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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