Sharp estimates for maximal operators associated to the wave equation

TL;DR

This paper establishes near-optimal conditions for the boundedness of the maximal operator linked to the wave equation, connecting Sobolev space initial data to Lebesgue space outputs.

Contribution

It provides almost sharp bounds for the maximal operator associated with the wave equation in Sobolev spaces, advancing understanding of its boundedness criteria.

Findings

Derived near-sharp conditions for boundedness

Identified the Sobolev space thresholds for the operator

Connected initial data regularity to output integrability

Abstract

We give almost sharp conditions under which the maximal operator associated with the wave equation with initial data in Sobolev space H^s(R^n) is bounded from H^s(R^n) to L^q(R^n).