Pointwise convergence of the Klein-Gordon flow

Abstract

We consider the PDEs version of the Carleson problem in the context of the cubic nonlinear Klein-Gordon equation. This means that we aim to establish the lowest regularity class for which one has almost everywhere pointwise convergence of the solutions to the initial data, as $t \to 0$. We prove sharp results for initial data in Sobolev spaces and for their randomized counterparts.