Some remarks on restriction of the Fourier transform for general measures

Abstract

In this paper we establish a formal connection between the average decay of the Fourier transform of functions with espect to a given measure and the Hausdorff behavior of that measure. We also present a generalization of the classical restriction theorem of Stein and Tomas replacing the sphere with sets of prefixed Hausdorff dimension n - 1 + [alpha], with 0 < [alpha] < 1.