{"title":"关于有界变化函数的一般链式法则","authors":"Camillo Brena , Nicola Gigli","doi":"10.1016/j.na.2024.113518","DOIUrl":null,"url":null,"abstract":"<div><p>We give an alternative proof of the general chain rule for functions of bounded variation (Ambrosio and Maso, 1990), which allows to compute the distributional differential of <span><math><mrow><mi>φ</mi><mo>∘</mo><mi>F</mi></mrow></math></span>, where <span><math><mrow><mi>φ</mi><mo>∈</mo><mi>LIP</mi><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>F</mi><mo>∈</mo><mi>BV</mi><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. In our argument we build on top of recently established links between “closability of certain differentiation operators” and “differentiability of Lipschitz functions in related directions” (Alberti et al., 2023): we couple this with the observation that “the map that takes <span><math><mi>φ</mi></math></span> and returns the distributional differential of <span><math><mrow><mi>φ</mi><mo>∘</mo><mi>F</mi></mrow></math></span> is closable” to conclude.</p><p>Unlike previous results in this direction, our proof can directly be adapted to the non-smooth setting of finite dimensional RCD spaces.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"About the general chain rule for functions of bounded variation\",\"authors\":\"Camillo Brena , Nicola Gigli\",\"doi\":\"10.1016/j.na.2024.113518\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We give an alternative proof of the general chain rule for functions of bounded variation (Ambrosio and Maso, 1990), which allows to compute the distributional differential of <span><math><mrow><mi>φ</mi><mo>∘</mo><mi>F</mi></mrow></math></span>, where <span><math><mrow><mi>φ</mi><mo>∈</mo><mi>LIP</mi><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>F</mi><mo>∈</mo><mi>BV</mi><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. In our argument we build on top of recently established links between “closability of certain differentiation operators” and “differentiability of Lipschitz functions in related directions” (Alberti et al., 2023): we couple this with the observation that “the map that takes <span><math><mi>φ</mi></math></span> and returns the distributional differential of <span><math><mrow><mi>φ</mi><mo>∘</mo><mi>F</mi></mrow></math></span> is closable” to conclude.</p><p>Unlike previous results in this direction, our proof can directly be adapted to the non-smooth setting of finite dimensional RCD spaces.</p></div>\",\"PeriodicalId\":49749,\"journal\":{\"name\":\"Nonlinear Analysis-Theory Methods & Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-02-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Analysis-Theory Methods & Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24000373\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Theory Methods & Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0362546X24000373","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
About the general chain rule for functions of bounded variation
We give an alternative proof of the general chain rule for functions of bounded variation (Ambrosio and Maso, 1990), which allows to compute the distributional differential of , where and . In our argument we build on top of recently established links between “closability of certain differentiation operators” and “differentiability of Lipschitz functions in related directions” (Alberti et al., 2023): we couple this with the observation that “the map that takes and returns the distributional differential of is closable” to conclude.
Unlike previous results in this direction, our proof can directly be adapted to the non-smooth setting of finite dimensional RCD spaces.
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