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 R2{{mathbb{R}}}^{2} (P. W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), no. 1, 1–15), in Rn{{mathbb{R}}}^{n} (K. Okikiolu, Characterization of subsets of rectifiable curves inRn{{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’βbeta -numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in
{"title":"Identifying 1-rectifiable measures in Carnot groups","authors":"Matthew Badger, Sean Li, Scott Zimmerman","doi":"10.1515/agms-2023-0102","DOIUrl":"https://doi.org/10.1515/agms-2023-0102","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. In particular, extending a theorem of M. Badger and R. Schul in Euclidean space, for an arbitrary <jats:italic>locally finite Borel measure</jats:italic> in an arbitrary <jats:italic>Carnot group</jats:italic>, 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 <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.p","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138533485","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}
Abstract This article analyses two schemes: Mann-type and viscosity-type proximal point algorithms. Using these schemes, we establish Δ-convergence and strong convergence theorems for finding a common solution of monotone vector field inclusion problems, a minimization problem, and a common fixed point of multivalued demicontractive mappings in Hadamard spaces. We apply our results to find mean and median values of probabilities, minimize energy of measurable mappings, and solve a kinematic problem in robotic motion control. We also include a numerical example to show the applicability of the schemes. Our findings corroborate some recent findings.
{"title":"Convergence theorems for monotone vector field inclusions and minimization problems in Hadamard spaces","authors":"S. Salisu, P. Kumam, Songpon Sriwongsa","doi":"10.1515/agms-2022-0150","DOIUrl":"https://doi.org/10.1515/agms-2022-0150","url":null,"abstract":"Abstract This article analyses two schemes: Mann-type and viscosity-type proximal point algorithms. Using these schemes, we establish Δ-convergence and strong convergence theorems for finding a common solution of monotone vector field inclusion problems, a minimization problem, and a common fixed point of multivalued demicontractive mappings in Hadamard spaces. We apply our results to find mean and median values of probabilities, minimize energy of measurable mappings, and solve a kinematic problem in robotic motion control. We also include a numerical example to show the applicability of the schemes. Our findings corroborate some recent findings.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42214708","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}
Abstract The main purpose of this article is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of commutator of multilinear fractional Calderón-Zygmund integral operators in the context of the variable exponent Lebesgue spaces. The authors do so by applying the techniques of Fourier series and multilinear fractional integral operator, as well as some pointwise estimates for the commutators. The key tool in obtaining such a pointwise estimate is a certain generalization of the classical sharp maximal operator.
{"title":"Characterization of Lipschitz functions via the commutators of multilinear fractional integral operators in variable Lebesgue spaces","authors":"Pu Zhang, Jiang-Long Wu","doi":"10.1515/agms-2022-0153","DOIUrl":"https://doi.org/10.1515/agms-2022-0153","url":null,"abstract":"Abstract The main purpose of this article is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of commutator of multilinear fractional Calderón-Zygmund integral operators in the context of the variable exponent Lebesgue spaces. The authors do so by applying the techniques of Fourier series and multilinear fractional integral operator, as well as some pointwise estimates for the commutators. The key tool in obtaining such a pointwise estimate is a certain generalization of the classical sharp maximal operator.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48740098","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}
Abstract Given M M and N N Hausdorff topological spaces, we study topologies on the space C 0 ( M ; N ) {C}^{0}left(M;hspace{0.33em}N) of continuous maps f : M → N f:Mto N . We review two classical topologies, the “strong” and the “weak” topology. We propose a definition of “mild topology” that is coarser than the “strong” and finer than the “weak” topology. We compare properties of these three topologies, in particular with respect to proper continuous maps f : M → N f:Mto N , and affine actions when N = R n N={{mathbb{R}}}^{n} . To define the “mild topology” we use “separation functions;” these “separation functions” are somewhat similar to the usual “distance function d ( x , y ) dleft(x,y) ” in metric spaces ( M , d ) left(M,d) , but have weaker requirements. Separation functions are used to define pseudo balls that are a global base for a T2 topology. Under some additional hypotheses, we can define “set separation functions” that prove that the topology is T6. Moreover, under further hypotheses, we will prove that the topology is metrizable. We provide some examples of uses of separation functions: one is the aforementioned case of the mild topology on C 0 ( M ; N ) {C}^{0}left(M;hspace{0.33em}N) . Other examples are the Sorgenfrey line and the topology of topological manifolds.
