On the weak Sard property

TL;DR

The paper investigates the relationships among strong, relaxed, and weak Sard properties for functions on domains of dimension 1 and 2, showing that the weak Sard property does not imply the relaxed Sard property even for $f$ in $\bigcap_{\alpha\in(0,1)} C^{1,\alpha}$, while monotone continuous maps equate the two. It provides explicit constructions in 1D and 2D demonstrating this separation via Cantor-type and fractal-square frameworks, and demonstrates that Hölder regularity is insufficient for the relaxed Sard property in dimension one. The results clarify the sharpness of Sard-type conclusions under varying regularity and monotonicity assumptions, with implications for geometric measure theory and the analysis of critical sets. The work collectively delineates the hierarchy of Sard-type properties and their dependence on dimension and regularity.

Abstract

If $f\colon [0,1]^2 \to \mathbb{R}$ is of class $C^2$ then Sard's theorem implies that $f$ has the following relaxed Sard property: the image under $f$ of the Lebesgue measure restricted to the critical set of $f$ is a singular measure. We show that for $C^{1,α}$ functions with $α<1$ this property is strictly stronger than the weak Sard property introduced by Alberti, Bianchini and Crippa, while for any monotone continuous function these two properties are equivalent. We also show that even in the one-dimensional setting Hölder regularity is not sufficient for the relaxed Sard property.

Figures (3)

• Figure 1: First two steps of function construction and their pushforward measures
• Figure 2: First two steps of function construction
• Figure 3: Critical sets of $f_1$ and $f_2$

Theorems & Definitions (22)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• proof : Proof of Theorem \ref{['wsp-eq-rsp-for-monotone-fn']}
• Proposition 2.1
• proof
• Proposition 2.2