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":"<p>We study connections between the <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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.</p>","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}
引用次数: 0
Abstract
We study connections between the -differentiability and the -differentiability of Sobolev functions. We prove that -differentiability implies the -differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the -differentiability of Sobolev functions -almost everywhere.