海森堡群低标度下的 C 1,α-可纠正性

Pub Date : 2024-02-22 DOI:10.1515/agms-2023-0105
Kennedy Obinna Idu, Francesco Paolo Maiale
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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}","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

摘要

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。
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C 1,α-rectifiability in low codimension in Heisenberg groups
A natural higher-order notion of C 1 , α {C}^{1,\alpha } -rectifiability, 0 < α 1 0\lt \alpha \le 1 , is introduced for subsets of the Heisenberg groups H n {{\mathbb{H}}}^{n} in terms of covering a set almost everywhere with a countable union of ( C H 1 , α , H ) \left({{\bf{C}}}_{H}^{1,\alpha },{\mathbb{H}}) -regular surfaces. Using this, we prove a geometric characterization of C 1 , α {C}^{1,\alpha } -rectifiable sets of low codimension in Heisenberg groups H n {{\mathbb{H}}}^{n} in terms of an almost everywhere existence of suitable approximate tangent paraboloids.
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