The Dubovitski\uı-Sard Theorem in Sobolev Spaces

Abstract

The Sard theorem from 1942 requires that a mapping $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is of class $C^k$, $k>\max (n-m,0)$. In 1957 Duvovitski\uı generalized Sard's theorem to the case of $C^k$ mappings for all $k$. Namely he proved that, for almost all $y\in {\mathbb{R}}^m$, ${\mathcal{H}}^{\ell }(C_f\cap f^{-1}(y))=0$ where $\ell =\max (n-m-k+1,0)$, ${\mathcal{H}}^{\ell }$ denotes the Hausdorff measure, and $C_f$ is the set of critical points of $f$. In 2001 De Pascale proved that the Sard theorem holds true for Sobolev mappings of the class $W_{\mathrm{loc}}^{k,p}({\mathbb{R}}^n,{\mathbb{R}}^m)$, $k>\max (n-m,0)$ and $p>n$. We will show that also Dubovitski\uı's theorem can be generalized to the case of $W_{\mathrm{loc}}^{k,p}({\mathbb{R}}^n,{\mathbb{R}}^m)$ mappings for all $k\in \mathbb{N}$ and $p>n$.