On the Connection Between Weighted Norm Inequalities, Commutators and Real Interpolation

On the Connection Between Weighted Norm Inequalities, Commutators and Real Interpolation

by Jesus Bastero, Francisco J. Ruiz, Mario Milman
     
 

ISBN-10: 0821827340

ISBN-13: 9780821827345

Pub. Date: 07/23/2001

Publisher: American Mathematical Society

We show that the class of weights $w$ for which the Calderon 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 Lions-Peetre methods. In particular we obtain extrapolation theorems (in the sense of Rubio de Francia's theory) and reiteration

Overview

We show that the class of weights $w$ for which the Calderon 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 Lions-Peetre 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 Calderon 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 Calderon type operators and their corresponding classes of weights.

Product Details

ISBN-13:
9780821827345
Publisher:
American Mathematical Society
Publication date:
07/23/2001
Series:
Memoirs of the American Mathematical Society Series, #154
Pages:
80

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