We characterize the simply connected domains $Omegasubsetneqmathbb{C}$ that exhibit the Denjoy-Wolff Property, meaning that every holomorphic self-map of $Omega$ without fixed points has a Denjoy-Wolff point. We demonstrate that this property holds if and only if every automorphism of $Omega$ without fixed points in $Omega$ has a Denjoy-Wolff point. Furthermore, we establish that the Denjoy-Wolff Property is equivalent to the existence of what we term an ``$H$-limit'' at each boundary point for a Riemann map associated with the domain. The $H$-limit condition is stronger than the existence of non-tangential limits but weaker than unrestricted limits. As an additional result of our work, we prove that there exist bounded simply connected domains where the Denjoy-Wolff Property holds but which are not visible in the sense of Bharali and Zimmer. Since visibility is a sufficient condition for the Denjoy-Wolff Property, this proves that in general it is not necessary.
{"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}
We study approximations of holomorphic functions of several complex variables by proper subrings of the polynomials. The subrings in question consist of polynomials of several complex variables whose exponents are restricted to a prescribed convex cone $mathbb{R}_+S$ for some compact convex $Sin mathbb{R}^n_+$. Analogous to the polynomial hull of a set, we denote the hull of $K$ with respect to the given ring by can define hulls of a set $K$ with respect to the given ring, here denoted $widehat K{}^S$. By studying an extremal function $V^S_K(z)$, we show a version of the Runge-Oka-Weil Theorem on approximation by these subrings on compact subsets of $mathbb{C}^{*n}$ that satisfy $K= widehat K{}^S$ and $V^{S*}_K|_K=0$. We show a sharper result for compact Reinhardt sets $K$, that a holomorphic function is uniformly approximable on $widehat K{}^S$ by members of the ring if and only if it is bounded on $widehat K{}^S$. We also show that if $K$ is a compact Reinhardt subsets of $mathbb{C}^{*n}$, then we have $V^S_K(z)=sup_{sin S} (langle s ,{operatorname{Log}, z}rangle- varphi_A(s)) $, where $varphi_A$ is the supporting function of $A=operatorname{Log}, K= {(log|z_1|,dots, log|z_n|) ,;, zin K}$.
{"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}
In 1995, Koll'ar conjectured that a complex projective $n$-fold $X$ with generically large fundamental group has Euler characteristic $chi(X, K_X)geq 0$. In this paper, we confirm the conjecture assuming $X$ has linear fundamental group, i.e., there exists an almost faithful representation $pi_1(X)to {rm GL}_N(mathbb{C})$. We deduce the conjecture by proving a stronger $L^2$ vanishing theorem: for the universal cover $widetilde{X}$ of such $X$, its $L^2$-Dolbeaut cohomology $H_{(2)}^{n,q}(widetilde{X})=0$ for $qneq 0$. The main ingredients of the proof are techniques from the linear Shafarevich conjecture along with some analytic methods.
{"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}
In this paper, we study an extremal problem concerning best approximation in the Hardy space $H^1$ on the unit disk $mathbb D$. Specifically, we consider weighted combinations of the Cauchy-Szeg"o kernel and its derivative, parametrized by an inner function $varphi$ and a complex number $lambda$, and provide explicit formula of the best approximation $e_{varphi,z}(lambda)$ by the subspace $H^1_0$. We also describe the extremal functions associated with this approximation.
{"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}
Motivated by invalidness of Liouville property for harmonic functions on the connected sum $#^varthetamathbb C^m$ with $varthetageq2,$ we study Nevanlinna theory on a complete K"ahler connected sum $$M=M_1#cdots# M_vartheta$$ with $vartheta$ non-parabolic ends. Based on the global Green function method, we extend the second main theorem of meromorphic mappings to $M.$ As a consequence, we obtain a Picard's little theorem provided that all $M_j^,s$ have non-negative Ricci curvature, which states that every meromorphic function on $M$ reduces to a constant if it omits three distinct values.In particular, it implies that Cauchy-Riemann equation supports a rigidity of Liouville property as an invariant under connected sums.
{"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}
Motivated by recent papers cite{For-Rong 2021} and cite{Ng-Rong 2024} we prove a boundary Schwarz lemma (Burns-Krantz rigidity type theorem) for non-smooth boundary points of the polydisc and symmetrized bidisc. Basic tool in the proofs is the phenomenon of invariance of complex geodesics (and their left inverses) being somehow regular at the boundary point under the mapping satisfying the property as in the Burns-Krantz rigidity theorem that lets the problem reduce to one dimensional problem. Additionally, we make a discussion on bounded symmetric domains and suggest a way to prove the Burns-Krantz rigidity type theorem in these domains that however cannot be applied for all bounded symmetric domains.
受近期论文（cite{For-Rong 2021}和（cite{Ng-Rong 2024}）的启发，我们证明了多圆盘和对称双圆盘非光滑边界点的边界施瓦茨定理（Burns-Krantz rigidity type theorem）。证明的基本工具是在满足伯恩斯-克兰茨刚性定理属性的映射下，复大地线（及其左反函数）在边界点处具有某种规则性，从而使问题简化为一维问题。此外，我们还讨论了有界对称域，并提出了在这些域中证明伯恩斯-克兰茨刚性定理的方法，但这一方法并不适用于所有有界对称域。
{"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}
Rarita-Schwinger equation plays an important role in theoretical physics. Burev s et al. generalized it to arbitrary spin $k/2$ in 2002 in the context of Clifford algebras. In this article, we introduce the mean value property, Cauchy's estimates, and Liouville's theorem for null solutions to Rarita-Schwinger operator in Euclidean spaces. Further, we investigate boundednesses to the Teodorescu transform and its derivatives. This gives rise to a Hodge decomposition of an $L^2$ spaces in terms of the kernel space of the Rarita-Schwinger operator and it also generalizes Bergman spaces in higher spin cases. end{abstract}
Burev s 等人于 2002 年在克利福德代数的背景下将其推广到任意自旋 $k/2$。在本文中，我们介绍了欧几里得空间中拉里塔-施文格算子空解的均值性质、考希估计和柳维尔定理。此外，我们还研究了 Teodorescu 变换及其导数的有界性。这就产生了以拉里塔-施文格算子的核空间为条件的 $L^2$ 空间的霍奇分解，而且它还概括了更高空间情况下的伯格曼空间。结束语
{"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}
In this article, we introduce some generalized Hardy spaces on fibrations of planar domains and fibrations of products of planar domains. We consider the kernel functions on these spaces, and we prove some weighted versions of Saitoh's conjecture in fibration cases.
{"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}
Our first aim of this article is to establish several new versions of refined Bohr inequalities for bounded analytic functions in the unit disk involving Schwarz functions. Secondly, %as applications of these results, we obtain several new multidimensional analogues of the refined Bohr inequalities for bounded holomorphic mappings on the unit ball in a complex Banach space involving higher dimensional Schwarz mappings. All the results are proved to be sharp.
{"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}
In this article, we present the symmetry group of a global slice Dirac operator and its iterated ones. Further, the explicit forms of intertwining operators of the iterated global slice Dirac operator are given. At the end, we introduce a variant of the global slice Dirac operator, which allows functions considered to be defined on the whole Euclidean space. The invariance property and the intertwining operators of this variant of the global slice Dirac operator are also presented.
{"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}
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