We prove Luzin N- and Morse–Sard properties for mappings \(v:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^d\) of the Sobolev–Lorentz class \(\mathrm {W}^{k}_{p,1}\), \(p=\frac{n}{k}\) (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 \(\mathrm {W}^{k}_{p,1}\) with exceptional set of small Hausdorff content.

中文翻译：

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Sobolev-Lorentz映射的尖锐情形的迹定理，Luzin N-和Morse-Sard属性。

我们证明了Sobolev–Lorentz类\（\ mathrm {W }的映射\（v：{\ mathbb {R}} ^ n \ rightarrow {\ mathbb {R}} ^ d \）的Luzin N-和Morse–Sard属性} ^ {k} _ {p，1} \），\（p = \ frac {n} {k} \）（这是保证映射连续性的尖锐案例）。我们的主要工具是极限情况\（q = p \）的Lorentz函数的Riesz势的新跟踪定理。使用这些结果，我们还会发现\（\ mathrm {W} ^ {k} _ {p，1} \）中具有特殊Hausdorff小内容集的函数的一些非常自然的逼近和微分性质。