纯不可约度量空间与Lipschitz函数的扰动

IF 5.4 3区 材料科学 Q2 CHEMISTRY, PHYSICAL
David Bate
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引用次数: 14

摘要

通过研究所有有界1-Lipschitz函数$f\colonX\to\mathbb R^m$的集合的元素相对于上确界范数的任意小扰动,我们刻画了具有有限Hausdorff$n$-测度的完备度量空间$X$的纯$n$-不可复约子集$S$。在一个这样的刻画中,它表明,如果$S$几乎在所有地方都具有正的较低密度,那么$\mathcal H^n(f(S))=0的所有$f$的集合是残差。相反,如果$E\subet X$是$n$-可纠正的,且$\mathcal H^n(E)>0$,则所有$f$的集合,且$\athcal H^ n(f(E))>0$是残差。这些结果为任意度量空间中的Besicovich-Federer投影定理提供了一个替代,该定理在欧几里得空间外是错误的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Purely unrectifiable metric spaces and perturbations of Lipschitz functions
We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $f\colon X \to \mathbb R^m$ with respect to the supremum norm. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, then the set of all $f$ with $\mathcal H^n(f(S))=0$ is residual. Conversely, if $E\subset X$ is $n$-rectifiable with $\mathcal H^n(E)>0$, the set of all $f$ with $\mathcal H^n(f(E))>0$ is residual. These results provide a replacement for the Besicovitch-Federer projection theorem in arbitrary metric spaces, which is known to be false outside of Euclidean spaces.
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来源期刊
ACS Applied Energy Materials
ACS Applied Energy Materials Materials Science-Materials Chemistry
CiteScore
10.30
自引率
6.20%
发文量
1368
期刊介绍: ACS Applied Energy Materials is an interdisciplinary journal publishing original research covering all aspects of materials, engineering, chemistry, physics and biology relevant to energy conversion and storage. The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrate knowledge in the areas of materials, engineering, physics, bioscience, and chemistry into important energy applications.
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