We present joint work with D. Beltran and J. Hickman on sharp local smoothing estimates for general Fourier integral operators.  Local smoothing bounds imply major estimates in harmonic analysis, including Bochner-Riesz estimates, oscillatory integral estimates and bounds for the size of Besicovitch sets and related problems involving Kakeya maximal functions.  Our work gives a sharp resolution to the most general form of the local smoothing problem formulated by the speaker in the early 1990s.  We rely on decoupling estimates of Bourgain and Demeter.

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