关于索波列函数与索波列规范的可微分性

Pub Date : 2024-07-22 DOI:10.1002/mana.202300545

Vladimir Gol'dshtein, Paz Hashash, Alexander Ukhlov

{"title":"关于索波列函数与索波列规范的可微分性","authors":"Vladimir Gol'dshtein, Paz Hashash, Alexander Ukhlov","doi":"10.1002/mana.202300545","DOIUrl":null,"url":null,"abstract":"We study connections between the <math>\n <semantics>\n <msubsup>\n <mi>W</mi>\n <mi>p</mi>\n <mn>1</mn>\n </msubsup>\n <annotation>$W^1_p$</annotation>\n </semantics></math>-differentiability and the <math>\n <semantics>\n <msub>\n <mi>L</mi>\n <mi>p</mi>\n </msub>\n <annotation>$L_p$</annotation>\n </semantics></math>-differentiability of Sobolev functions. We prove that <math>\n <semantics>\n <msubsup>\n <mi>W</mi>\n <mi>p</mi>\n <mn>1</mn>\n </msubsup>\n <annotation>$W^1_p$</annotation>\n </semantics></math>-differentiability implies the <math>\n <semantics>\n <msub>\n <mi>L</mi>\n <mi>p</mi>\n </msub>\n <annotation>$L_p$</annotation>\n </semantics></math>-differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the <math>\n <semantics>\n <msubsup>\n <mi>W</mi>\n <mi>p</mi>\n <mn>1</mn>\n </msubsup>\n <annotation>$W^1_p$</annotation>\n </semantics></math>-differentiability of Sobolev functions <math>\n <semantics>\n <msub>\n <mo>cap</mo>\n <mi>p</mi>\n </msub>\n <annotation>$\\operatorname{cap}_p$</annotation>\n </semantics></math>-almost everywhere.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202300545","citationCount":"0","resultStr":"{\"title\":\"On differentiability of Sobolev functions with respect to the Sobolev norm\",\"authors\":\"Vladimir Gol'dshtein, Paz Hashash, Alexander Ukhlov\",\"doi\":\"10.1002/mana.202300545\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study connections between the <math>\\n <semantics>\\n <msubsup>\\n <mi>W</mi>\\n <mi>p</mi>\\n <mn>1</mn>\\n </msubsup>\\n <annotation>$W^1_p$</annotation>\\n </semantics></math>-differentiability and the <math>\\n <semantics>\\n <msub>\\n <mi>L</mi>\\n <mi>p</mi>\\n </msub>\\n <annotation>$L_p$</annotation>\\n </semantics></math>-differentiability of Sobolev functions. We prove that <math>\\n <semantics>\\n <msubsup>\\n <mi>W</mi>\\n <mi>p</mi>\\n <mn>1</mn>\\n </msubsup>\\n <annotation>$W^1_p$</annotation>\\n </semantics></math>-differentiability implies the <math>\\n <semantics>\\n <msub>\\n <mi>L</mi>\\n <mi>p</mi>\\n </msub>\\n <annotation>$L_p$</annotation>\\n </semantics></math>-differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the <math>\\n <semantics>\\n <msubsup>\\n <mi>W</mi>\\n <mi>p</mi>\\n <mn>1</mn>\\n </msubsup>\\n <annotation>$W^1_p$</annotation>\\n </semantics></math>-differentiability of Sobolev functions <math>\\n <semantics>\\n <msub>\\n <mo>cap</mo>\\n <mi>p</mi>\\n </msub>\\n <annotation>$\\\\operatorname{cap}_p$</annotation>\\n </semantics></math>-almost everywhere.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-07-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202300545\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300545\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300545","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

我们研究索波列函数的-可微性与-可微性之间的联系。我们证明了-可微分性意味着-可微分性，但相反的暗示并不成立。我们还讨论了近似可微分性的概念。此外，我们还考虑了几乎无处不在的 Sobolev 函数的可微性。

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

On differentiability of Sobolev functions with respect to the Sobolev norm

We study connections between the $W_{p}^{1}$ -differentiability and the $L_{p}$ -differentiability of Sobolev functions. We prove that $W_{p}^{1}$ -differentiability implies the $L_{p}$ -differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the $W_{p}^{1}$ -differentiability of Sobolev functions ${cap}_{p}$ -almost everywhere.

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