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

{"title":"保留渐近 Lipschitz 边界的平滑近似值","authors":"Enrico Pasqualetto","doi":"arxiv-2409.01772","DOIUrl":null,"url":null,"abstract":"The goal of this note is to prove that every real-valued Lipschitz function\non a Banach space can be pointwise approximated on a given $\\sigma$-compact set\nby smooth cylindrical functions whose asymptotic Lipschitz constants are\ncontrolled. This result has applications in the study of metric Sobolev and BV\nspaces: it implies that smooth cylindrical functions are dense in energy in\nthese kinds of functional spaces defined over any weighted Banach space.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"392 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Smooth approximations preserving asymptotic Lipschitz bounds\",\"authors\":\"Enrico Pasqualetto\",\"doi\":\"arxiv-2409.01772\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The goal of this note is to prove that every real-valued Lipschitz function\\non a Banach space can be pointwise approximated on a given $\\\\sigma$-compact set\\nby smooth cylindrical functions whose asymptotic Lipschitz constants are\\ncontrolled. This result has applications in the study of metric Sobolev and BV\\nspaces: it implies that smooth cylindrical functions are dense in energy in\\nthese kinds of functional spaces defined over any weighted Banach space.\",\"PeriodicalId\":501444,\"journal\":{\"name\":\"arXiv - MATH - Metric Geometry\",\"volume\":\"392 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Metric Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.01772\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.01772","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

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.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

arXiv - MATH - Metric Geometry

自引率

0.00%

发文量