The trace theorem, the Luzin N- and Morse-Sard properties for the sharp case of Sobolev-Lorentz mappings

<p>We prove Luzin <i>N</i>- and Morse–Sard properties for mappings <i>v</i>:ℝ<i>ⁿ</i>→ℝ<i>^d</i> of the Sobolev–Lorentz class W^<i>k</i><sub style="position: relative; left: -.8em;"><i>p</i>,1</sub> <i>p</i> = <i>n/k</i> (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 <i>q</i>=<i>p</i>. Using these results, we find also some very natural approximation and differentiability properties for functions in W^<i>k</i><sub style="position: relative; left: -.8em;"><i>p</i>,1</sub> with exceptional set of small Hausdorff content.</p>