Wolff’s inequality for hypersurfaces

We extend Wolff’s “local smoothing ” inequality [19] to a wider class of not necessarily conical hypersurfaces of codimension 1. This class in-cludes surfaces with nonvanishing curvature, as well as certain surfaces with more than one flat direction. An immediate consequence is the Lp-boundedness of the corresponding Fourier multiplier operators. The purpose of this article is to extend the “local smoothing ” inequality, proved in [19, 6] for circular cones in Rd, d ≥ 2, and in [10] for more general conical surfaces in R3, to a wider class of bounded surfaces of codimension 1 in Rd+1, d ≥ 3