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The Denjoy-Wolff Theorem in simply connected domains 简单相连域中的登乔伊-沃尔夫定理
arXiv - MATH - Complex Variables Pub Date : 2024-09-18 DOI: arxiv-2409.11722
Anna Miriam Benini, Filippo Bracci
{"title":"The Denjoy-Wolff Theorem in simply connected domains","authors":"Anna Miriam Benini, Filippo Bracci","doi":"arxiv-2409.11722","DOIUrl":"https://doi.org/arxiv-2409.11722","url":null,"abstract":"We characterize the simply connected domains $Omegasubsetneqmathbb{C}$\u0000that exhibit the Denjoy-Wolff Property, meaning that every holomorphic self-map\u0000of $Omega$ without fixed points has a Denjoy-Wolff point. We demonstrate that\u0000this property holds if and only if every automorphism of $Omega$ without fixed\u0000points in $Omega$ has a Denjoy-Wolff point. Furthermore, we establish that the\u0000Denjoy-Wolff Property is equivalent to the existence of what we term an\u0000``$H$-limit'' at each boundary point for a Riemann map associated with the\u0000domain. The $H$-limit condition is stronger than the existence of\u0000non-tangential limits but weaker than unrestricted limits. As an additional\u0000result of our work, we prove that there exist bounded simply connected domains\u0000where the Denjoy-Wolff Property holds but which are not visible in the sense of\u0000Bharali and Zimmer. Since visibility is a sufficient condition for the\u0000Denjoy-Wolff Property, this proves that in general it is not necessary.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265790","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}
引用次数: 0
Holomorphic approximation by polynomials with exponents restricted to a convex cone 多项式的全纯近似，其指数限制在凸锥范围内
arXiv - MATH - Complex Variables Pub Date : 2024-09-18 DOI: arxiv-2409.12132
Álfheiður Edda Sigurðardóttir
{"title":"Holomorphic approximation by polynomials with exponents restricted to a convex cone","authors":"Álfheiður Edda Sigurðardóttir","doi":"arxiv-2409.12132","DOIUrl":"https://doi.org/arxiv-2409.12132","url":null,"abstract":"We study approximations of holomorphic functions of several complex variables\u0000by proper subrings of the polynomials. The subrings in question consist of\u0000polynomials of several complex variables whose exponents are restricted to a\u0000prescribed convex cone $mathbb{R}_+S$ for some compact convex $Sin\u0000mathbb{R}^n_+$. Analogous to the polynomial hull of a set, we denote the hull\u0000of $K$ with respect to the given ring by can define hulls of a set $K$ with\u0000respect to the given ring, here denoted $widehat K{}^S$. By studying an\u0000extremal function $V^S_K(z)$, we show a version of the Runge-Oka-Weil Theorem\u0000on approximation by these subrings on compact subsets of $mathbb{C}^{*n}$ that\u0000satisfy $K= widehat K{}^S$ and $V^{S*}_K|_K=0$. We show a sharper result for\u0000compact Reinhardt sets $K$, that a holomorphic function is uniformly\u0000approximable on $widehat K{}^S$ by members of the ring if and only if it is\u0000bounded on $widehat K{}^S$. We also show that if $K$ is a compact Reinhardt\u0000subsets of $mathbb{C}^{*n}$, then we have $V^S_K(z)=sup_{sin S} (langle s\u0000,{operatorname{Log}, z}rangle- varphi_A(s)) $, where $varphi_A$ is the\u0000supporting function of $A=operatorname{Log}, K= {(log|z_1|,dots,\u0000log|z_n|) ,;, zin K}$.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265731","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}
引用次数: 0
$L^2$-vanishing theorem and a conjecture of Kollár L^2$ 消失定理和科拉尔猜想
arXiv - MATH - Complex Variables Pub Date : 2024-09-17 DOI: arxiv-2409.11399
Ya Deng, Botong Wang
{"title":"$L^2$-vanishing theorem and a conjecture of Kollár","authors":"Ya Deng, Botong Wang","doi":"arxiv-2409.11399","DOIUrl":"https://doi.org/arxiv-2409.11399","url":null,"abstract":"In 1995, Koll'ar conjectured that a complex projective $n$-fold $X$ with\u0000generically large fundamental group has Euler characteristic $chi(X, K_X)geq\u00000$. In this paper, we confirm the conjecture assuming $X$ has linear\u0000fundamental group, i.e., there exists an almost faithful representation\u0000$pi_1(X)to {rm GL}_N(mathbb{C})$. We deduce the conjecture by proving a\u0000stronger $L^2$ vanishing theorem: for the universal cover $widetilde{X}$ of\u0000such $X$, its $L^2$-Dolbeaut cohomology $H_{(2)}^{n,q}(widetilde{X})=0$ for\u0000$qneq 0$. The main ingredients of the proof are techniques from the linear\u0000Shafarevich conjecture along with some analytic methods.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"197 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265791","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}
