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

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