Reversed Strichartz estimates for wave on non-trapping asymptotically hyperbolic manifolds and applications

TL;DR

This paper establishes reversed Strichartz estimates for wave equations on non-trapping asymptotically hyperbolic manifolds, solving an open problem in global well-posedness and providing new spectral and maximal operator estimates.

Contribution

It introduces reversed Strichartz estimates in this geometric setting and applies them to nonlinear wave equations, addressing previously unresolved endpoint cases.

Findings

Reversed Strichartz estimates proven for wave equations on asymptotically hyperbolic manifolds.

Solved the open problem regarding endpoint global well-posedness for nonlinear waves.

Derived new spectral projector and maximal wave operator estimates.

Abstract

We provide reversed Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds using cluster estimates for spectral projectors proved previously in such generality. As a consequence, we solve a problem left open in \cite{SSWZ} about the endpoint case for global well-posedness of nonlinear wave equations. We also provide estimates in this context for the maximal wave operator.