The aim of the paper is to study the level sets of the solutions of Dirichlet problems for the Levi operator on strongly pseudoconvex domains $Omega$ in $mathbb C^2$. Such solutions are generically non smooth, and the geometric properties of their level sets are characterized by means of hulls of their intersections with $bOmega$, using as main tool the local maximum property introduced by Slodkowski (PJM, 1988). The same techniques are then employed to study the behavior of the complete Levi operator for graphs in $mathbb C^2$.
{"title":"Levi equation and local maximum property","authors":"Giuseppe Della Sala, Giuseppe Tomassini","doi":"arxiv-2409.05776","DOIUrl":"https://doi.org/arxiv-2409.05776","url":null,"abstract":"The aim of the paper is to study the level sets of the solutions of Dirichlet\u0000problems for the Levi operator on strongly pseudoconvex domains $Omega$ in\u0000$mathbb C^2$. Such solutions are generically non smooth, and the geometric\u0000properties of their level sets are characterized by means of hulls of their\u0000intersections with $bOmega$, using as main tool the local maximum property\u0000introduced by Slodkowski (PJM, 1988). The same techniques are then employed to\u0000study the behavior of the complete Levi operator for graphs in $mathbb C^2$.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201817","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 extend the uniform $L^2$-estimate of $bar{partial}$-equations for flat nontrivial line bundles, proved for compact K"ahler manifolds in the previous work, to compact complex manifolds. In the proof, by tracing the Dolbeault isomorphism in detail, we derive the desired $L^2$-estimate directly from Ueda's lemma.
{"title":"Uniform $L^2$-estimates for flat nontrivial line bundles on compact complex manifolds","authors":"Yoshinori Hashimoto, Takayuki Koike, Shin-ichi Matsumura","doi":"arxiv-2409.05300","DOIUrl":"https://doi.org/arxiv-2409.05300","url":null,"abstract":"In this paper, we extend the uniform $L^2$-estimate of\u0000$bar{partial}$-equations for flat nontrivial line bundles, proved for compact\u0000K\"ahler manifolds in the previous work, to compact complex manifolds. In the\u0000proof, by tracing the Dolbeault isomorphism in detail, we derive the desired\u0000$L^2$-estimate directly from Ueda's lemma.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201819","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 $(M,kappa)$ be a closed and connected real-analytic Riemannian manifold, acted upon by a compact Lie group of isometries $G$. We consider the following two kinds of equivariant asymptotics along a fixed Grauer tube boundary $X^tau$ of $(M,kappa)$. 1): Given the induced unitary representation of $G$ on the eigenspaces of the Laplacian of $(M,kappa)$, these split over the irreducible representations of $G$. On the other hand, the eigenfunctions of the Laplacian of $(M,kappa)$ admit a simultaneous complexification to some Grauert tube. We study the asymptotic concentration along $X^tau$ of the complexified eigenfunctions pertaining to a fixed isotypical component. 2): There are furthermore an induced action of $G$ as a group of CR and contact automorphisms on $X^tau$, and a corresponding unitary representation on the Hardy space $H(X^tau)$. The action of $G$ on $X^tau$ commutes with the homogeneous lq geogesic flowrq, and the representation on the Hardy space commutes with the elliptic self-adjoint Toeplitz operator induced by the generator of the goedesic flow. Hence each eigenspace of the latter also splits over the irreducible representations of $G$. We study the asymptotic concentration of the eigenfunctions in a given isotypical component. We also give some applications of these asymptotics.
