The Trace Theorem, the Luzin N- and Morse-Sard Properties for the Sharp Case of Sobolev-Lorentz Mappings.

Abstract

We prove Luzin N- and Morse-Sard properties for mappings $v:R^n\to R^d$ of the Sobolev-Lorentz class $W_{p,1}^k$ , $p=\frac{n}{k}$ (this is the sharp case that guaranties the continuity of mappings). Our main tool is a new trace theorem for Riesz potentials of Lorentz functions for the limiting case $q=p$ . Using these results, we find also some very natural approximation and differentiability properties for functions in $W_{p,1}^k$ with exceptional set of small Hausdorff content.