Convex Hypersurfaces and Fourier Transforms

Abstract

Hilbert transforms and maximal functions along curves and surfaces, spectral synthesis problems, and the study of certain operators related to hyperbolic partial differential and pseudodifferential operators. The problem of estimating such Fourier transforms has a long history. See for example Hlawka [3], Herz [2], Littman [4], Randol [9], [10], Svensson [15], Sogge and Stein [12], [13], Stein [14], Marshall [5], etc. The results of these investigations suggest a strong relation between the local curvature properties of the surface S and the decay of the function a'. We are mainly interested in how the geometry of S near a fixed point xO affects the decay of a6, for by using a partition of unity on S, it is clearly enough to study integrals of the type