For any intrinsic Gromov hyperbolic space we establish a Gehring-Hayman type theorem for conformally deformed spaces. As an application, we prove that any intrinsic hyperbolic space with atleast two points in the Gromov boundary can be uniformized by densities induced by Busemann functions.
{"title":"Uniformization of intrinsic Gromov hyperbolic spaces with Busemann functions","authors":"Vasudevarao Allu, Alan P Jose","doi":"arxiv-2408.01412","DOIUrl":"https://doi.org/arxiv-2408.01412","url":null,"abstract":"For any intrinsic Gromov hyperbolic space we establish a Gehring-Hayman type\u0000theorem for conformally deformed spaces. As an application, we prove that any\u0000intrinsic hyperbolic space with atleast two points in the Gromov boundary can\u0000be uniformized by densities induced by Busemann functions.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141941834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ondrěj Bouchala, Jarmo Jääskeläinen, Pekka Koskela, Haiqing Xu, Xilin Zhou
Each homeomorphic parametrization of a Jordan curve via the unit circle extends to a homeomorphism of the entire plane. It is a natural question to ask if such a homeomorphism can be chosen so as to have some Sobolev regularity. This prompts the simplified question: for a homeomorphic embedding of the unit circle into the plane, when can we find a homeomorphism from the unit disk that has the same boundary values and integrable first-order distributional derivatives? We give the optimal geometric criterion for the interior Jordan domain so that there exists a Sobolev homeomorphic extension for any homeomorphic parametrization of the Jordan curve. The problem is partially motivated by trying to understand which boundary values can correspond to deformations of finite energy.
{"title":"Homeomorphic Sobolev extensions of parametrizations of Jordan curves","authors":"Ondrěj Bouchala, Jarmo Jääskeläinen, Pekka Koskela, Haiqing Xu, Xilin Zhou","doi":"arxiv-2408.00506","DOIUrl":"https://doi.org/arxiv-2408.00506","url":null,"abstract":"Each homeomorphic parametrization of a Jordan curve via the unit circle\u0000extends to a homeomorphism of the entire plane. It is a natural question to ask\u0000if such a homeomorphism can be chosen so as to have some Sobolev regularity.\u0000This prompts the simplified question: for a homeomorphic embedding of the unit\u0000circle into the plane, when can we find a homeomorphism from the unit disk that\u0000has the same boundary values and integrable first-order distributional\u0000derivatives? We give the optimal geometric criterion for the interior Jordan domain so\u0000that there exists a Sobolev homeomorphic extension for any homeomorphic\u0000parametrization of the Jordan curve. The problem is partially motivated by\u0000trying to understand which boundary values can correspond to deformations of\u0000finite energy.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"192 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Based on the full similarity in algebraic properties and differentiation rules between quaternionic (H-) holomorphic and complex (C-) holomorphic functions, we assume that there exists one holistic notion of a holomorphic function that has a H-representation in the case of quaternions and a C-representation in the case of complex variables. We get the essential definitions and criteria for a quaternionic power series convergence, adapting complex analogues to the quaternion case. It is established that the power series expansions of any holomorphic function in C- and H-representations are similar and converge with identical convergence radiuses. We define a H-analytic function and prove that every H-holomorphic function is H-analytic. Some examples of power series expansions are given.
