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

TL;DR

This paper establishes a universal restriction estimate in three dimensions for measures supported on surfaces with integrable gradients and extends to measures on sets with Hausdorff dimension greater than two, using geometric methods linked to the Falconer distance problem.

Contribution

It introduces a universal L^2(mu) to L^4(R^3) restriction estimate for measures on certain surfaces and extends the result to measures supported on sets with large Hausdorff dimension.

Findings

Proves a universal restriction estimate for measures on graphs of W^1_1 functions.

Extends the estimate to Frostman measures on sets with Hausdorff dimension > 2.

Uses geometric approach connected to the Falconer distance problem.

Abstract

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.