The HADES seminar on Tuesday, September 24th, will be at 2:00pm in Room 740.

Speaker: Semyon Dyatlov

Abstract: How fast can the Fourier transform of a nontrivial compactly supported function decay at infinity? Certainly it can be faster than any polynomial (any smooth compactly supported function has such Fourier decay), but it cannot decay exponentially (as this would imply real analyticity of the function). The Beurling–Malliavin Theorem gives a partial answer to this question in dimension 1: if $\omega:\mathbb R\to [0,\infty)$ is a Lipschitz continuous weight such that $\int_{\mathbb R}\frac{\omega(x)\,dx}{1+x^2} <\infty$, then there exists a nontrivial function $u$ with $|\hat u|\leq e^{-\omega}$. This theorem has been used by Bourgain and Dyatlov in the proof of Fractal Uncertainty Principle (FUP) in dimension 1.

I will first give a proof of a weaker version of the Beurling–Malliavin Theorem which is still sufficient for the application to FUP. Then I will discuss the generalization of this theorem to higher dimensions, which has recently been used by Alex Cohen in his proof of higher-dimensional FUP.