This work begins by introducing the groundbreaking concept of log-p-analytic functions. Following this introduction, we proceed to delineate four distinct formulations of Landau-type theorems, specifically crafted for the domain of poly-analytic functions. Among these, two theorems are distinguished by their exactitude, and a third theorem offers a refinement to the existing work of Abdulhadi and Hajj. Concluding the paper, we present four specialized versions of Landau-type theorems applicable to a subset of bounded log-p-analytic functions, resulting in the derivation of two precise outcomes.
{"title":"Advancements in Log-P-Analytic Functions: Landau-Type Theorems and Their Refinements","authors":"Hanghang Zhao, Ming-Sheng Liu, Kit Ian Kou","doi":"arxiv-2409.09624","DOIUrl":"https://doi.org/arxiv-2409.09624","url":null,"abstract":"This work begins by introducing the groundbreaking concept of log-p-analytic\u0000functions. Following this introduction, we proceed to delineate four distinct\u0000formulations of Landau-type theorems, specifically crafted for the domain of\u0000poly-analytic functions. Among these, two theorems are distinguished by their\u0000exactitude, and a third theorem offers a refinement to the existing work of\u0000Abdulhadi and Hajj. Concluding the paper, we present four specialized versions\u0000of Landau-type theorems applicable to a subset of bounded log-p-analytic\u0000functions, resulting in the derivation of two precise outcomes.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265798","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 first present the mixed Hilbert-Samuel multiplicities of analytic local rings over mathbb{C} as generalized Lelong numbers and further represent them as intersection numbers in the context of modifications. As applications, we give estimates or an exact formula for the multiplicities of isolated singularities that given by the Grauert blow-downs of negative holomorphic vector bundles.
{"title":"Multiplicites and modifications, and singularities associated to blowing down negative vector bundles","authors":"Fusheng Deng, Yinji Li, Qunhuan Liu, Xiangyu Zhou","doi":"arxiv-2409.09407","DOIUrl":"https://doi.org/arxiv-2409.09407","url":null,"abstract":"We first present the mixed Hilbert-Samuel multiplicities of analytic local\u0000rings over mathbb{C} as generalized Lelong numbers and further represent them\u0000as intersection numbers in the context of modifications. As applications, we\u0000give estimates or an exact formula for the multiplicities of isolated\u0000singularities that given by the Grauert blow-downs of negative holomorphic\u0000vector bundles.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265797","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 prove that the irreducible desingularization of a singularity given by the Grauert blow down of a negative holomorphic vector bundle over a compact complex manifold is unique up to isomorphism, and as an application, we show that two negative line bundles over compact complex manifolds are isomorphic if and only if their Grauert blow downs have isomorphic germs near the singularities. We also show that there is a unique way to modify a submanifold of a complex manifold to a hypersurface, namely, the blow up of the ambient manifold along the submanifold.
{"title":"Uniqueness of irreducible desingularization of singularities associated to negative vector bundles","authors":"Fusheng Deng, Yinji Li, Qunhuan Liu, Xiangyu Zhou","doi":"arxiv-2409.09402","DOIUrl":"https://doi.org/arxiv-2409.09402","url":null,"abstract":"We prove that the irreducible desingularization of a singularity given by the\u0000Grauert blow down of a negative holomorphic vector bundle over a compact\u0000complex manifold is unique up to isomorphism, and as an application, we show\u0000that two negative line bundles over compact complex manifolds are isomorphic if\u0000and only if their Grauert blow downs have isomorphic germs near the\u0000singularities. We also show that there is a unique way to modify a submanifold\u0000of a complex manifold to a hypersurface, namely, the blow up of the ambient\u0000manifold along the submanifold.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265799","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 extend the methods of Lewicka - Pakzad, Sz'ekelyhidi - Cao and Li - Qiu to study the notion of very weak solutions to the complex $sigma_2$ equation in domains in $mathbb C^n, ngeq 2$. As a by-product we sharpen the regularity threshold of the counterexamples obtained by Li and Qiu in the real case.
