Sharp local smoothing estimates for Fourier integral operators

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.