Electronic ISBN:  9781470403249 
Product Code:  MEMO/154/731.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 154; 2001; 80 ppMSC: Primary 46;
We show that the class of weights \(w\) for which the Calderón operator is bounded on \(L^p(w)\) can be used to develop a theory of real interpolation which is more general and exhibits new features when compared to the usual variants of the LionsPeetre methods. In particular we obtain extrapolation theorems (in the sense of Rubio de Francia's theory) and reiteration theorems for these methods. We also consider interpolation methods associated with the classes of weights for which the Calderón operator is bounded on weighted Lorentz spaces and obtain similar results. We extend the commutator theorems associated with the real method of interpolation in several directions. We obtain weighted norm inequalities for higher order commutators as well as commutators of fractional order. One application of our results gives new weighted norm inequalities for higher order commutators of singular integrals with multiplications by BMO functions. We also introduce analogs of the space BMO in order to consider the relationship between commutators for Calderón type operators and their corresponding classes of weights.
ReadershipGraduate students and research mathematicians interested in functional analysis.

Table of Contents

Chapters

1. Introduction

2. Calderón weights

3. Applications to real interpolation: reiteration and extrapolation

4. Other classes of weights

5. Extrapolation of weighted norm inequalities via extrapolation theory

6. Applications to function spaces

7. Commutators defined by the $K$method

8. Generalized commutators

9. The quasi Banach case

10. Applications to harmonic analysis

11. BMO type spaces associated to Calderón weights

12. Atomic decompositions and duality


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We show that the class of weights \(w\) for which the Calderón operator is bounded on \(L^p(w)\) can be used to develop a theory of real interpolation which is more general and exhibits new features when compared to the usual variants of the LionsPeetre methods. In particular we obtain extrapolation theorems (in the sense of Rubio de Francia's theory) and reiteration theorems for these methods. We also consider interpolation methods associated with the classes of weights for which the Calderón operator is bounded on weighted Lorentz spaces and obtain similar results. We extend the commutator theorems associated with the real method of interpolation in several directions. We obtain weighted norm inequalities for higher order commutators as well as commutators of fractional order. One application of our results gives new weighted norm inequalities for higher order commutators of singular integrals with multiplications by BMO functions. We also introduce analogs of the space BMO in order to consider the relationship between commutators for Calderón type operators and their corresponding classes of weights.
Graduate students and research mathematicians interested in functional analysis.

Chapters

1. Introduction

2. Calderón weights

3. Applications to real interpolation: reiteration and extrapolation

4. Other classes of weights

5. Extrapolation of weighted norm inequalities via extrapolation theory

6. Applications to function spaces

7. Commutators defined by the $K$method

8. Generalized commutators

9. The quasi Banach case

10. Applications to harmonic analysis

11. BMO type spaces associated to Calderón weights

12. Atomic decompositions and duality