Multilinear Calderón–Zygmund Singular Integrals

Abstract

It is quite common for linear operators to depend on several functions of which only one is thought of as the main variable and the remaining ones are usually treated as parameters. Examples of such operators are ubiquitous in harmonic analysis: multiplier operators, homogeneous singular integrals associated with functions Ω on the sphere, Littlewood–Paley operators, Calderón commutators, and the Cauchy integral along Lipschitz curves. Treating the additional functions that arise in these operators as frozen parameters often provides limited results that could be thought analogous to those that one obtains by studying calculus of functions of several variables by freezing variables. In this article, we advocate a more flexible point of view in the study of linear operators, analogous to that employed in pure multivariable calculus. Unfreezing the additional functions and treating them as input variables provides a more robust approach that often yields sharper results in terms of regularity of the input functions.

This chapter contains the material covered in a minicourse given by the author at the Centre de Recerca Matemàtica in Barcelona during the period May 4–8, 2009. The course was an expanded version of a series of three lectures delivered by the author two weeks earlier (April 23–25) at the New Mexico Analysis Seminar held at the University of New Mexico in Albuquerque. The author would like to thank 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 Carlos Pèrez in May 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 Xavier Tolsa and Joan Verdera. The author would also like to thank the Centre de Recerca Matemàtica for providing an inspiring environment for research to the participants during these workshops. This research was partially supported by the NSF under grant DMS 0900946.