摘要给定M M和N N Hausdorff拓扑空间，我们研究了空间C0（M；N）｛C｝^｛0｝left（M；hspace）上的拓扑{0.33em}N)连续映射f:M→ N f:M到N。我们回顾了两种经典拓扑，“强”拓扑和“弱”拓扑。我们提出了一个“温和拓扑”的定义，它比“强”拓扑更粗糙，比“弱”拓扑更精细。我们比较了这三种拓扑的性质，特别是关于适当的连续映射f:M→ Nf:M到N，以及当N＝R N N＝{｛mathbb｛R｝｝}^｛N｝时的仿射作用。为了定义“温和拓扑”，我们使用“分离函数”；这些“分离函数“与度量空间（M，d）left（M，d）中通常的“距离函数d（x，y）dleft（x，y）”有些相似，但要求较弱。分离函数用于定义伪球，伪球是T2拓扑的全局基础。在一些额外的假设下，我们可以定义“集合分离函数”来证明拓扑是T6。此外，在进一步的假设下，我们将证明拓扑是可度量的。我们提供了分离函数使用的一些例子：一个是C 0（M；N）｛C｝^｛0｝left（M；hspace上的温和拓扑的前面提到的情况{0.33em}N)。其他的例子是索根弗雷线和拓扑流形的拓扑。
{"title":"Separation functions and mild topologies","authors":"A. Mennucci","doi":"10.1515/agms-2022-0149","DOIUrl":"https://doi.org/10.1515/agms-2022-0149","url":null,"abstract":"Abstract Given M M and N N Hausdorff topological spaces, we study topologies on the space C 0 ( M ; N ) {C}^{0}left(M;hspace{0.33em}N) of continuous maps f : M → N f:Mto N . We review two classical topologies, the “strong” and the “weak” topology. We propose a definition of “mild topology” that is coarser than the “strong” and finer than the “weak” topology. We compare properties of these three topologies, in particular with respect to proper continuous maps f : M → N f:Mto N , and affine actions when N = R n N={{mathbb{R}}}^{n} . To define the “mild topology” we use “separation functions;” these “separation functions” are somewhat similar to the usual “distance function d ( x , y ) dleft(x,y) ” in metric spaces ( M , d ) left(M,d) , but have weaker requirements. Separation functions are used to define pseudo balls that are a global base for a T2 topology. Under some additional hypotheses, we can define “set separation functions” that prove that the topology is T6. Moreover, under further hypotheses, we will prove that the topology is metrizable. We provide some examples of uses of separation functions: one is the aforementioned case of the mild topology on C 0 ( M ; N ) {C}^{0}left(M;hspace{0.33em}N) . Other examples are the Sorgenfrey line and the topology of topological manifolds.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42164775","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}
Abstract The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space ( M , g , e − ϕ d V g , m ) left(M,g,{e}^{-phi }{rm{d}}{V}_{g},m) , the weighted Yamabe problem consists on finding another smooth metric measure space conformal to ( M , g , e − ϕ d V g , m ) left(M,g,{e}^{-phi }{rm{d}}{V}_{g},m) such that its weighted scalar curvature is equal to λ + μ e − ϕ ∕ m lambda +mu {e}^{-phi /m} for some constants μ mu and λ lambda , satisfying a certain condition. In this article, we consider the problem of prescribing the weighted scalar curvature. We first prove some uniqueness and nonuniqueness results and then some existence result about prescribing the weighted scalar curvature. We also estimate the first nonzero eigenvalue of the weighted Laplacian of ( M , g , e − ϕ d V g , m ) left(M,g,{e}^{-phi }{rm{d}}{V}_{g},m) . On the other hand, we prove a version of the conformal Schwarz lemma on ( M , g , e − ϕ d V g , m ) left(M,g,{e}^{-phi }{rm{d}}{V}_{g},m) . All these results are achieved by using geometric flows related to the weighted Yamabe flow. We also prove the backward uniqueness of the weighted Yamabe flow. Finally, we consider weighted Yamabe solitons, which are the self-similar solutions of the weighted Yamabe flow.
