On differentiability of Sobolev functions with respect to the Sobolev norm

IF 0.8 3区数学 Q2 MATHEMATICS Mathematische Nachrichten Pub Date : 2024-07-22 DOI:10.1002/mana.202300545

Vladimir Gol'dshtein, Paz Hashash, Alexander Ukhlov

{"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":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 10","pages":"3681-3699"},"PeriodicalIF":0.8000,"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":"Mathematische Nachrichten","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300545","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

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.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

关于索波列函数与索波列规范的可微分性

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。

求助全文

约1分钟内获得全文去求助

来源期刊

Mathematische Nachrichten 数学-数学

CiteScore

1.50

自引率

0.00%

发文量

157

审稿时长

4-8 weeks

期刊介绍： Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index

期刊最新文献

Issue Information Contents Solvability of invariant systems of differential equations on H 2 $\mathbb {H}^2$ and beyond Issue Information Contents