Pure and Applied Analysis

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

Yuanlong Chen and Hart F. Smith

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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 W 1 , 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.

Article information

Source
Pure Appl. Anal., Volume 1, Number 1 (2019), 101-148.

Dates
Received: 8 August 2018
Revised: 12 September 2018
Accepted: 22 October 2018
First available in Project Euclid: 4 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.paa/1549297979

Digital Object Identifier
doi:10.2140/paa.2019.1.101

Mathematical Reviews number (MathSciNet)
MR3900030

Zentralblatt MATH identifier
07027486

Subjects
Primary: 58J45: Hyperbolic equations [See also 35Lxx]
Secondary: 35L15: Initial value problems for second-order hyperbolic equations

Keywords
wave equation dispersive estimates

Citation

Chen, Yuanlong; Smith, Hart F. Dispersive estimates for the wave equation on Riemannian manifolds of bounded curvature. Pure Appl. Anal. 1 (2019), no. 1, 101--148. doi:10.2140/paa.2019.1.101. https://projecteuclid.org/euclid.paa/1549297979


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