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Weights, Extrapolation and the Theory of Rubio de Francia

:	David Cruz-Uribe, José Maria Martell, Carlos Pérez
:	Weights, Extrapolation and the Theory of Rubio de Francia
:	Birkhäuser Basel
:	9783034800723
:	1
:	CHF 84.70
:

:	Analysis
:	English

:	282
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

This book provides a systematic development of the Rubio de Francia theory of extrapolation, its many generalizations and its applications to one and two-weight norm inequalities. The book is based upon a new and elementary proof of the classical extrapolation theorem that fully develops the power of the Rubio de Francia iteration algorithm. This technique allows us to give a unified presentation of the theory and to give important generalizations to Banach function spaces and to two-weight inequalities. We provide many applications to the classical operators of harmonic analysis to illustrate our approach, giving new and simpler proofs of known results and proving new theorems. The book is intended for advanced graduate students and researchers in the area of weighted norm inequalities, as well as for mathematicians who want to apply extrapolation to other areas such as partial differential equations.

	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