Multilinear singular integrals

Abstract

We survey the thoery of multilinear singular integral operators with modulation symmetry. The basic example for this theory is the bilinear Hilbert transform and its multilinear variants. We outline a proof of boundedness of Carleson's operator which shows the close connection of this operator to multilinear singular inte- grals. We discuss particular multilinear singular integrals which historically arose in the study of eigenfunctions of Schrodinger operators. This survey article arose from a series of three expository lectures given at the 6th International Conference on Harmonic Analysis and Partial Di! erential Equations at El Escorial 2000. I would like to thank the organizers for organizing this stimulating and successful conference. The article is divided into three chapters following the three lectures at El Escorial. The first section gives an introduction into the subject of multilinear singular integrals with modulation symmetries as it has evolved over the past five years. The second section presents a proof of boundedness of the Carleson operator, the essential ingredient in the proof of Carleson's theorem on almost everywhere convergence of Fourier series. We follow closely the work (29), with additional comments as pre- sented during the lecture. While Carleson's operator is not a multilinear operator itself, it serves as a good model for the type of arguments used in the theory: thus the theory extends to other objects than multilinear operators. The third section is again purely expository and gives an overview of how the discussed theory of multilinear singular integrals plays a role in eigenfunction expansions of one dimensional Schrodinger operators. In fact, some of the most prominent objects in the theory such as the bilinear Hilbert transform and Carleson's operator enter directly into these multilinear expansions.