The Morse-Sard theorem revisited

Abstract

Let $n,m,k$ be positive integers with $k=n-m+1$. We establish an abstract Morse-Sard-type theorem which allows us to deduce, on the one hand, a previous result of De Pascale's for Sobolev $W_{\text{loc}}^{k,p}({\mathbb{R}}^n,{\mathbb{R}}^m)$ functions with $p>n$ and, on the other hand, also the following new result: if $f\in C^{k-1}({\mathbb{R}}^n,{\mathbb{R}}^m)$ satisfies $$\underset{h\to 0}{\mathrm{limsup}}\frac{|D^{k-1}f(x+h)-D^{k-1}f(x)|}{|h|}<\infty$$ for every $x\in {\mathbb{R}}^n$ (that is, $D^{k-1}f$ is a Stepanov function), then the set of critical values of $f$ is Lebesgue-null in ${\mathbb{R}}^m$. In the case that $m=1$ we also show that this limiting condition holding for every $x\in {\mathbb{R}}^n\setminus \mathcal{N}$, where $\mathcal{N}$ is a set of zero $(n-2+\alpha )$-dimensional Hausdorff measure for some $0<\alpha <1$, is sufficient to guarantee the same conclusion.

arXiv commentWe corrected some misprints and made some changes in the introduction