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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.

Inhaltsverzeichnis

Frontmatter

One-Weight Extrapolation

Frontmatter

Chapter 1. Introduction to Norm Inequalities and Extrapolation

Abstract
The extrapolation theorem of Rubio de Francia is one of the deepest results in the study of weighted norm inequalities in harmonic analysis: it is simple to state but has profound and diverse applications. The goal of this book is to give a systematic development of the theory of extrapolation, one which unifies known results and expands them in new directions. In addition, we want to show how extrapolation theory, broadly defined, can be applied to the theory of weighted norm inequalities. We describe new and simpler proofs of known results, and then prove new results and show how these lead to additional open questions.
David V. Cruz-Uribe, José María Martell, Carlos Pérez

Chapter 2. The Essential Theorem

Abstract
In this chapter we give our new proof of the Rubio de Francia extrapolation theorem, Theorem 1.4, and discuss how our proof allows a number of powerful generalizations. For the convenience of the reader we restate it here.
David V. Cruz-Uribe, José María Martell, Carlos Pérez

Chapter 3. Extrapolation for Muckenhoupt Bases

Abstract
In this chapter we prove our fundamental generalization of the Rubio de Francia extrapolation theorem. Our result combines the two most important generalizations we discussed in Chapter 2: generalized maximal operators and the elimination of the operator. We then consider the results we get by rescaling, particularly A extrapolation. Next, we prove four variants of our main result: extrapolation with sharp constants, off-diagonal extrapolation, extrapolation with pairs of positive operators in place of the maximal operator (which we apply to one-sided A p weights), and limited range extrapolation. Finally, we survey some of the many possible applications of our results. Extrapolation in the context of Banach function spaces and modular spaces will be discussed in Chapter 4.
David V. Cruz-Uribe, José María Martell, Carlos Pérez

Chapter 4. Extrapolation on Function Spaces

Abstract
In this chapter we extend our theory of extrapolation to get norm inequalities on Banach function spaces starting from inequalities in weighted L p.
David V. Cruz-Uribe, José María Martell, Carlos Pérez

Two-Weight Factorization and Extrapolation

Frontmatter

Chapter 5. Preliminary Results

Abstract
In this chapter we gather together many of the basic facts we will use throughout Part II: Orlicz spaces, A p-type conditions, maximal operators, and fractional maximal operators. Fractional maximal operators are treated separately even though a unified presentation is possible; there is some duplication but we believe that the exposition is made clearer this way. Some of the results we present are not new and we give complete references. In some cases we give the proofs, either for completeness or because we can improve upon the original.
David V. Cruz-Uribe, José María Martell, Carlos Pérez

Chapter 6. Two-Weight Factorization

Abstract
Our primary goal in this chapter is to define the appropriate analogs of A 1 weights and prove a “reverse factorization” property for pairs of weights that satisfy the A p bump conditions defined in Chapter 5. As we discussed in Chapter 2, reverse factorization is an essential tool in the proof of the Rubio de Francia extrapolation theorem; in the two-weight case these ideas play a similar but less direct role, as we will discuss in Chapter 7.
David V. Cruz-Uribe, José María Martell, Carlos Pérez

Chapter 7. Two-Weight Extrapolation

Abstract
In this chapter we give our generalization of the Rubio de Francia extrapolation theorem to the two-weight setting.
David V. Cruz-Uribe, José María Martell, Carlos Pérez

Chapter 8. Endpoint and A ∞ Extrapolation

Abstract
In this chapter we consider further variations of the two-weight extrapolation theorem proved in Chapter 7.
David V. Cruz-Uribe, José María Martell, Carlos Pérez

Chapter 9. Applications of Two-Weight Extrapolation

Abstract
In this chapter and the next we apply the two-weight extrapolation theorems in Chapters 7 and 8 to the theory of two-weight norm inequalities.
David V. Cruz-Uribe, José María Martell, Carlos Pérez

Chapter 10. Further Applications of Two-Weight Extrapolation

Abstract
In this chapter we continue to apply the extrapolation theorems in Chapters 7 and 8 to the theory of two-weight norm inequalities. We consider two operators: the dyadic square function and the vector-valued maximal operator. There are two different approaches to these operators. On the one hand, they can be treated as vector-valued singular integrals, and so the natural conjectures for the weak and strong type inequalities would be the analogs of those for singular integrals in Section 9.2.
David V. Cruz-Uribe, José María Martell, Carlos Pérez

Backmatter

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