Analysis and Geometry in Metric Spaces最新文献

{"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":"17 1","pages":""},"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}

{"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":"27 1","pages":""},"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}

{"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>&gt;</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":"13 1","pages":""},"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}

{"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":"66 1","pages":""},"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}

{"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 &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0001_eq_001.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:mo&gt;:&lt;/m:mo&gt; &lt;m:mi&gt;X&lt;/m:mi&gt; &lt;m:mo&gt;→&lt;/m:mo&gt; &lt;m:mi&gt;Y&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;f:Xto Y&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; between &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0001_eq_002.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;Q&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;Q&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-dimensional spaces &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0001_eq_003.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;X&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;Y&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;X,Y&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, we show that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0001_eq_004.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;f&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0001_eq_005.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;f&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; belongs to the Sobolev class &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0001_eq_006.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;N&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"normal\"&gt;loc&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;X&lt;/m:mi&gt; &lt;m:mo&gt;;&lt;/m:mo&gt; &lt;m:mspace width=\"0.33em\" /&gt; &lt;m:mi&gt;Y&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;{N}_{{rm{loc}}}^{1,p}left(X;hspace{0.33em}Y)&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, where &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2024-0001_eq_007.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;m:mo&gt;≤&lt;/m:mo&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;m:mo&gt;≤&lt;/m:mo&gt; &lt;m:mi&gt;Q&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;1le ple Q&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, and also implies one direction of the geometric definition of quasiconformality. Unlik","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"8 1","pages":""},"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}

{"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":"12 1","pages":""},"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}

{"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 &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0105_eq_001.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;C&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;{C}^{1,alpha }&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-rectifiability, &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0105_eq_002.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;m:mo&gt;&lt;&lt;/m:mo&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;m:mo&gt;≤&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;0lt alpha le 1&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, is introduced for subsets of the Heisenberg groups &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0105_eq_003.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;H&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;{{mathbb{H}}}^{n}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; in terms of covering a set almost everywhere with a countable union of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0105_eq_004.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"bold\"&gt;C&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;H&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msubsup&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;H&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;left({{bf{C}}}_{H}^{1,alpha },{mathbb{H}})&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-regular surfaces. Using this, we prove a geometric characterization of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0105_eq_005.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;C&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;{C}^{1,alpha }&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-rectifiable sets of low codimension in Heisenberg groups &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0105_eq_006.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;H&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;{{mathbb{H}}}^{n}&lt;/jats:tex-","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"48 1","pages":""},"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}

{"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":"11 1","pages":""},"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}

{"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 &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_001.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;X&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;left(X,d)&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is the maximal subgroup of the symmetric group &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_002.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"normal\"&gt;Sym&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;X&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;{rm{Sym}}left(X)&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; whose elements preserve the four-point equality &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_003.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;y&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;dleft(x,y)=dleft(u,v)&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. Let us denote by &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_004.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"script\"&gt;ℐP&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;{mathcal{ {mathcal I} P}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; the class of all pseudometric spaces &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_005.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;X&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;left(X,d)&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; for which every combinatorial self-similarity &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_006.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"normal\"&gt;Φ&lt;/m:mi&gt; &lt;m:mo&gt;:&lt;/m:mo&gt; &lt;m:mi&gt;X&lt;/m:mi&gt; &lt;m:mo&gt;→&lt;/m:mo&gt; &lt;m:mi&gt;X&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;Phi :Xto X&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; satisfies the equality &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"168-169 1","pages":""},"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}

{"title":"Estimates for bilinear θ-type generalized fractional integral and its commutator on new non-homogeneous generalized Morrey spaces","authors":"Guanghui Lu, Miaomiao Wang, Shuangping Tao","doi":"10.1515/agms-2023-0101","DOIUrl":"https://doi.org/10.1515/agms-2023-0101","url":null,"abstract":"Let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_001.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"script\"&gt;X&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;μ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;left({mathcal{X}},d,mu )&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; 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 &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_002.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;M&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;μ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;{M}_{p}^{u}left(mu )&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, where &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_003.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;m:mo&gt;≤&lt;/m:mo&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;m:mo&gt;&lt;&lt;/m:mo&gt; &lt;m:mi&gt;∞&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;1le plt infty &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_004.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;r&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;:&lt;/m:mo&gt; &lt;m:mi mathvariant=\"script\"&gt;X&lt;/m:mi&gt; &lt;m:mo&gt;×&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;∞&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;→&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;∞&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;uleft(x,r):{mathcal{X}}times left(0,infty )to left(0,infty )&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is a Lebesgue measurable function. Furthermore, under assumption that the measurable functions &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0101_eq_005.png\" /&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:msub&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;{u}_{1},{u}_{2}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, and &lt;jats:inline","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"9 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138533437","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}