引用次数: 0
Best approximations for the weighted combination of the Cauchy--Szegö kernel and its derivative in the mean Cauchy-Szegö 核的加权组合及其均值导数的最佳近似值
arXiv - MATH - Complex Variables Pub Date : 2024-09-17 DOI: arxiv-2409.10833
Viktor V. Savchuk, Maryna V. Savchuk
{"title":"Best approximations for the weighted combination of the Cauchy--Szegö kernel and its derivative in the mean","authors":"Viktor V. Savchuk, Maryna V. Savchuk","doi":"arxiv-2409.10833","DOIUrl":"https://doi.org/arxiv-2409.10833","url":null,"abstract":"In this paper, we study an extremal problem concerning best approximation in\u0000the Hardy space $H^1$ on the unit disk $mathbb D$. Specifically, we consider\u0000weighted combinations of the Cauchy-Szeg\"o kernel and its derivative,\u0000parametrized by an inner function $varphi$ and a complex number $lambda$, and\u0000provide explicit formula of the best approximation $e_{varphi,z}(lambda)$ by\u0000the subspace $H^1_0$. We also describe the extremal functions associated with\u0000this approximation.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265789","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}
引用次数: 0
Nevanlinna Theory on Complete Kähler Connected Sums With Non-parabolic Ends 具有非抛物线末端的完整 Kähler 连接和的 Nevanlinna 理论
arXiv - MATH - Complex Variables Pub Date : 2024-09-16 DOI: arxiv-2409.10243
Xianjing Dong
{"title":"Nevanlinna Theory on Complete Kähler Connected Sums With Non-parabolic Ends","authors":"Xianjing Dong","doi":"arxiv-2409.10243","DOIUrl":"https://doi.org/arxiv-2409.10243","url":null,"abstract":"Motivated by invalidness of Liouville property for harmonic functions on the\u0000connected sum $#^varthetamathbb C^m$ with $varthetageq2,$ we study\u0000Nevanlinna theory on a complete K\"ahler connected sum $$M=M_1#cdots# M_vartheta$$ with $vartheta$ non-parabolic ends. Based on\u0000the global Green function method, we extend the second main theorem of\u0000meromorphic mappings to $M.$ As a consequence, we obtain a Picard's little\u0000theorem provided that all $M_j^,s$ have non-negative Ricci curvature, which\u0000states that every meromorphic function on $M$ reduces to a constant if it omits\u0000three distinct values.In particular, it implies that Cauchy-Riemann equation\u0000supports a rigidity of Liouville property as an invariant under connected sums.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265792","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}
引用次数: 0
Burns-Krantz rigidity in non-smooth domains 非光滑域中的伯恩斯-克兰兹刚度
arXiv - MATH - Complex Variables Pub Date : 2024-09-16 DOI: arxiv-2409.10700
Włodzimierz Zwonek
{"title":"Burns-Krantz rigidity in non-smooth domains","authors":"Włodzimierz Zwonek","doi":"arxiv-2409.10700","DOIUrl":"https://doi.org/arxiv-2409.10700","url":null,"abstract":"Motivated by recent papers cite{For-Rong 2021} and cite{Ng-Rong 2024} we\u0000prove a boundary Schwarz lemma (Burns-Krantz rigidity type theorem) for\u0000non-smooth boundary points of the polydisc and symmetrized bidisc. Basic tool\u0000in the proofs is the phenomenon of invariance of complex geodesics (and their\u0000left inverses) being somehow regular at the boundary point under the mapping\u0000satisfying the property as in the Burns-Krantz rigidity theorem that lets the\u0000problem reduce to one dimensional problem. Additionally, we make a discussion\u0000on bounded symmetric domains and suggest a way to prove the Burns-Krantz\u0000rigidity type theorem in these domains that however cannot be applied for all\u0000bounded symmetric domains.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"119 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265793","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}
引用次数: 0
