Singular integrals meet modulation invariance

TL;DR

This paper explores the intersection of singular integrals and modulation invariance in Fourier analysis, highlighting recent advances and ongoing research inspired by classical and modern developments.

Contribution

It surveys recent progress on modulation invariant singular integrals, emphasizing new techniques and results in the context of Fourier analysis and harmonic analysis.

Findings

Revitalization of modulation invariant singular integral theory

Advances in bilinear Hilbert transform analysis

Connections to classical Fourier analysis concepts

Abstract

Many concepts of Fourier analysis on Euclidean spaces rely on the specification of a frequency point. For example classical Littlewood Paley theory decomposes the spectrum of functions into annuli centered at the origin. In the presence of structures which are invariant under translation of the spectrum (modulation) these concepts need to be refined. This was first done by L. Carleson in his proof of almost everywhere convergence of Fourier series in 1966. The work of M. Lacey and the author in the 1990's on the bilinear Hilbert transform, a prototype of a modulation invariant singular integral, has revitalized the theme. It is now subject of active research which will be surveyed in the lecture. Most of the recent related work by the author is joint with C. Muscalu and T. Tao.