We prove that any Lipschitz map that satisfies a condition inspired by the work of G. David may be decomposed into countably many bi-Lipschitz pieces.
我们证明，任何满足 G. 戴维工作启发条件的利普斯奇兹映射都可以分解成可数的双利普斯奇兹片段。
{"title":"Qualitative Lipschitz to bi-Lipschitz decomposition","authors":"David Bate","doi":"10.1515/agms-2024-0005","DOIUrl":"https://doi.org/10.1515/agms-2024-0005","url":null,"abstract":"We prove that any Lipschitz map that satisfies a condition inspired by the work of G. David may be decomposed into countably many bi-Lipschitz pieces.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sihua Liang, Patrizia Pucci, Yueqiang Song, Xueqi Sun
This article is devoted to the study of a critical Choquard-Kirchhoff pp-sub-Laplacian equation on the entire Heisenberg group Hn{{mathbb{H}}}^{n}, where the Kirchhoff function KK can be zero at zero, i.e., the equation can be degenerate, and involving a nonlinearity, which is critical in the sense of the Hardy-Littlewood-Sobolev inequality. We first establish the concentration-compactness principle for the pp-sub-Laplacian Choquard equation on the Heisenberg group, and we then prove existence results.
本文致力于研究整个海森堡群 H n {{mathbb{H}}}^{n} 上的临界乔夸德-基尔霍夫 p p -次拉普拉斯方程。 ，其中基尔霍夫函数 K K 在零点可能为零，即方程可能是退化的，并且涉及非线性，在哈代-利特尔伍德-索博列夫不等式的意义上是临界的。我们首先建立了海森堡群上 p p-子拉普拉奇乔夸德方程的集中-紧凑性原理，然后证明了存在性结果。
{"title":"On a critical Choquard-Kirchhoff p-sub-Laplacian equation in ℍ n","authors":"Sihua Liang, Patrizia Pucci, Yueqiang Song, Xueqi Sun","doi":"10.1515/agms-2024-0006","DOIUrl":"https://doi.org/10.1515/agms-2024-0006","url":null,"abstract":"This article is devoted to the study of a critical Choquard-Kirchhoff <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0006_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>p</m:mi> </m:math> <jats:tex-math>p</jats:tex-math> </jats:alternatives> </jats:inline-formula>-sub-Laplacian equation on the entire Heisenberg group <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0006_eq_002.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>, where the Kirchhoff function <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0006_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>K</m:mi> </m:math> <jats:tex-math>K</jats:tex-math> </jats:alternatives> </jats:inline-formula> can be zero at zero, i.e., the equation can be degenerate, and involving a nonlinearity, which is critical in the sense of the Hardy-Littlewood-Sobolev inequality. We first establish the concentration-compactness principle for the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0006_eq_004.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>p</m:mi> </m:math> <jats:tex-math>p</jats:tex-math> </jats:alternatives> </jats:inline-formula>-sub-Laplacian Choquard equation on the Heisenberg group, and we then prove existence results.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, we characterize the geodesic dimension NGEO{N}_{{rm{GEO}}} and give a new lower bound to the curvature exponent NCE{N}_{{rm{CE}}} on Sard-regular Carnot groups. As an application, we give an example of step-two Carnot group on which NCE>NGEO{N}_{{rm{CE}}}gt {N}_{{rm{GEO}}}; this answers a question posed by Rizzi (Measure contraction properties of Carnot groups. Calc. Var. Partial Differential Equations 55 (2016), no. 3, Art. 60, 20).
在本研究中，我们描述了测地维 N GEO {N}_{rm{GEO}} 的特征，并给出了沙特规则卡诺群上曲率指数 N CE {N}_{rm{CE}} 的新下限。作为应用，我们给出了一个阶二卡诺群的例子，其中 N CE > N GEO {N}_{{rm{CE}}}gt {N}_{{rm{GEO}}} ；这回答了里齐提出的一个问题（卡诺群的度量收缩性质.Calc.Calc.Partial Differential Equations 55 (2016), no.3, Art.60, 20).
