Dispersive estimates for 2D-wave equations with critical potentials

TL;DR

This paper establishes sharp dispersive and Strichartz estimates for the 2D-wave equation with critical electromagnetic potentials, addressing the challenges posed by scaling invariance and singularities.

Contribution

It provides the first sharp dispersive estimates for 2D-wave equations with critical electromagnetic potentials, including the full model with electric and magnetic fields.

Findings

Sharp time-decay estimates in the purely magnetic case

Strichartz estimates for the full critical electromagnetic model

Handling of singular potentials not locally integrable

Abstract

We study the 2D-wave equation with a scaling-critical electromagnetic potential. This problem is doubly critical, because of the scaling invariance of the model and the singularities of the potentials, which are not locally integrable. In particular, the diamagnetic phenomenon allows to consider negative electric potential which can be singular in the same fashion as the inverse-square potential. We prove sharp time-decay estimates in the purely magnetic case, and Strichartz estimates for the complete model, involving a critical electromagnetic field.