基于四元（H-）全态函数和复元（C-）全态函数在代数性质和微分规则上的完全相似性，我们假定存在一个全态函数的整体概念，它在四元情况下有 H 表示，在复变情况下有 C 表示。我们得到了四元幂级数收敛的基本定义和标准，并将复变类比于四元的情况。我们确定，任何全形函数在 C- 表示和 H- 表示中的幂级数展开都是相似的，并以相同的收敛半径收敛。我们定义了 H- 解析函数，并证明了每个 H-holomorphic 函数都是 H- 解析函数。
{"title":"On Decompositions of H-holomorphic functions into quaternionic power series","authors":"Michael Parfenov","doi":"arxiv-2407.21474","DOIUrl":"https://doi.org/arxiv-2407.21474","url":null,"abstract":"Based on the full similarity in algebraic properties and differentiation\u0000rules between quaternionic (H-) holomorphic and complex (C-) holomorphic\u0000functions, we assume that there exists one holistic notion of a holomorphic\u0000function that has a H-representation in the case of quaternions and a\u0000C-representation in the case of complex variables. We get the essential\u0000definitions and criteria for a quaternionic power series convergence, adapting\u0000complex analogues to the quaternion case. It is established that the power\u0000series expansions of any holomorphic function in C- and H-representations are\u0000similar and converge with identical convergence radiuses. We define a\u0000H-analytic function and prove that every H-holomorphic function is H-analytic.\u0000Some examples of power series expansions are given.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A proper subdomain $G$ of the unit disk $mathbb{D}$ is horocyclically convex (horo-convex) if, for every $omega in mathbb{D}cap partial G$, there exists a horodisk $H$ such that $omega in partial H$ and $Gcap H=emptyset$. In this paper we give an internal characterization of these domains, namely, that $G$ is horo-convex if and only if any two points can be joined inside $G$ by a $C^1$ curve composed with finitely many Jordan arcs with hyperbolic curvature in $(-2,2)$. We also give a lower bound for the hyperbolic metric of horo-convex regions and some consequences.
{"title":"On an Internal Characterization of Horocyclically Convex Domains in the Unit Disk","authors":"Juan Arango, Hugo Arbeláez, Diego Mejía","doi":"arxiv-2407.21271","DOIUrl":"https://doi.org/arxiv-2407.21271","url":null,"abstract":"A proper subdomain $G$ of the unit disk $mathbb{D}$ is horocyclically convex\u0000(horo-convex) if, for every $omega in mathbb{D}cap partial G$, there\u0000exists a horodisk $H$ such that $omega in partial H$ and $Gcap\u0000H=emptyset$. In this paper we give an internal characterization of these\u0000domains, namely, that $G$ is horo-convex if and only if any two points can be\u0000joined inside $G$ by a $C^1$ curve composed with finitely many Jordan arcs with\u0000hyperbolic curvature in $(-2,2)$. We also give a lower bound for the hyperbolic\u0000metric of horo-convex regions and some consequences.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Non-normable spaces of analytic functions","authors":"Iván Jiménez, Dragan Vukotić","doi":"arxiv-2407.21212","DOIUrl":"https://doi.org/arxiv-2407.21212","url":null,"abstract":"For each value of $p$ such that $0<p<1$, we give a specific example of two\u0000functions in the Hardy space $H^p$ and in the Bergman space $A^p$ that do not\u0000satisfy the triangle inequality. For Hardy spaces, this provides a much simpler\u0000proof than the one due to Livingston that involves abstract functional analysis\u0000arguments and an approximation theorem. For Bergman spaces, we have not been\u0000able to locate any examples or proofs in the existing literature.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Walton Green, Kévin Le Balc'h, Jérémy Martin, Marcu-Antone Orsoni
We consider the heat equation set on a bounded $C^1$ domain of $mathbb R^n$ with Dirichlet boundary conditions. The first purpose of this paper is to prove that the heat equation is observable from any measurable set $omega$ with positive $(n-1+delta)$-Hausdorff content, for $delta >0$ arbitrary small. The proof relies on a new spectral estimate for linear combinations of Laplace eigenfunctions, obtained via a Remez type inequality, and the use of the so-called Lebeau-Robbiano's method. Even if this observability result is sharp with respect to the scale of Hausdorff dimension, our second goal is to construct families of sets $omega$ which have codimension greater than or equal to $1$ for which the heat equation remains observable.
{"title":"Observability of the heat equation from very small sets","authors":"A. Walton Green, Kévin Le Balc'h, Jérémy Martin, Marcu-Antone Orsoni","doi":"arxiv-2407.20954","DOIUrl":"https://doi.org/arxiv-2407.20954","url":null,"abstract":"We consider the heat equation set on a bounded $C^1$ domain of $mathbb R^n$\u0000with Dirichlet boundary conditions. The first purpose of this paper is to prove\u0000that the heat equation is observable from any measurable set $omega$ with\u0000positive $(n-1+delta)$-Hausdorff content, for $delta >0$ arbitrary small. The\u0000proof relies on a new spectral estimate for linear combinations of Laplace\u0000eigenfunctions, obtained via a Remez type inequality, and the use of the\u0000so-called Lebeau-Robbiano's method. Even if this observability result is sharp\u0000with respect to the scale of Hausdorff dimension, our second goal is to\u0000construct families of sets $omega$ which have codimension greater than or\u0000equal to $1$ for which the heat equation remains observable.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the existence and non-existence of entire solutions of certain non-linear delay-differential equations.
