弱Sard性质

IF 1.2 3区数学 Q1 MATHEMATICS

Journal of Mathematical Analysis and Applications Pub Date : 2025-09-01 DOI:10.1016/j.jmaa.2025.130022

Roman V. Dribas, Andrew S. Golovnev, Nikolay A. Gusev

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引用次数: 0

摘要

如果f:[0,1]2→R属于C2类，则Sard定理表明f具有以下松弛的Sard性质：限定于f的临界集的Lebesgue测度在f下的像是一个奇异测度。我们证明了对于具有α<；1的C1，α函数，这个性质严格强于由Alberti， Bianchini和Crippa引入的弱Sard性质，而对于任何单调连续函数，这两个性质是等价的。我们还表明，即使在一维情况下Hölder正则性对于松弛Sard性质也是不充分的。

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On the weak Sard property

f : {[0, 1]}^{2} \to 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.

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来源期刊

Journal of Mathematical Analysis and Applications 数学-数学

CiteScore

2.50

自引率

7.70%

发文量

790

审稿时长

6 months

期刊介绍： The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Papers are sought which employ one or more of the following areas of classical analysis: • Analytic number theory • Functional analysis and operator theory • Real and harmonic analysis • Complex analysis • Numerical analysis • Applied mathematics • Partial differential equations • Dynamical systems • Control and Optimization • Probability • Mathematical biology • Combinatorics • Mathematical physics.