非阿尔弗斯正则空间中的度量准正则性和索波列夫正则性

IF 0.9 3区 数学 Q2 MATHEMATICS
Panu Lahti, Xiaodan Zhou
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Unlike previous results, we only assume a pointwise version of Ahlfors <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0001_eq_008.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>Q</m:mi> </m:math> <jats:tex-math>Q</jats:tex-math> </jats:alternatives> </jats:inline-formula>-regularity, which in particular enables various weighted spaces to be included in the theory. Notably, even in the classical Euclidean setting, we are able to obtain new results using this approach. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0001_eq_009.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\"normal\">loc</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>Q</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mspace width=\"0.33em\" /> <m:mi>Y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>f\\in {N}_{{\\rm{loc}}}^{1,Q}\\left(X;\\hspace{0.33em}Y)</jats:tex-math> </jats:alternatives> </jats:inline-formula> without the strong assumption of the infinitesimal distortion <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0001_eq_010.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mrow> <m:mi>f</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{h}_{f}</jats:tex-math> </jats:alternatives> </jats:inline-formula> belonging to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0001_eq_011.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{L}^{\\infty }\\left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"8 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces\",\"authors\":\"Panu Lahti, Xiaodan Zhou\",\"doi\":\"10.1515/agms-2024-0001\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given a homeomorphism <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2024-0001_eq_001.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>Y</m:mi> </m:math> <jats:tex-math>f:X\\\\to Y</jats:tex-math> </jats:alternatives> </jats:inline-formula> between <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2024-0001_eq_002.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>Q</m:mi> </m:math> <jats:tex-math>Q</jats:tex-math> </jats:alternatives> </jats:inline-formula>-dimensional spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2024-0001_eq_003.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>Y</m:mi> </m:math> <jats:tex-math>X,Y</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2024-0001_eq_004.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>f</m:mi> </m:math> <jats:tex-math>f</jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2024-0001_eq_005.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>f</m:mi> </m:math> <jats:tex-math>f</jats:tex-math> </jats:alternatives> </jats:inline-formula> belongs to the Sobolev class <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2024-0001_eq_006.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msubsup> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\\\"normal\\\">loc</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mspace width=\\\"0.33em\\\" /> <m:mi>Y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{N}_{{\\\\rm{loc}}}^{1,p}\\\\left(X;\\\\hspace{0.33em}Y)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2024-0001_eq_007.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mi>Q</m:mi> </m:math> <jats:tex-math>1\\\\le p\\\\le Q</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2024-0001_eq_008.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>Q</m:mi> </m:math> <jats:tex-math>Q</jats:tex-math> </jats:alternatives> </jats:inline-formula>-regularity, which in particular enables various weighted spaces to be included in the theory. Notably, even in the classical Euclidean setting, we are able to obtain new results using this approach. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2024-0001_eq_009.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\\\"normal\\\">loc</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>Q</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mspace width=\\\"0.33em\\\" /> <m:mi>Y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>f\\\\in {N}_{{\\\\rm{loc}}}^{1,Q}\\\\left(X;\\\\hspace{0.33em}Y)</jats:tex-math> </jats:alternatives> </jats:inline-formula> without the strong assumption of the infinitesimal distortion <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2024-0001_eq_010.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mrow> <m:mi>f</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{h}_{f}</jats:tex-math> </jats:alternatives> </jats:inline-formula> belonging to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2024-0001_eq_011.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{L}^{\\\\infty }\\\\left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula>.\",\"PeriodicalId\":48637,\"journal\":{\"name\":\"Analysis and Geometry in Metric Spaces\",\"volume\":\"8 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-04-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Analysis and Geometry in Metric Spaces\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/agms-2024-0001\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis and Geometry in Metric Spaces","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/agms-2024-0001","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

摘要

给定 Q Q 维空间 X, Y X,Y 之间的同构 f : X → Y f:X\to Y,我们证明在合适的例外集之外满足准同构性度量定义的 f f 意味着 f f 属于 Sobolev 类 N loc 1 , p ( X ; Y ) {N}_{{\rm{loc}}}^{1,p}\left(X;\hspace{0.33em}Y) ,其中 1 ≤ p ≤ Q 1\le p\le Q,也意味着几何定义中准形式性的一个方向。与之前的结果不同,我们只假定了阿赫弗斯 Q Q 规则性的一个点对点版本,这尤其使得各种加权空间都能包含在理论中。值得注意的是,即使在经典欧几里得环境中,我们也能利用这种方法获得新结果。特别是,在包括卡诺群的空间中,我们能够证明 Sobolev 正则性 f∈ N loc 1 , Q ( X ; Y ) f\in {N}_{{\rm{loc}}}^{1,Q}\left(X;\hspace{0.33em}Y) 而无需强假设无穷小变形 h f {h}_{f} 属于 L ∞ ( X ) {L}^{\infty }\left(X) 。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces
Given a homeomorphism f : X Y f:X\to Y between Q Q -dimensional spaces X , Y X,Y , we show that f f satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that f f belongs to the Sobolev class N loc 1 , p ( X ; Y ) {N}_{{\rm{loc}}}^{1,p}\left(X;\hspace{0.33em}Y) , where 1 p Q 1\le p\le Q , and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors Q Q -regularity, which in particular enables various weighted spaces to be included in the theory. Notably, even in the classical Euclidean setting, we are able to obtain new results using this approach. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity f N loc 1 , Q ( X ; Y ) f\in {N}_{{\rm{loc}}}^{1,Q}\left(X;\hspace{0.33em}Y) without the strong assumption of the infinitesimal distortion h f {h}_{f} belonging to L ( X ) {L}^{\infty }\left(X) .
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来源期刊
Analysis and Geometry in Metric Spaces
Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology
CiteScore
1.80
自引率
0.00%
发文量
8
审稿时长
16 weeks
期刊介绍: Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.
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