Some remarks on restriction of the Fourier transform for general measures

In this paper we establish a formal connection between the average decay of the Fourier transform of functions with respect 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 $0 < \alpha < 1$.