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The Morse-Sard theorem revisited (1511.05822v5)

Published 18 Nov 2015 in math.CA, math.AP, and math.DG

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.

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