Wolff’s inequality for hypersurfaces

We extend Wolff’s “local smoothing” inequality [18] to a wider class of not necessarily conical hypersurfaces of codimension 1. This class includes 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. Mathematics Subject Classification: 42B08, 42B15. The purpose of this article is to extend the “local smoothing” inequality of [18], [5], proved there for circular cones, to a wider class of bounded surfaces of codimension 1 in R, d ≥ 2. Recall that Wolff’s inequality [18] states that if f is a function with Fourier transform supported in a δ-neighbourhood of the segment of the circular cone with 1 ≤ |x| ≤ 2, then ∀ǫ ∃Cǫ : ‖f‖p ≤ Cǫδ d−1 2 + d p ‖f‖p,δ, (1) with d = 2 and p > 74; this was then extended in [5] to d ≥ 3, p > 2 + 32 3d−7 , and d ≥ 4, p > 2 + 8 d−3 . Here the norm on the right side is defined by