The Morse-Sard theorem revisited

TL;DR

This paper develops an abstract Morse-Sard theorem framework that extends previous results to functions with Stepanov-type regularity, showing that critical values are Lebesgue-null under broader conditions.

Contribution

It introduces a new abstract Morse-Sard theorem that applies to Stepanov functions, broadening the class of functions for which critical value nullity is established.

Findings

Reproduces De Pascale's result for Sobolev functions with p>n

Establishes critical value nullity for Stepanov functions with bounded difference quotients

Shows sufficiency of a measure condition outside a small Hausdorff measure set when m=1

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 $W^{k,p}_{\textrm{loc}}(\mathbb{R}^n, \mathbb{R}^m)$ 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 $$\limsup_{h\to 0}\frac{|D^{k-1}f(x+h)-D^{k-1}f(x)|}{|h|}<\infty$$ 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.