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

TL;DR

This paper extends sharp Wolff-type decoupling estimates to variable coefficient wave equations on manifolds, leading to new sharp local smoothing results that are optimal in odd dimensions, advancing understanding of Fourier integral operators.

Contribution

It introduces variable coefficient Wolff-type inequalities and applies them to establish sharp local smoothing estimates for wave equations on manifolds, generalizing previous constant coefficient results.

Findings

Extended Wolff-type inequalities to variable coefficients

Established sharp local smoothing estimates on Riemannian manifolds

Results are sharp in odd dimensions for regularity and Lebesgue exponents

Abstract

The sharp Wolff-type decoupling estimates of Bourgain--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.