Vol. 13, No. 2, 2020

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Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds

David Beltran, Jonathan Hickman and Christopher D. Sogge

Vol. 13 (2020), No. 2, 403–433
Abstract

The sharp Wolff-type decoupling estimates of Bourgain and Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent.

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Keywords
local smoothing, variable coefficient, Fourier integral operators, decoupling inequalities
Mathematical Subject Classification 2010
Primary: 35S30
Secondary: 35L05
Milestones
Received: 15 February 2018
Revised: 27 December 2018
Accepted: 23 February 2019
Published: 19 March 2020
Authors
David Beltran
Basque Center for Applied Mathematics
Bilbao
Spain
Jonathan Hickman
Mathematical Institute
University of St Andrews
St Andrews
United Kingdom
Christopher D. Sogge
Department of Mathematics
Johns Hopkins University
Baltimore, MD
United States