09/17/2025
By Joris Roos
Title: Convergence of the inverse Fourier transform and the family of bilinear Hilbert transforms
Time: 11 a.m. to Noon
Room: Southwick Hall, Room 350W
Everyone is welcome!
Abstract: The Fourier transform decomposes a function into frequencies, and it is a powerful tool in analysis: it "maps" convolutions into multiplications, and derivatives into polynomials. When applied twice to a smooth and rapidly decaying function f, the Fourier transform reproduces f pointwise, up to a reflection about the origin. Therefore, it can be thought of as its own inverse. But how does the inversion converge to the initial datum for general function? How well is a signal f on the real line reconstructed as a superposition of increasingly many of its frequencies?
The problem was solved by Carleson in 1966 and it is essentially governed by the Carleson maximal operator. This operator exhibits the same type of symmetries of the family of bilinear Hilbert transforms (BHTs), which appear also in other contexts in analysis, e.g. studying Cauchy intergrals on Lipschitz curves. In this talk, I will give a gentle introduction to the time-frequency analysis of these operators, reviewing some classical results in the area. In particular, we will consider how the operators generalize when we increase the dimension of the domain of f.
The new results are based on joint work with Olli Saari (Universitat Politècnica de Catalunya) and Christoph Thiele (Universität Bonn).