Some remarks about The Morse-Sard theorem and approximate differentiability

2017/05/16 by Daniel Azagra, Azagra, Daniel, Miguel García‐Bravo +2
Mathematics · #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Mathematical Dynamics and Fractals #math.CA #math.FA

paper · pdf · doi:10.48550/arxiv.1705.05624

17 pages

arxiv created 2017/05/16 · openalex publication_date 2017/05/16 · arxiv updated 2017/05/17 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

Let n, m be positive integers, n≥ m. We make several remarks on the relationship between approximate differentiability of higher order and Morse-Sard properties. For instance, among other things we show that if a function f:ℝⁿ→ℝ^m is locally Lipschitz and is approximately differentiable of order i almost everywhere with respect to the Hausdorff measure H^i+m-2, for every i=2, …, n-m+1, then f has the Morse-Sard property (that is to say, the image of the critical set of f is null with respect to the Lebesgue measure in ℝ^m).

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