.2 Extensions of the extrapolation theorem 34
Generalized maximal operators 34
Elimination of the operator 35
Sharp constants 37
Off-diagonal extrapolation 38
Extrapolation for arbitrary pairs of operators 38
Limited range extrapolation 39
Extrapolation to Banach function spaces 39
Chapter 3 Extrapolation for Muckenhoupt Bases 41
3.1 Preliminaries 41
Muckenhoupt bases 41
Pairs of functions 44
A technical reduction 45
3.2 Ap extrapolation 47
3.3 Rescaling and extrapolation 50
A1 extrapolation 53
3.4 Sharp extrapolation constants 54
3.5 Off-diagonal extrapolation 58
3.6 Extrapolation for pairs of positive operators 63
Extrapolation for one-sided weights 63
Extrapolation for pairs of positive operators 65
3.7 Limited range extrapolation 68
3.8 Applications 75
Norm inequalities for operators 75
Vector-valued inequalities 75
Coifman-Fefferman inequalities 76
Chapter 4 Extrapolation on Function Spaces 78
4.1 Preliminaries 79
Banach function spaces 79
Examples of function spaces 82
Modular spaces 83
Examples of modular spaces 85
4.2 Extrapolation on Banach function spaces 85
General function spaces 85
Rearrangement invariant spaces 88
4.3 Extrapolation on modular spaces 91
4.4 Applications 97
Modular spaces and r.i. function spaces 98
Variable Lebesgue spaces 98
Part II Two-Weight Factorization and Extrapolation 107
Chapter 5 Preliminary Results 108
5.1 Weights 108
5.2 Orlicz spaces 108
5.3 Orlicz maximal operators 110
5.4 Generalizations of the Ap condition 114
Log bumps 116
Log-log bumps 117
Exponential log bumps 118
Power bumps 119
5.5 The composition of maximal operators 121
5.6 Orlicz fractional maximal operators 125
5.7 Composition of fractional maximal operators 127
Chapter 6 Two-Weight Factorization 133
6.1 Reverse factorization and factored weights 134
6.2 Factorization of weights 136
6.3 Inserting Ap weights 140
6.4 Weights for fractional operators 141
Reverse factorization and factored weights 141
Factorization of weights 143
Chapter 7 Two-Weight Extrapolation 145
7.1 Two-weight extrapolation 147
Extrapolation and families of Orlicz bumps 148
No bump condition 148
Bp bumps 148
Log bumps 149
Exponential log bumps 150
Power bumps 150
7.2 Proof of two-weight extrapolation 152
7.3 Two-weight, weak type extrapolation 158
7.4 Extrapolation for factored weights 160
7.5 Extrapolation for fractional weights 164
7.6 Appendix: A one case proof of extrapolation 166
Chapter 8 Endpoint and A8 Extrapolation 173
8.1 Endpoint extrapolation 175
8.2 Three special cases for the pairs (u,Mu) 177
8.3 The converse of endpoint extrapolation 179
8.4 Endpoint extrapolation for fractional operators 182
Chapter 9 Applications of Two-Weight Extrapolation 185
9.1 The sharp maximal operator 186
Coifman-Fefferman type inequalities 190
Proof of Lemma 9.2 194
9.2 Singular integral operators 196
The conjectures 197
Strong (p, p) inequalities 198
Weak (p, p) inequalities 202
Inequalities for factored weights 204
9.3 Fractional integral operators 206
The conjectures 207
Weak (p, p) inequalities 207
Inequalities for factored weights 209
Chapter 10 Further Applications of Two-Weight Extrapolation 211
10.1 The dyadic square function 212
The conjectures 213
Strong (p, p) inequalities 215
Inequalities for factored weights 219
Proof of Theorems 10.13, 10.14 and 10.19 222
Proof of Theorem 10.12 223
Proof of Theorem 10.16 224
Proof of Theorem 10.18 232
Coifman-Fefferman inequalities 234
10.2 Vector-valued maximal operators 237
The conjectures 237
Strong (p, p) inequalities 239
Weak (p, p) inequalities 244
Inequalities for factored weights 245
Appendix A The Calderón-Zygmund Decomposition 247
A.1 The Calderón-Zygmund decomposition for MF 247
A.2 A weighted Calderón-Zygmund decomposition 250
A.3 A fractional Calderón-Zygmund decomposition 251
A.4 A Calderón-Zygmund decomposition for Borel measures 253
Bibliography 263
Index of Symbols 279
Author Index 283
Subject Index 286