{"title":"度量空间中的准共形、Lipschitz 和 BV 映射","authors":"Panu Lahti","doi":"10.1515/acv-2022-0071","DOIUrl":null,"url":null,"abstract":"Consider a mapping <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>Y</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0071_eq_0735.png\" /> <jats:tex-math>{f\\colon X\\to Y}</jats:tex-math> </jats:alternatives> </jats:inline-formula> between two metric measure spaces. We study generalized versions of the local Lipschitz number <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Lip</m:mi> <m:mo></m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0071_eq_0649.png\" /> <jats:tex-math>{\\operatorname{Lip}f}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, as well as of the distortion number <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>H</m:mi> <m:mi>f</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0071_eq_0480.png\" /> <jats:tex-math>{H_{f}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> that is used to define quasiconformal mappings. Using these numbers, we give sufficient conditions for <jats:italic>f</jats:italic> being a BV mapping <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>BV</m:mi> <m:mi>loc</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mi>Y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0071_eq_0759.png\" /> <jats:tex-math>{f\\in\\mathrm{BV}_{\\mathrm{loc}}(X;Y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or a Newton–Sobolev mapping <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msubsup> <m:mi>N</m:mi> <m:mi>loc</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mi>Y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0071_eq_0751.png\" /> <jats:tex-math>{f\\in N_{\\mathrm{loc}}^{1,p}(X;Y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0071_eq_0395.png\" /> <jats:tex-math>{1\\leq p<\\infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"45 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Quasiconformal, Lipschitz, and BV mappings in metric spaces\",\"authors\":\"Panu Lahti\",\"doi\":\"10.1515/acv-2022-0071\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Consider a mapping <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>Y</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_acv-2022-0071_eq_0735.png\\\" /> <jats:tex-math>{f\\\\colon X\\\\to Y}</jats:tex-math> </jats:alternatives> </jats:inline-formula> between two metric measure spaces. We study generalized versions of the local Lipschitz number <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>Lip</m:mi> <m:mo></m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_acv-2022-0071_eq_0649.png\\\" /> <jats:tex-math>{\\\\operatorname{Lip}f}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, as well as of the distortion number <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>H</m:mi> <m:mi>f</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_acv-2022-0071_eq_0480.png\\\" /> <jats:tex-math>{H_{f}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> that is used to define quasiconformal mappings. Using these numbers, we give sufficient conditions for <jats:italic>f</jats:italic> being a BV mapping <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>BV</m:mi> <m:mi>loc</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mi>Y</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_acv-2022-0071_eq_0759.png\\\" /> <jats:tex-math>{f\\\\in\\\\mathrm{BV}_{\\\\mathrm{loc}}(X;Y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or a Newton–Sobolev mapping <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msubsup> <m:mi>N</m:mi> <m:mi>loc</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mi>Y</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_acv-2022-0071_eq_0751.png\\\" /> <jats:tex-math>{f\\\\in N_{\\\\mathrm{loc}}^{1,p}(X;Y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi mathvariant=\\\"normal\\\">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_acv-2022-0071_eq_0395.png\\\" /> <jats:tex-math>{1\\\\leq p<\\\\infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.\",\"PeriodicalId\":49276,\"journal\":{\"name\":\"Advances in Calculus of Variations\",\"volume\":\"45 1\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-01-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Calculus of Variations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/acv-2022-0071\",\"RegionNum\":3,\"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":"Advances in Calculus of Variations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/acv-2022-0071","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
考虑两个度量空间之间的映射 f : X → Y {f\colon X\to Y} 。我们研究局部 Lipschitz 数 Lip f {\operatorname{Lip}f} 的广义版本，以及用于定义准共形映射的变形数 H f {H_{f}} 的广义版本。利用这些数字，我们给出了 f 是 BV 映射 f∈ BV loc ( X ; Y ) {f\in\mathrm{BV}_{\mathrm{loc}}(X. Y)} 或牛顿映射 f∈ BV loc ( X ; Y ) {f\in\mathrm{BV}_{\mathrm{loc}}(X. Y)} 的充分条件；Y)} 或者牛顿-索博列夫映射 f∈ N loc 1 , p ( X ; Y ) {f\in N_{\mathrm{loc}}^{1,p}(X;Y)} , 其中 1 ≤ p < ∞ {1\leq p<\infty} 。
Quasiconformal, Lipschitz, and BV mappings in metric spaces
Consider a mapping f:X→Y{f\colon X\to Y} between two metric measure spaces. We study generalized versions of the local Lipschitz number Lipf{\operatorname{Lip}f}, as well as of the distortion number Hf{H_{f}} that is used to define quasiconformal mappings. Using these numbers, we give sufficient conditions for f being a BV mapping f∈BVloc(X;Y){f\in\mathrm{BV}_{\mathrm{loc}}(X;Y)} or a Newton–Sobolev mapping f∈Nloc1,p(X;Y){f\in N_{\mathrm{loc}}^{1,p}(X;Y)}, with 1≤p<∞{1\leq p<\infty}.
期刊介绍：
Advances in Calculus of Variations publishes high quality original research focusing on that part of calculus of variation and related applications which combines tools and methods from partial differential equations with geometrical techniques.
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