Case引入的加权Yamabe问题是Gagliardo-Nirenberg不等式对光滑度量测度空间的推广。更准确地说，给定光滑度量测度空间（M，g，e−ξd V g，M）left（M，g，｛e｝^｛-phi｝｛rm｛d｝｝）{V}_{g} ，m），加权Yamabe问题在于找到与（m，g，e−ξd V g，m）left（m，g，｛e｝^｛-phi｝｛rm｛d｝｝共形的另一个光滑度量测度空间{V}_{g} ，m），使得其加权标量曲率等于λ+μe−ξ/mlambda+mu｛e｝^｛-phi/m｝，对于一些常数μmu和λlambda，满足一定条件。在本文中，我们考虑了指定加权标量曲率的问题。我们首先证明了关于加权标量曲率的一些唯一性和非唯一性结果，然后证明了一些存在性结果。我们还估计了加权拉普拉斯算子的第一个非零特征值，即（M，g，e−ξd V g，M）left（M，g，｛e｝^｛-phi｝｛rm｛d｝｝）{V}_{g} ，m）。另一方面，我们证明了（M，g，e−ξd V g，M）left（M，g，｛e｝^｛-phi｝｛rm｛d｝｝）上共形Schwarz引理的一个版本{V}_{g} ，m）。所有这些结果都是通过使用与加权Yamabe流相关的几何流来实现的。我们还证明了加权Yamabe流的后向唯一性。最后，我们考虑加权Yamabe孤子，这是加权Yamabe流的自相似解。
{"title":"On the problem of prescribing weighted scalar curvature and the weighted Yamabe flow","authors":"P. Ho, Jin‐Hyuk Shin","doi":"10.1515/agms-2022-0152","DOIUrl":"https://doi.org/10.1515/agms-2022-0152","url":null,"abstract":"Abstract The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space ( M , g , e − ϕ d V g , m ) left(M,g,{e}^{-phi }{rm{d}}{V}_{g},m) , the weighted Yamabe problem consists on finding another smooth metric measure space conformal to ( M , g , e − ϕ d V g , m ) left(M,g,{e}^{-phi }{rm{d}}{V}_{g},m) such that its weighted scalar curvature is equal to λ + μ e − ϕ ∕ m lambda +mu {e}^{-phi /m} for some constants μ mu and λ lambda , satisfying a certain condition. In this article, we consider the problem of prescribing the weighted scalar curvature. We first prove some uniqueness and nonuniqueness results and then some existence result about prescribing the weighted scalar curvature. We also estimate the first nonzero eigenvalue of the weighted Laplacian of ( M , g , e − ϕ d V g , m ) left(M,g,{e}^{-phi }{rm{d}}{V}_{g},m) . On the other hand, we prove a version of the conformal Schwarz lemma on ( M , g , e − ϕ d V g , m ) left(M,g,{e}^{-phi }{rm{d}}{V}_{g},m) . All these results are achieved by using geometric flows related to the weighted Yamabe flow. We also prove the backward uniqueness of the weighted Yamabe flow. Finally, we consider weighted Yamabe solitons, which are the self-similar solutions of the weighted Yamabe flow.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41634326","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}
Abstract We study the families of measures on Carnot groups that have vanishing p p -module, which we call M p {M}_{p} -exceptional families. We found necessary and sufficient Conditions for the family of intrinsic Lipschitz surfaces passing through a common point to be M p {M}_{p} -exceptional for p ≥ 1 pge 1 . We describe a wide class of M p {M}_{p} -exceptional intrinsic Lipschitz surfaces for p ∈ ( 0 , ∞ ) pin left(0,infty ) .
摘要研究了具有消失p p -模的卡诺群上的测度族，我们称之为M p {M＿p} -{例外族。我们得到了通过一个公共点的本征Lipschitz曲面族为M p M＿p的充分必要条件- }p{≥1 p }{}ge 1{例外}。对于p∈(0，∞)p {}inleft (0, infty)，我们描述了一类广义的M p M＿p -例外内禀Lipschitz曲面。
{"title":"Exceptional families of measures on Carnot groups","authors":"B. Franchi, I. Markina","doi":"10.1515/agms-2022-0148","DOIUrl":"https://doi.org/10.1515/agms-2022-0148","url":null,"abstract":"Abstract We study the families of measures on Carnot groups that have vanishing p p -module, which we call M p {M}_{p} -exceptional families. We found necessary and sufficient Conditions for the family of intrinsic Lipschitz surfaces passing through a common point to be M p {M}_{p} -exceptional for p ≥ 1 pge 1 . We describe a wide class of M p {M}_{p} -exceptional intrinsic Lipschitz surfaces for p ∈ ( 0 , ∞ ) pin left(0,infty ) .","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45320844","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}
Abstract Let (𝒳, d, μ) be a non-homogeneous metric measure space satisfying the upper doubling and geometrically doubling conditions in the sense of Hytönen. Under assumption that θ and dominating function λ satisfy certain conditions, the authors prove that fractional type Marcinkiewicz integral operator M˜ tilde M α,lρ,q associated with θ-type generalized fractional kernel is bounded from the generalized Morrey space ℒr,ϕp/r,κ (μ) into space ℒp,ϕ,κ (μ), and bounded from the Lebesgue space Lr(μ) into space Lp(μ). Furthermore, the boundedness of commutator M˜ tilde M α,l,ρq,b generated by b∈RBMO˜(μ) b in widetilde {RBMO}left( mu right) and the M˜ tilde M α,l,ρq,b on space ℒp(μ) and on space ℒp,ϕ,κ (μ) is also obtained.