我们扩展了 Lewicka - Pakzad, Sz'ekelyhidi - Cao 和 Li - Qiuto 的方法，研究了复$sigma_2$方程在$mathbb C^n, ngeq 2$域中的极弱解的概念。作为副产品，我们提高了李和邱在实例中得到的反例的规律性临界值。
{"title":"Very weak solutions of quadratic Hessian equations","authors":"Sławomir Dinew, Szymon Myga","doi":"arxiv-2409.08852","DOIUrl":"https://doi.org/arxiv-2409.08852","url":null,"abstract":"We extend the methods of Lewicka - Pakzad, Sz'ekelyhidi - Cao and Li - Qiu\u0000to study the notion of very weak solutions to the complex $sigma_2$ equation\u0000in domains in $mathbb C^n, ngeq 2$. As a by-product we sharpen the\u0000regularity threshold of the counterexamples obtained by Li and Qiu in the real\u0000case.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265800","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 establish three Landau-type theorems for certain bounded poly-analytic functions, which generalize the corresponding result for bi-analytic functions given by Liu and Ponnusamy [Canad. Math. Bull. 67(1): 2024, 152-165]. Further, we prove three bi-Lipschitz theorems for these subclasses of poly-analytic functions.
在本文中，我们为某些有界多解析函数建立了三个朗道型定理，它们概括了 Liu 和 Ponnusamy [Canad. Math. Bull. 67(1):2024, 152-165] 所给出的禁止解析函数的相应结果。此外，我们还证明了多解析函数子类的三个双李普西茨定理。
{"title":"Landau-type theorems for certain subclasses of poly-analytic functions","authors":"Vasudevarao Allu, Rohit Kumar","doi":"arxiv-2409.08029","DOIUrl":"https://doi.org/arxiv-2409.08029","url":null,"abstract":"In this paper, we establish three Landau-type theorems for certain bounded\u0000poly-analytic functions, which generalize the corresponding result for\u0000bi-analytic functions given by Liu and Ponnusamy [Canad. Math. Bull. 67(1):\u00002024, 152-165]. Further, we prove three bi-Lipschitz theorems for these\u0000subclasses of poly-analytic functions.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225823","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}
The ratios $sqrt{8/9}=2sqrt{2}/3approx 0.9428$ and $sqrt{3}/2 approx 0.866$ appear in various contexts of black hole physics, as values of the charge-to-mass ratio $Q/M$ or the rotation parameter $a/M$ for Reissner-Nordstr"om and Kerr black holes, respectively. In this work, in the Reissner-Nordstr"om case, I relate these ratios with the quantization of the horizon area, or equivalently of the entropy. Furthermore, these ratios are related to a century-old work of Kasner, in which he conjectured that certain sequences arising from complex analysis may have a quantum interpretation. These numbers also appear in the case of Kerr black holes, but the explanation is not as straightforward. The Kasner ratio may also be relevant for understanding the random matrix and random graph approaches to black hole physics, such as fast scrambling of quantum information, via a bound related to Ramanujan graph. Intriguingly, some other pure mathematical problems in complex analysis, notably complex interpolation in the unit disk, appear to share some mathematical expressions with the black hole problem and thus also involve the Kasner ratio.
{"title":"Black Holes, Complex Curves, and Graph Theory: Revising a Conjecture by Kasner","authors":"Yen Chin Ong","doi":"arxiv-2409.08236","DOIUrl":"https://doi.org/arxiv-2409.08236","url":null,"abstract":"The ratios $sqrt{8/9}=2sqrt{2}/3approx 0.9428$ and $sqrt{3}/2 approx\u00000.866$ appear in various contexts of black hole physics, as values of the\u0000charge-to-mass ratio $Q/M$ or the rotation parameter $a/M$ for\u0000Reissner-Nordstr\"om and Kerr black holes, respectively. In this work, in the\u0000Reissner-Nordstr\"om case, I relate these ratios with the quantization of the\u0000horizon area, or equivalently of the entropy. Furthermore, these ratios are\u0000related to a century-old work of Kasner, in which he conjectured that certain\u0000sequences arising from complex analysis may have a quantum interpretation.\u0000These numbers also appear in the case of Kerr black holes, but the explanation\u0000is not as straightforward. The Kasner ratio may also be relevant for\u0000understanding the random matrix and random graph approaches to black hole\u0000physics, such as fast scrambling of quantum information, via a bound related to\u0000Ramanujan graph. Intriguingly, some other pure mathematical problems in complex\u0000analysis, notably complex interpolation in the unit disk, appear to share some\u0000mathematical expressions with the black hole problem and thus also involve the\u0000Kasner ratio.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201813","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 construct entire curves in projective spaces, which are frequently hypercyclic simultaneously for countably many given translations, with optimal slow growth rates.