Some properties and integral transforms in higher spin Clifford analysis 高自旋克利福德分析中的一些性质和积分变换
arXiv - MATH - Complex Variables Pub Date : 2024-09-16 DOI: arxiv-2409.09952
Chao Ding
{"title":"Some properties and integral transforms in higher spin Clifford analysis","authors":"Chao Ding","doi":"arxiv-2409.09952","DOIUrl":"https://doi.org/arxiv-2409.09952","url":null,"abstract":"Rarita-Schwinger equation plays an important role in theoretical physics.\u0000Burev s et al. generalized it to arbitrary spin $k/2$ in 2002 in the context\u0000of Clifford algebras. In this article, we introduce the mean value property,\u0000Cauchy's estimates, and Liouville's theorem for null solutions to\u0000Rarita-Schwinger operator in Euclidean spaces. Further, we investigate\u0000boundednesses to the Teodorescu transform and its derivatives. This gives rise\u0000to a Hodge decomposition of an $L^2$ spaces in terms of the kernel space of the\u0000Rarita-Schwinger operator and it also generalizes Bergman spaces in higher spin\u0000cases. end{abstract}","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"99 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265795","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}
引用次数: 0
Weighted versions of Saitoh's conjecture in fibration cases 斋藤猜想在纤维情况下的加权版本
arXiv - MATH - Complex Variables Pub Date : 2024-09-16 DOI: arxiv-2409.10002
Qi'an Guan, Gan Li, Zheng Yuan
{"title":"Weighted versions of Saitoh's conjecture in fibration cases","authors":"Qi'an Guan, Gan Li, Zheng Yuan","doi":"arxiv-2409.10002","DOIUrl":"https://doi.org/arxiv-2409.10002","url":null,"abstract":"In this article, we introduce some generalized Hardy spaces on fibrations of\u0000planar domains and fibrations of products of planar domains. We consider the\u0000kernel functions on these spaces, and we prove some weighted versions of\u0000Saitoh's conjecture in fibration cases.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269518","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}
引用次数: 0
Multidimensional analogues of the refined versions of Bohr inequalities involving Schwarz mappings 涉及施瓦茨映射的玻尔不等式精炼版的多维类似物
arXiv - MATH - Complex Variables Pub Date : 2024-09-16 DOI: arxiv-2409.10091
Shanshan Jia, Ming-Sheng Liu, Saminathan Ponnusamy
{"title":"Multidimensional analogues of the refined versions of Bohr inequalities involving Schwarz mappings","authors":"Shanshan Jia, Ming-Sheng Liu, Saminathan Ponnusamy","doi":"arxiv-2409.10091","DOIUrl":"https://doi.org/arxiv-2409.10091","url":null,"abstract":"Our first aim of this article is to establish several new versions of refined\u0000Bohr inequalities for bounded analytic functions in the unit disk involving\u0000Schwarz functions. Secondly, %as applications of these results, we obtain\u0000several new multidimensional analogues of the refined Bohr inequalities for\u0000bounded holomorphic mappings on the unit ball in a complex Banach space\u0000involving higher dimensional Schwarz mappings. All the results are proved to be\u0000sharp.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265794","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}
引用次数: 0
Invariance of iterated global differential operator for slice monogenic functions 切片单原函数迭代全局微分算子的不变性
arXiv - MATH - Complex Variables Pub Date : 2024-09-16 DOI: arxiv-2409.09949
Chao Ding, Zhenghua Xu
{"title":"Invariance of iterated global differential operator for slice monogenic functions","authors":"Chao Ding, Zhenghua Xu","doi":"arxiv-2409.09949","DOIUrl":"https://doi.org/arxiv-2409.09949","url":null,"abstract":"In this article, we present the symmetry group of a global slice Dirac\u0000operator and its iterated ones. Further, the explicit forms of intertwining\u0000operators of the iterated global slice Dirac operator are given. At the end, we\u0000introduce a variant of the global slice Dirac operator, which allows functions\u0000considered to be defined on the whole Euclidean space. The invariance property\u0000and the intertwining operators of this variant of the global slice Dirac\u0000operator are also presented.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265796","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}
引用次数: 0
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