本文研究了某些非线性延迟微分方程全解的存在与不存在问题。
{"title":"On solutions of certain non-linear delay differential equations","authors":"Nidhi Gahlian","doi":"arxiv-2407.19855","DOIUrl":"https://doi.org/arxiv-2407.19855","url":null,"abstract":"In this paper, we study the existence and non-existence of entire solutions\u0000of certain non-linear delay-differential equations.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide several families of compact complex curves embedded in smooth complex surfaces such that no neighborhood of the curve can be embedded in an algebraic surface. Different constructions are proposed, by patching neighborhoods of curves in projective surfaces, and blowing down exceptional curves. These constructions generalize examples recently given by S. Lvovski. One of our non algebraic argument is based on an extension theorem of S. Ivashkovich.
{"title":"Non-algebraizable neighborhoods of curves","authors":"Maycol Falla Luza, Frank Loray, Paulo Sad","doi":"arxiv-2407.20206","DOIUrl":"https://doi.org/arxiv-2407.20206","url":null,"abstract":"We provide several families of compact complex curves embedded in smooth\u0000complex surfaces such that no neighborhood of the curve can be embedded in an\u0000algebraic surface. Different constructions are proposed, by patching\u0000neighborhoods of curves in projective surfaces, and blowing down exceptional\u0000curves. These constructions generalize examples recently given by S. Lvovski.\u0000One of our non algebraic argument is based on an extension theorem of S.\u0000Ivashkovich.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathbb{D}$ be the unit disk in the complex plane. Among other results, we prove the following curious result for a finite Blaschke product: $$B(z)=e ^{is}prod_{k=1}^d frac{z-a_k}{1-z overline{a_k}}.$$ The Lebesgue measure of the sublevel set of $B$ satisfies the following sharp inequality for $t in [0,1]$: $$|{zin mathbb{D}:|B(z)|
{"title":"A sharp estimate of area for sublevel-set of Blaschke products","authors":"David Kalaj","doi":"arxiv-2407.19539","DOIUrl":"https://doi.org/arxiv-2407.19539","url":null,"abstract":"Let $mathbb{D}$ be the unit disk in the complex plane. Among other results,\u0000we prove the following curious result for a finite Blaschke product: $$B(z)=e\u0000^{is}prod_{k=1}^d frac{z-a_k}{1-z overline{a_k}}.$$ The Lebesgue measure of\u0000the sublevel set of $B$ satisfies the following sharp inequality for $t in\u0000[0,1]$: $$|{zin mathbb{D}:|B(z)|<t}|le pi t^{2/d},$$ with equality at a\u0000single point $tin(0,1)$ if and only if $a_k=0$ for every $k$. In that case the\u0000equality is attained for every $t$.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Omega$ be a convex domain in $mathbb{C}^n$ and $varphi$ a convex function on $Omega$. We prove that $log{K_{Omega,varphi}(z)}$ is a convex function (might be identically $-infty$) on $Omega$, where $K_{Omega,varphi}$ is the weighted Bergman kernel.
{"title":"Convexity of the Bergman Kernels on Convex Domains","authors":"Yuanpu Xiong","doi":"arxiv-2407.19254","DOIUrl":"https://doi.org/arxiv-2407.19254","url":null,"abstract":"Let $Omega$ be a convex domain in $mathbb{C}^n$ and $varphi$ a convex\u0000function on $Omega$. We prove that $log{K_{Omega,varphi}(z)}$ is a convex\u0000function (might be identically $-infty$) on $Omega$, where\u0000$K_{Omega,varphi}$ is the weighted Bergman kernel.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：