The Zygmund Morse-Sard theorem
The classical Morse-Sard Theorem says that the set of critical values of \(f : \mathbb{R}^{n + k} \to \mathbb{R}^ n\) has Lebesgue measure zero if \(f \in C^{k + 1}\). We show the \(C^{k + 1}\) smoothness requirement can be weakened to \(C^{k + \text{Zygmund}}\). This is corollary to the following theorem: For integers \(n>m>r>0\), let \(s=(n-r)/(m-r)\); if \(f : \mathbb{R}^ n \to \mathbb{R}^ m\) belongs to the Lipschitz class \(\Lambda_ s\) and \(E\) is a set of rank \(r\) for \(f\), then \(f(E)\) has measure zero.
scientific article; zbMATH DE number 653346