The Dubovitski\u{\i}-Sard Theorem in Sobolev Spaces

TL;DR

This paper extends the Dubovitskif1i-Sard theorem, originally for smooth functions, to Sobolev space mappings, broadening its applicability in analysis and geometric measure theory.

Contribution

It generalizes Dubovitskif1i's theorem to Sobolev spaces for all integer orders and integrability exponents greater than the dimension, expanding the theorem's scope.

Findings

The theorem holds for Sobolev mappings with k in natural numbers and p > n.

Critical sets have Hausdorff measure zero for almost all y.

The generalization applies to a wider class of functions than previously established.

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\u{\i} 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 $W_{\rm loc}^{k,p}(\mathbb{R}^n,\mathbb{R}^m)$, $k>\max(n-m,0)$ and $p>n$. We will show that also Dubovitski\u{\i}'s theorem can be generalized to the case of $W_{\rm loc}^{k,p}(\mathbb{R}^n,\mathbb{R}^m)$ mappings for all $k\in\mathbb{N}$ and $p>n$.