Singular Integrals and Weights

Abstract

This chapter is an expanded version of the material covered in a minicourse given at the Centre de Recerca Matemàtica in Barcelona during the week May 4–8, 2009.We provide details and different proofs of known results as well as new ones. We also survey on several recent results related to the core of this course, namely weighted optimal bounds for Calderón–Zygmund operators with weights. The basic topics covered by the lectures revolved around the Rubio de Francia iteration algorithm, the extrapolation theorem with optimal bounds, the Coifman–Fefferman estimate, the Besicovitch covering lemma, and rearrangements of functions. These notes can be seen as a modern introduction to the A _p theory of weights.

The author would like to thank Professors Joan Mateu and Joan Orobitg for coordinating two special minicourses at the Centre de Recerca Matemàtica on Multilinear Harmonic Analysis and Weights presented by the author and Professor Loukas Grafakos during the period May 4–8, 2009. These minicourses were part of a special research program for the academic year 2008–2009 entitled Harmonic Analysis, Geometric Measure Theory, and Quasiconformal Mappings, coordinated by Professors Xavier Tolsa and Joan Verdera. The author would also like to thank the Centre de Recerca Matemàtica for the invitation to spend the semester there and to give this course. The author acknowledges support from the Spanish Ministry of Science and Innovation through grant MTM2009-08934 and from Junta de Andalucía through the proyecto de excelencia FQM-4745.