Dispersive estimates for the wave equation on Riemannian manifolds of bounded curvature

TL;DR

This paper proves dispersive estimates for wave equations on compact Riemannian manifolds with bounded curvature, extending known results to less smooth metrics with minimal regularity assumptions.

Contribution

It establishes dispersive estimates for wave equations on manifolds with bounded curvature under weak regularity conditions on the metric tensor.

Findings

Dispersive estimates hold for solutions on compact manifolds with bounded curvature.

Results extend to non-compact manifolds with uniform conditions.

Estimates are valid for bounded time intervals with minimal regularity assumptions.

Abstract

We establish space-time dispersive estimates for solutions to the wave equation on compact Riemannian manifolds with bounded sectional curvature, with the same exponents as for $C^\infty$ metrics. The estimates are for bounded time intervals, so by finite propagation velocity the results apply also on non-compact manifolds under appropriate uniform conditions. We assume a priori that in local coordinates the metric tensor components satisfy ${\rm g}_{ij}\in W^{1,p}$ for some $p>d$, which ensures that the curvature tensor is well defined in the weak sense, but this can be relaxed to any assumption that suffices for the local harmonic coordinate calculations in the paper.