设(f, d， μ)是满足Hytönen意义上的上加倍和几何加倍条件的非齐次度量度量空间。在θ和主导函数λ满足一定条件的假设下，证明了与θ型广义分数型核相关的分数型Marcinkiewicz积分算子M ～ tilde M α，lρ，q从广义Morrey空间∑，ϕ /r，κ (μ)有界到∑，φ，κ (μ)空间，并从Lebesgue空间Lr(μ)有界到∑(μ)空间。此外，还得到了由b∈RBMO≈(μ) b inwidetilde RBMO left ({}muright)生成的换向子M ～ tilde M α，l，ρq,b和M ～ tilde M α，l，ρq,b在空间＿＿p (μ)和空间＿＿p， φ，κ (μ)上的有界性。
{"title":"Fractional Type Marcinkiewicz Integral Operator Associated with Θ-Type Generalized Fractional Kernel and Its Commutator on Non-homogeneous Spaces","authors":"G. Lu, S. Tao, Miaomiao Wang","doi":"10.1515/agms-2022-0137","DOIUrl":"https://doi.org/10.1515/agms-2022-0137","url":null,"abstract":"Abstract Let (𝒳, d, μ) be a non-homogeneous metric measure space satisfying the upper doubling and geometrically doubling conditions in the sense of Hytönen. Under assumption that θ and dominating function λ satisfy certain conditions, the authors prove that fractional type Marcinkiewicz integral operator M˜ tilde M α,lρ,q associated with θ-type generalized fractional kernel is bounded from the generalized Morrey space ℒr,ϕp/r,κ (μ) into space ℒp,ϕ,κ (μ), and bounded from the Lebesgue space Lr(μ) into space Lp(μ). Furthermore, the boundedness of commutator M˜ tilde M α,l,ρq,b generated by b∈RBMO˜(μ) b in widetilde {RBMO}left( mu right) and the M˜ tilde M α,l,ρq,b on space ℒp(μ) and on space ℒp,ϕ,κ (μ) is also obtained.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48430432","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}
Abstract Liu and Lu [27] investigated a generalized Gauss curvature flow and obtained an even solution to the dual Orlicz-Minkowski problem under some appropriate assumptions. The present paper investigates a inverse Gauss curvature flow, and achieves the long-time existence and convergence of this flow via a different C0-estimate technique under weaker conditions. As an application of this inverse Gauss curvature flow, the present paper first arrives at a non-even smooth solution to the Orlicz Minkowski problem.
{"title":"Inverse Gauss Curvature Flows and Orlicz Minkowski Problem","authors":"Bin Chen, Jingshi Cui, P. Zhao","doi":"10.1515/agms-2022-0146","DOIUrl":"https://doi.org/10.1515/agms-2022-0146","url":null,"abstract":"Abstract Liu and Lu [27] investigated a generalized Gauss curvature flow and obtained an even solution to the dual Orlicz-Minkowski problem under some appropriate assumptions. The present paper investigates a inverse Gauss curvature flow, and achieves the long-time existence and convergence of this flow via a different C0-estimate technique under weaker conditions. As an application of this inverse Gauss curvature flow, the present paper first arrives at a non-even smooth solution to the Orlicz Minkowski problem.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41572686","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}
Abstract Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Herewe extend Ancona’s potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schrödinger operators on Gromov hyperbolic metric measure spaces, unifying these settings in a common framework ready for applications to singular spaces such as RCD spaces or minimal hypersurfaces. Results include boundary Harnack inequalities and a complete classification of positive harmonic functions in terms of the Martin boundary which is identified with the geometric Gromov boundary.
{"title":"Potential Theory on Gromov Hyperbolic Spaces","authors":"Matthias Kemper, J. Lohkamp","doi":"10.1515/agms-2022-0147","DOIUrl":"https://doi.org/10.1515/agms-2022-0147","url":null,"abstract":"Abstract Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Herewe extend Ancona’s potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schrödinger operators on Gromov hyperbolic metric measure spaces, unifying these settings in a common framework ready for applications to singular spaces such as RCD spaces or minimal hypersurfaces. Results include boundary Harnack inequalities and a complete classification of positive harmonic functions in terms of the Martin boundary which is identified with the geometric Gromov boundary.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46951909","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}
Abstract The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally bi-Lipschitz to the original domain near its boundary, but transforms the space into a bounded domain. We will show that if the original metric space is a uniform domain with respect to its completion, then the transformed space is also a uniform domain.
{"title":"Conformal Transformation of Uniform Domains Under Weights That Depend on Distance to The Boundary","authors":"Ryan Gibara, N. Shanmugalingam","doi":"10.1515/agms-2022-0141","DOIUrl":"https://doi.org/10.1515/agms-2022-0141","url":null,"abstract":"Abstract The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally bi-Lipschitz to the original domain near its boundary, but transforms the space into a bounded domain. We will show that if the original metric space is a uniform domain with respect to its completion, then the transformed space is also a uniform domain.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41339360","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}
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