{"title":"C 1,α-rectifiability in low codimension in Heisenberg groups","authors":"Kennedy Obinna Idu, Francesco Paolo Maiale","doi":"10.1515/agms-2023-0105","DOIUrl":null,"url":null,"abstract":"A natural higher-order notion of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0105_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{C}^{1,\\alpha }</jats:tex-math> </jats:alternatives> </jats:inline-formula>-rectifiability, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0105_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>α</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:math> <jats:tex-math>0\\lt \\alpha \\le 1</jats:tex-math> </jats:alternatives> </jats:inline-formula>, is introduced for subsets of the Heisenberg groups <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0105_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{\\mathbb{H}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in terms of covering a set almost everywhere with a countable union of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0105_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msubsup> <m:mrow> <m:mi mathvariant=\"bold\">C</m:mi> </m:mrow> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:mi mathvariant=\"double-struck\">H</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\left({{\\bf{C}}}_{H}^{1,\\alpha },{\\mathbb{H}})</jats:tex-math> </jats:alternatives> </jats:inline-formula>-regular surfaces. Using this, we prove a geometric characterization of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0105_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{C}^{1,\\alpha }</jats:tex-math> </jats:alternatives> </jats:inline-formula>-rectifiable sets of low codimension in Heisenberg groups <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0105_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{\\mathbb{H}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in terms of an almost everywhere existence of suitable approximate tangent paraboloids.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/agms-2023-0105","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
A natural higher-order notion of C1,α{C}^{1,\alpha }-rectifiability, 0<α≤10\lt \alpha \le 1, is introduced for subsets of the Heisenberg groups Hn{{\mathbb{H}}}^{n} in terms of covering a set almost everywhere with a countable union of (CH1,α,H)\left({{\bf{C}}}_{H}^{1,\alpha },{\mathbb{H}})-regular surfaces. Using this, we prove a geometric characterization of C1,α{C}^{1,\alpha }-rectifiable sets of low codimension in Heisenberg groups Hn{{\mathbb{H}}}^{n} in terms of an almost everywhere existence of suitable approximate tangent paraboloids.
C 1 的一个自然的高阶概念,α {C}^{1,\alpha } -0 < α ≤ 1 0\lt \alpha \le 1,是针对海森堡群 H n {{mathbb{H}}}^{n} 的子集引入的,即几乎无处不在地用 ( C H 1 , α , H ) \left({{\bf{C}}_{H}}^{1,\alpha },{\mathbb{H}}) 不规则曲面的可数联合覆盖一个集合。利用这一点,我们证明了 C 1 , α {C}^{1,\alpha } 的几何特征。 -在海森堡群 H n {{\mathbb{H}}}^{n} 中,几乎无处不存在合适的近似切线抛物面,从而证明了低标度可正集的几何特征。