Hausdorff measures and the Morse-Sard theorem (Q5940007)

The paper generalizes the classical Morse-Sard theorem by sizing the image sets of critical points of a mapping in terms of Hausdorff measures. NEWLINENEWLINENEWLINEFrom the abstract: ``Let \(F:U\subset {\mathbb R}^n\rightarrow {\mathbb R}^n\) be a differentiable function and \(p<m\) an integer. If \(k\geq 1\) is an integer, \(\alpha \in [0,1]\) and \(F\in C^{k+(\alpha)}\), if we set \(C_p(F)=\{x\in U: \text{rank} Df(x)\leq p\}\), then the Hausdorff measure of dimension \((p+{n-p\over k+\alpha})\) of \(C_p(F)\) is zero. NEWLINENEWLINENEWLINEHere \(C^{k+(\alpha)}\) stands for the class of \(C^k\)-functions in \(U\) whose \(k\)-th derivative at each \(x\in C_p(F)\) satisfies a local \(\alpha\)-Hölder condition.