Bourgain's counterexample for the Schrödinger maximal function

Speaker: Dr Terry Harris
Affiliation: School of Mathematics and Physics, UQ

Abstract

The Carleson problem for the free Schrödinger equation asks for the minimum regularity on initial data to ensure that, as time t tends to zero, solutions u(x,t) converge a.e. to their initial data u_0(x). Despite conservation of L^2 norms,  L^2 is not sufficient. For the 1D Schrödinger equation, 1/4 of a derivative in L^2 is necessary and sufficient, and it was long thought that 1/4 would be sufficient in higher dimensions. Surprisingly, in 2016 Bourgain gave an example showing that an exponent at least n/(2(n+1)) is necessary for dimension n, instead of 1/4. Bourgain's example uses number theory involving Gauss sums. 

In this talk, I will introduce the problem and explain some ideas from a new proof of the necessary condition based on the X-ray transform. The proof considers a seemingly more general problem for which it is easier to obtain counterexamples, and then shows the general problem is equivalent to the original one. The key tool in the proof is an identity for X-ray transforms which originated in the problem of recovering a 3D object from its X-rays.

If time permits, I will also discuss some techniques from Du-Zhang's 2018 proof that s>n/(2(n+1)) is sufficient, such as multilinear Fourier restriction and Fourier decoupling.