保留渐近 Lipschitz 边界的平滑近似值

arXiv - MATH - Metric Geometry Pub Date : 2024-09-03 DOI:arxiv-2409.01772

Enrico Pasqualetto

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引用次数: 0

摘要

本论文的目的是证明，巴拿赫空间上的每个实值 Lipschitz 函数都可以在给定的 $\sigma$-compact 集合上通过光滑圆柱函数进行点逼近，而光滑圆柱函数的渐近 Lipschitz 常量是受控的。这一结果在度量索波列夫空间和 BV 空间的研究中具有应用价值：它意味着光滑圆柱函数在这些定义于任意加权巴拿赫空间的函数空间中能量密集。

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Smooth approximations preserving asymptotic Lipschitz bounds

The goal of this note is to prove that every real-valued Lipschitz function on a Banach space can be pointwise approximated on a given $\sigma$-compact set by smooth cylindrical functions whose asymptotic Lipschitz constants are controlled. This result has applications in the study of metric Sobolev and BV spaces: it implies that smooth cylindrical functions are dense in energy in these kinds of functional spaces defined over any weighted Banach space.

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arXiv - MATH - Metric Geometry

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