A universal Stein-Tomas restriction estimate for measures in three dimensions

We study restriction estimates in R^3 for surfaces given as graphs of W^1_1(R^2) (integrable gradient) functions. We obtain a "universal" L^2(mu) -> L^4(R^3, L^2(SO(3))) estimate for the extension operator f -> \hat{f mu} in three dimensions. We also prove that the three dimensional estimate holds for any Frostman measure supported on a compact set of Hausdorff dimension greater than two. The approach is geometric and is influenced by a connection with the Falconer distance problem.