{"title":"Curvature exponent and geodesic dimension on Sard-regular Carnot groups","authors":"Sebastiano Nicolussi Golo, Ye Zhang","doi":"10.1515/agms-2024-0004","DOIUrl":"https://doi.org/10.1515/agms-2024-0004","url":null,"abstract":"In this study, we characterize the geodesic dimension <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0004_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\"normal\">GEO</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{N}_{{rm{GEO}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and give a new lower bound to the curvature exponent <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0004_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\"normal\">CE</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{N}_{{rm{CE}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> on Sard-regular Carnot groups. As an application, we give an example of step-two Carnot group on which <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0004_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\"normal\">CE</m:mi> </m:mrow> </m:msub> <m:mo>></m:mo> <m:msub> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\"normal\">GEO</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{N}_{{rm{CE}}}gt {N}_{{rm{GEO}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>; this answers a question posed by Rizzi (<jats:italic>Measure contraction properties of Carnot groups</jats:italic>. Calc. Var. Partial Differential Equations 55 (2016), no. 3, Art. 60, 20).","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive several properties of the heat equation with the Hodge operator associated with the Rumin’s complex on Heisenberg groups and prove several properties of the fundamental solution. As an application, we use the heat kernel for Rumin’s differential forms to construct a Calderón reproducing formula on Rumin’s forms.
{"title":"On the heat kernel of the Rumin complex and Calderón reproducing formula","authors":"Paolo Ciatti, Bruno Franchi, Yannick Sire","doi":"10.1515/agms-2024-0002","DOIUrl":"https://doi.org/10.1515/agms-2024-0002","url":null,"abstract":"We derive several properties of the heat equation with the Hodge operator associated with the Rumin’s complex on Heisenberg groups and prove several properties of the fundamental solution. As an application, we use the heat kernel for Rumin’s differential forms to construct a Calderón reproducing formula on Rumin’s forms.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141170276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a homeomorphism f:X→Yf:Xto Y between QQ-dimensional spaces X,YX,Y, we show that ff satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that ff belongs to the Sobolev class Nloc1,p(X;Y){N}_{{rm{loc}}}^{1,p}left(X;hspace{0.33em}Y), where 1≤p≤Q1le ple Q, and also implies one direction of the geometric definition of quasiconformality. Unlik
给定 Q Q 维空间 X, Y X,Y 之间的同构 f : X → Y f:Xto 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 1le ple 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 ) fin {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) 。
{"title":"Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces","authors":"Panu Lahti, Xiaodan Zhou","doi":"10.1515/agms-2024-0001","DOIUrl":"https://doi.org/10.1515/agms-2024-0001","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:Xto 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>1le ple Q</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and also implies one direction of the geometric definition of quasiconformality. Unlik","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140629597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A ΛLambda -tree is a ΛLambda -metric space satisfying three axioms (1), (2), and (3). We give a characterization of those ordered abelian groups ΛLambda for which axioms (1) and (2) imply axiom (3). As a special case, it follows that for the important class of ordered abelian groups ΛLambda that satisfy Λ=2ΛLambda =2Lambda , (3) follows from (1) and (2). For some ordered abelian groups ΛLambda , we show that axiom (2) is independent of axioms (1) and (3) and ask whether this holds for all ordered abelian groups. Part of this work has been formalized in the proof assistant Lean{mathsf{Lean}}.
{"title":"(In)dependence of the axioms of Λ-trees","authors":"Raphael Appenzeller","doi":"10.1515/agms-2023-0106","DOIUrl":"https://doi.org/10.1515/agms-2023-0106","url":null,"abstract":"A <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Λ</m:mi> </m:math> <jats:tex-math>Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula>-tree is a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Λ</m:mi> </m:math> <jats:tex-math>Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula>-metric space satisfying three axioms (1), (2), and (3). We give a characterization of those ordered abelian groups <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Λ</m:mi> </m:math> <jats:tex-math>Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula> for which axioms (1) and (2) imply axiom (3). As a special case, it follows that for the important class of ordered abelian groups <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Λ</m:mi> </m:math> <jats:tex-math>Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula> that satisfy <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Λ</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> <m:mi mathvariant=\"normal\">Λ</m:mi> </m:math> <jats:tex-math>Lambda =2Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula>, (3) follows from (1) and (2). For some ordered abelian groups <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Λ</m:mi> </m:math> <jats:tex-math>Lambda </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that axiom (2) is independent of axioms (1) and (3) and ask whether this holds for all ordered abelian groups. Part of this work has been formalized in the proof assistant <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0106_eq_007.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"sans-serif\">Lean</m:mi> </m:math> <jats:tex-math>{mathsf{Lean}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140584567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A natural higher-order notion of C1,α{C}^{1,alpha }-rectifiability, 0<α≤10lt 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}
C 1 的一个自然的高阶概念，α {C}^{1,alpha } -0 < α ≤ 1 0lt 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。
{"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":"https://doi.org/10.1515/agms-2023-0105","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>0lt 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-","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139954321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the notion of an extremal subset in a geodesically complete space with curvature bounded above, i.e., a GCBA space. This is an analog of an extremal subset in an Alexandrov space with curvature bounded below introduced by Perelman and Petrunin. We prove that under an additional assumption, the set of topological singularities in a GCBA space forms an extremal subset. We also exhibit some structural properties of extremal subsets in GCBA spaces.
