Vol. 1, No. 1, 2019

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

Yuanlong Chen and Hart F. Smith

Vol. 1 (2019), No. 1, 101–148
Abstract

We prove space-time dispersive estimates for solutions to the wave equation on compact Riemannian manifolds with bounded curvature tensor, where we assume that the metric tensor is of W1,p regularity for some p > d, which ensures that the curvature tensor is well-defined in the weak sense. The estimates are established for the same range of Lebesgue and Sobolev exponents that hold in the case of smooth metrics. Our results are for bounded time intervals, so by finite propagation velocity they hold also on noncompact manifolds under appropriate uniform geometry conditions.

Keywords
wave equation, dispersive estimates
Mathematical Subject Classification 2010
Primary: 58J45
Secondary: 35L15
Milestones
Received: 8 August 2018
Revised: 12 September 2018
Accepted: 22 October 2018
Published: 21 November 2018
Authors
Yuanlong Chen
Department of Mathematics
University of Washington
Seattle, WA
United States
Hart F. Smith
Department of Mathematics
University of Washington
Seattle, WA
United States