{"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":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"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\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-01-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/acv-2022-0071\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/acv-2022-0071","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","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}.
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