{"title":"Extremal subsets in geodesically complete spaces with curvature bounded above","authors":"Tadashi Fujioka","doi":"10.1515/agms-2023-0104","DOIUrl":"https://doi.org/10.1515/agms-2023-0104","url":null,"abstract":"We introduce the notion of an extremal subset in a geodesically complete space with curvature bounded above, i.e., a GCBA space. This is an analog of an extremal subset in an Alexandrov space with curvature bounded below introduced by Perelman and Petrunin. We prove that under an additional assumption, the set of topological singularities in a GCBA space forms an extremal subset. We also exhibit some structural properties of extremal subsets in GCBA spaces.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139753927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The group of combinatorial self-similarities of a pseudometric space (X,d)left(X,d) is the maximal subgroup of the symmetric group Sym(X){rm{Sym}}left(X) whose elements preserve the four-point equality d(x,y)=d(u,v)dleft(x,y)=dleft(u,v). Let us denote by ℐP{mathcal{ {mathcal I} P}} the class of all pseudometric spaces (X,d)left(X,d) for which every combinatorial self-similarity Φ:X→XPhi :Xto X satisfies the equality
伪几何空间 ( X , d ) left(X,d)的组合自相似性群是对称群 Sym ( X ) {rm{Sym}}left(X) 的最大子群，其元素保持四点相等 d ( x , y ) = d ( u , v ) dleft(x,y)=dleft(u,v) 。让我们用 ℐP {mathcal{ {mathcal I} P}} 表示所有伪几何空间 ( X , d ) 的类 left(X,d)，其中每个组合自相似性 Φ : X → X Phi ：Xto X 满足等式 d ( x , Φ ( x ) ) = 0 , dleft(x,Phi left(x))=0，但是 ( X , d ) left(X,d)的度量反射的所有排列都是这种反射的组合自相似性。对ℐP {mathcal{ {mathcal I} P}} 的结构进行了全面描述。 -空间的结构得到了充分描述。
{"title":"Pseudometric spaces: From minimality to maximality in the groups of combinatorial self-similarities","authors":"Viktoriia Bilet, Oleksiy Dovgoshey","doi":"10.1515/agms-2023-0103","DOIUrl":"https://doi.org/10.1515/agms-2023-0103","url":null,"abstract":"The group of combinatorial self-similarities of a pseudometric space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>left(X,d)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the maximal subgroup of the symmetric group <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Sym</m:mi> <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>{rm{Sym}}left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula> whose elements preserve the four-point equality <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>dleft(x,y)=dleft(u,v)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let us denote by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℐP</m:mi> </m:math> <jats:tex-math>{mathcal{ {mathcal I} P}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> the class of all pseudometric spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>left(X,d)</jats:tex-math> </jats:alternatives> </jats:inline-formula> for which every combinatorial self-similarity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Φ</m:mi> <m:mo>:</m:mo> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>X</m:mi> </m:math> <jats:tex-math>Phi :Xto X</jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfies the equality <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139753826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let (X,d,μ)left({mathcal{X}},d,mu ) be a non-homogeneous metric measure space satisfying the geometrically doubling and upper doubling conditions. In this setting, we first introduce a generalized Morrey space Mpu(μ){M}_{p}^{u}left(mu ), where 1≤p<∞1le plt infty and u(x,r):X×(0,∞)→(0,∞)uleft(x,r):{mathcal{X}}times left(0,infty )to left(0,infty ) is a Lebesgue measurable function. Furthermore, under assumption that the measurable functions u1,u2{u}_{1},{u}_{2}, and
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