{"title":"Equivariant scaling asymptotics for Poisson and Szegő kernels on Grauert tube boundaries","authors":"Simone Gallivanone, Roberto Paoletti","doi":"arxiv-2409.04753","DOIUrl":"https://doi.org/arxiv-2409.04753","url":null,"abstract":"Let $(M,kappa)$ be a closed and connected real-analytic Riemannian manifold,\u0000acted upon by a compact Lie group of isometries $G$. We consider the following\u0000two kinds of equivariant asymptotics along a fixed Grauer tube boundary\u0000$X^tau$ of $(M,kappa)$. 1): Given the induced unitary representation of $G$ on the eigenspaces of the\u0000Laplacian of $(M,kappa)$, these split over the irreducible representations of\u0000$G$. On the other hand, the eigenfunctions of the Laplacian of $(M,kappa)$\u0000admit a simultaneous complexification to some Grauert tube. We study the\u0000asymptotic concentration along $X^tau$ of the complexified eigenfunctions\u0000pertaining to a fixed isotypical component. 2): There are furthermore an induced action of $G$ as a group of CR and\u0000contact automorphisms on $X^tau$, and a corresponding unitary representation\u0000on the Hardy space $H(X^tau)$. The action of $G$ on $X^tau$ commutes with the\u0000homogeneous lq geogesic flowrq, and the representation on the Hardy space\u0000commutes with the elliptic self-adjoint Toeplitz operator induced by the\u0000generator of the goedesic flow. Hence each eigenspace of the latter also splits\u0000over the irreducible representations of $G$. We study the asymptotic\u0000concentration of the eigenfunctions in a given isotypical component. We also give some applications of these asymptotics.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201816","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 investigate holomorphic mappings $F$ on the unit ball $mathbb{B}$ of a complex Banach space of the form $F(x)=f(x)x$, where $f$ is a holomorphic function on $mathbb{B}$. First, we investigate criteria for univalence, starlikeness and quasi-convexity of type $B$ on $mathbb{B}$. Next, we investigate a generalized Bieberbach conjecture, a covering theorem and a distortion theorem, the Fekete-Szeg"{o} inequality, lower bound for the Bloch constant, and Alexander's type theorem for such mappings.
{"title":"Bieberbach conjecture, Bohr radius, Bloch constant and Alexander's theorem in infinite dimensions","authors":"Hidetaka Hamada, Gabriela Kohr, Mirela Kohr","doi":"arxiv-2409.04028","DOIUrl":"https://doi.org/arxiv-2409.04028","url":null,"abstract":"In this paper, we investigate holomorphic mappings $F$ on the unit ball\u0000$mathbb{B}$ of a complex Banach space of the form $F(x)=f(x)x$, where $f$ is a\u0000holomorphic function on $mathbb{B}$. First, we investigate criteria for\u0000univalence, starlikeness and quasi-convexity of type $B$ on $mathbb{B}$. Next,\u0000we investigate a generalized Bieberbach conjecture, a covering theorem and a\u0000distortion theorem, the Fekete-Szeg\"{o} inequality, lower bound for the Bloch\u0000constant, and Alexander's type theorem for such mappings.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201821","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 finish the basic development of the discrete octonionic analysis by presenting a Weyl calculus-based approach to bounded domains in $mathbb{R}^{8}$. In particular, we explicitly prove the discrete Stokes formula for a bounded cuboid, and then we generalise this result to arbitrary bounded domains in interior and exterior settings by the help of characteristic functions. After that, discrete interior and exterior Borel-Pompeiu and Cauchy formulae are introduced. Finally, we recall the construction of discrete octonionic Hardy spaces for bounded domains. Moreover, we explicitly explain where the non-associativity of octonionic multiplication is essential and where it is not. Thus, this paper completes the basic framework of the discrete octonionic analysis introduced in previous papers, and, hence, provides a solid foundation for further studies in this field.
