The Zygmund Morse-Sard theorem
type: Item
instance of: scholarly article
title: The Zygmund Morse-Sard theorem
zbMATH Open document ID: 0802.58010
DOI: 10.1007/BF02921589
author: Alec Kercheval
published in: The Journal of Geometric Analysis
publication date: 1994-12-19T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 26B35
zbMATH DE Number: 653346
zbMATH Keywords: Morse-Sard theorem
zbMATH Keywords: Lipschitz space
zbMATH Keywords: modulus of continuity
zbMATH Keywords: critical values
zbMATH Keywords: measure zero
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 653346