Wave Front Sets, Local Smoothing and Bourgain's Circular Maximal Theorem

The purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of Bourgain's circular maximal theorem [1]. We use easy wave front analysis along with techniques previously used in proofs of the Carleson-Sj6lin theorem (see [3],[5],[7]) and in the proof of sharp regularity properties of Fourier integral operators [13]. The circular means operators are defined by

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