我们构建了投影空间中的整条曲线，这些曲线对于可数的给定平移来说，经常同时具有低增长率。
{"title":"Frequently hypercyclic meromorphic curves with slow growth","authors":"Bin Guo, Song-Yan Xie, Zhangchi Chen","doi":"arxiv-2409.08048","DOIUrl":"https://doi.org/arxiv-2409.08048","url":null,"abstract":"We construct entire curves in projective spaces, which are frequently\u0000hypercyclic simultaneously for countably many given translations, with optimal\u0000slow growth rates.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"84 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201815","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}
Given a strictly pseudoconvex CR manifold $M$ of dimension three and positive CR Yamabe class, and a positive smooth function $K:Mtomathbf{R}$ verifying some mild and generic hypotheses, we prove the compactness of the set of solutions of the Webster curvature prescription problem associated to $K$, and we compute the Leray-Schauder degree in terms of the critical points of $K$. As a corollary, we get an existence result which generalizes the ones existent in the literature.
{"title":"Blow-up analysis and degree theory for the Webster curvature prescription problem in three dimensions","authors":"Claudio Afeltra","doi":"arxiv-2409.07334","DOIUrl":"https://doi.org/arxiv-2409.07334","url":null,"abstract":"Given a strictly pseudoconvex CR manifold $M$ of dimension three and positive\u0000CR Yamabe class, and a positive smooth function $K:Mtomathbf{R}$ verifying\u0000some mild and generic hypotheses, we prove the compactness of the set of\u0000solutions of the Webster curvature prescription problem associated to $K$, and\u0000we compute the Leray-Schauder degree in terms of the critical points of $K$. As\u0000a corollary, we get an existence result which generalizes the ones existent in\u0000the literature.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201811","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}
Steven R. Bell, Loredana Lanzani, Nathan A. Wagner
We decompose $p$ - integrable functions on the boundary of a simply connected Lipschitz domain $Omega subset mathbb C$ into the sum of the boundary values of two, uniquely determined holomorphic functions, where one is holomorphic in $Omega$ while the other is holomorphic in $mathbb Csetminus overline{Omega}$ and vanishes at infinity. This decomposition has been described previously for smooth functions on the boundary of a smooth domain. Uniqueness of the decomposition is elementary in the smooth case, but extending it to the $L^p$ setting relies upon a regularity result for the holomorphic Hardy space $h^p(bOmega)$ which appears to be new even for smooth $Omega$. An immediate consequence of our result will be a new characterization of the kernel of the Cauchy transform acting on $L^p(bOmega)$. These results give a new perspective on the classical Dirichlet problem for harmonic functions and the Poisson formula even in the case of the disc. Further applications are presented along with directions for future work.
{"title":"A new way to express boundary values in terms of holomorphic functions on planar Lipschitz domains","authors":"Steven R. Bell, Loredana Lanzani, Nathan A. Wagner","doi":"arxiv-2409.06611","DOIUrl":"https://doi.org/arxiv-2409.06611","url":null,"abstract":"We decompose $p$ - integrable functions on the boundary of a simply connected\u0000Lipschitz domain $Omega subset mathbb C$ into the sum of the boundary values\u0000of two, uniquely determined holomorphic functions, where one is holomorphic in\u0000$Omega$ while the other is holomorphic in $mathbb Csetminus\u0000overline{Omega}$ and vanishes at infinity. This decomposition has been\u0000described previously for smooth functions on the boundary of a smooth domain.\u0000Uniqueness of the decomposition is elementary in the smooth case, but extending\u0000it to the $L^p$ setting relies upon a regularity result for the holomorphic\u0000Hardy space $h^p(bOmega)$ which appears to be new even for smooth $Omega$. An\u0000immediate consequence of our result will be a new characterization of the\u0000kernel of the Cauchy transform acting on $L^p(bOmega)$. These results give a\u0000new perspective on the classical Dirichlet problem for harmonic functions and\u0000the Poisson formula even in the case of the disc. Further applications are\u0000presented along with directions for future work.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201812","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}
The Wiman-Valiron inequality relates the maximum modulus of an analytic function to its Taylor coefficients via the maximum term. After a short overview of the known results, we obtain a general version of this inequality that seems to have been overlooked in the literature so far. We end the paper with an open problem.
{"title":"A note on the Wiman-Valiron inequality","authors":"Karl-G. Grosse-Erdmann","doi":"arxiv-2409.06499","DOIUrl":"https://doi.org/arxiv-2409.06499","url":null,"abstract":"The Wiman-Valiron inequality relates the maximum modulus of an analytic\u0000function to its Taylor coefficients via the maximum term. After a short\u0000overview of the known results, we obtain a general version of this inequality\u0000that seems to have been overlooked in the literature so far. We end the paper\u0000with an open problem.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201814","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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