APDE seminar@EHU: Multiple convolutions and multilinear fractal Fourier extension

Abstract

The classical Stein-Tomas theorem extends from the theory of linear Fourier restriction estimates for smooth manifolds to the one of fractal measures exhibiting Fourier decay. In the multilinear “smooth” setting, transversality (a certain geometric condition) allows for estimates beyond those implied by the linear theory. The goal of this talk is to investigate the open-ended question “how does transversality manifest itself in the fractal world?”. The absence of a Gauss map in the fractal setting (and of any other relevant notion coming from differential geometry) makes this question very challenging.     

We will show, for instance, that transversality could manifest itself through integrability properties of the multiple convolution of the measures involved, but that is just the beginning of the story. In the special case of Cantor-type fractals, we will construct multilinear Knapp examples which, in some cases, will give more restrictive necessary conditions for a multilinear theorem to hold than those previously available in the literature. This is joint work with Ana de Orellana (University of St. Andrews, Scotland).