确定卡诺群中的1-可纠正措施

IF 0.9 3区 数学 Q2 MATHEMATICS
Matthew Badger, Sean Li, Scott Zimmerman
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Our main result is entwined with an extension of <jats:italic>analyst’s traveling salesman theorem</jats:italic>, which characterizes the subsets of rectifiable curves in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0102_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{\\mathbb{R}}}^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (P. W. Jones, <jats:italic>Rectifiable sets and the traveling salesman problem</jats:italic>, Invent. Math. 102 (1990), no. 1, 1–15), in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0102_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{\\mathbb{R}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (K. Okikiolu, <jats:italic>Characterization of subsets of rectifiable curves in</jats:italic> <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0102_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"bold\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{\\bf{R}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, J. London Math. Soc. (2) 46 (1992), no. 2, 336–348), or in an arbitrary Carnot group (S. Li) in terms of local geometric least-squares data called <jats:italic>Jones’</jats:italic> <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0102_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>β</m:mi> </m:math> <jats:tex-math>\\beta </jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:italic>numbers</jats:italic>. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0102_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{\\mathbb{R}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> that charges a rectifiable curve in an arbitrary <jats:italic>complete, doubling, locally quasiconvex metric space</jats:italic>.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"8 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"Identifying 1-rectifiable measures in Carnot groups\",\"authors\":\"Matthew Badger, Sean Li, Scott Zimmerman\",\"doi\":\"10.1515/agms-2023-0102\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. 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Our main result is entwined with an extension of <jats:italic>analyst’s traveling salesman theorem</jats:italic>, which characterizes the subsets of rectifiable curves in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0102_eq_001.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{\\\\mathbb{R}}}^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (P. W. Jones, <jats:italic>Rectifiable sets and the traveling salesman problem</jats:italic>, Invent. Math. 102 (1990), no. 1, 1–15), in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0102_eq_002.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{\\\\mathbb{R}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (K. Okikiolu, <jats:italic>Characterization of subsets of rectifiable curves in</jats:italic> <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0102_eq_003.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mrow> <m:mi mathvariant=\\\"bold\\\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{\\\\bf{R}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, J. London Math. Soc. (2) 46 (1992), no. 2, 336–348), or in an arbitrary Carnot group (S. 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In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0102_eq_005.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{\\\\mathbb{R}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> that charges a rectifiable curve in an arbitrary <jats:italic>complete, doubling, locally quasiconvex metric space</jats:italic>.\",\"PeriodicalId\":48637,\"journal\":{\"name\":\"Analysis and Geometry in Metric Spaces\",\"volume\":\"8 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-11-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Analysis and Geometry in Metric Spaces\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/agms-2023-0102\",\"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-2023-0102","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: 6

摘要

我们继续在几何测度理论中发展一个程序,旨在确定空间中的测度如何与空间中的规范集族相互作用。特别地,推广了M. Badger和R. Schul在欧几里得空间中的一个定理,对于任意Carnot群中的任意局部有限Borel测度,我们给出了判别由可校正曲线承载的测度部分和对可校正曲线奇异的测度部分的检验。我们的主要结果与分析的旅行推销员定理的一个扩展相关联,该定理表征了r2 {{\mathbb{R}}}^{2} (P. W. Jones,可整流集和旅行推销员问题,发明)中的可整流曲线子集。数学,102(1990),第1期。{{\bf{R}}}}^{n} {{n}}的可整流曲线子集的刻画,J.伦敦数学。Soc。(2) 46(1992)号;2,336 - 348),或者在任意卡诺群(S. Li)中,用局部几何最小二乘数据(称为Jones ' β \ β -数)表示。在次要结果中,我们实现了rn {{\mathbb{R}}}^{n}中的加倍测度的Garnett-Killip-Schul构造,该构造在任意完备的、加倍的、局部拟凸度量空间中收费一条可校正曲线。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Identifying 1-rectifiable measures in Carnot groups
We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of M. Badger and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of analyst’s traveling salesman theorem, which characterizes the subsets of rectifiable curves in R 2 {{\mathbb{R}}}^{2} (P. W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), no. 1, 1–15), in R n {{\mathbb{R}}}^{n} (K. Okikiolu, Characterization of subsets of rectifiable curves in R n {{\bf{R}}}^{n} , J. London Math. Soc. (2) 46 (1992), no. 2, 336–348), or in an arbitrary Carnot group (S. Li) in terms of local geometric least-squares data called Jones’ β \beta -numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R n {{\mathbb{R}}}^{n} that charges a rectifiable curve in an arbitrary complete, doubling, locally quasiconvex metric space.
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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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