{"title":"Application of the Weyl calculus perspective on discrete octonionic analysis in bounded domains","authors":"Rolf Sören Kraußhar, Anastasiia Legatiuk, Dmitrii Legatiuk","doi":"arxiv-2409.04285","DOIUrl":"https://doi.org/arxiv-2409.04285","url":null,"abstract":"In this paper, we finish the basic development of the discrete octonionic\u0000analysis by presenting a Weyl calculus-based approach to bounded domains in\u0000$mathbb{R}^{8}$. In particular, we explicitly prove the discrete Stokes\u0000formula for a bounded cuboid, and then we generalise this result to arbitrary\u0000bounded domains in interior and exterior settings by the help of characteristic\u0000functions. After that, discrete interior and exterior Borel-Pompeiu and Cauchy\u0000formulae are introduced. Finally, we recall the construction of discrete\u0000octonionic Hardy spaces for bounded domains. Moreover, we explicitly explain\u0000where the non-associativity of octonionic multiplication is essential and where\u0000it is not. Thus, this paper completes the basic framework of the discrete\u0000octonionic analysis introduced in previous papers, and, hence, provides a solid\u0000foundation for further studies in this field.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201822","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 several results on approximation and interpolation of holomorphic Legendrian curves in convex domains in $mathbb{C}^{2n+1}$, $n geq 2$, with the standard contact structure. Namely, we show that such a curve, defined on a compact bordered Riemann surface $M$, whose image lies in the interior of a convex domain $mathscr{D} subset mathbb{C}^{2n+1}$, may be approximated uniformly on compacts in the interior $mathrm{Int} , M$ by holomorphic Legendrian curves $mathrm{Int} , M to mathscr{D}$ such that the approximants are proper, complete, agree with the starting curve on a given finite set in $mathrm{Int} , M$ to a given finite order, and hit a specified diverging discrete set in the convex domain. We first show approximation of this kind on bounded strongly convex domains and then generalise it to arbitrary convex domains. As a consequence we show that any bordered Riemann surface properly embeds into a convex domain as a complete holomorphic Legendrian curve under a suitable geometric condition on the boundary of the codomain.
{"title":"Holomorphic Legendrian curves in convex domains","authors":"Andrej Svetina","doi":"arxiv-2409.04197","DOIUrl":"https://doi.org/arxiv-2409.04197","url":null,"abstract":"We prove several results on approximation and interpolation of holomorphic\u0000Legendrian curves in convex domains in $mathbb{C}^{2n+1}$, $n geq 2$, with\u0000the standard contact structure. Namely, we show that such a curve, defined on a\u0000compact bordered Riemann surface $M$, whose image lies in the interior of a\u0000convex domain $mathscr{D} subset mathbb{C}^{2n+1}$, may be approximated\u0000uniformly on compacts in the interior $mathrm{Int} , M$ by holomorphic\u0000Legendrian curves $mathrm{Int} , M to mathscr{D}$ such that the\u0000approximants are proper, complete, agree with the starting curve on a given\u0000finite set in $mathrm{Int} , M$ to a given finite order, and hit a specified\u0000diverging discrete set in the convex domain. We first show approximation of\u0000this kind on bounded strongly convex domains and then generalise it to\u0000arbitrary convex domains. As a consequence we show that any bordered Riemann\u0000surface properly embeds into a convex domain as a complete holomorphic\u0000Legendrian curve under a suitable geometric condition on the boundary of the\u0000codomain.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201820","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 investigate Chebyshev polynomials corresponding to Jacobi weights and determine monotonicity properties of their related Widom factors. This complements work by Bernstein from 1930-31 where the asymptotical behavior of the related Chebyshev norms was established. As a part of the proof, we analyze a Bernstein-type inequality for Jacobi polynomials due to Chow et al. Our findings shed new light on the asymptotical uniform bounds of Jacobi polynomials. We also show a relation between weighted Chebyshev polynomials on the unit circle and Jacobi weighted Chebyshev polynomials on [-1,1]. This generalizes work by Lachance et al. In order to complete the picture we provide numerical experiments on the remaining cases that our proof does not cover.
{"title":"Chebyshev polynomials related to Jacobi weights","authors":"Jacob S. Christiansen, Olof Rubin","doi":"arxiv-2409.02623","DOIUrl":"https://doi.org/arxiv-2409.02623","url":null,"abstract":"We investigate Chebyshev polynomials corresponding to Jacobi weights and\u0000determine monotonicity properties of their related Widom factors. This\u0000complements work by Bernstein from 1930-31 where the asymptotical behavior of\u0000the related Chebyshev norms was established. As a part of the proof, we analyze\u0000a Bernstein-type inequality for Jacobi polynomials due to Chow et al. Our\u0000findings shed new light on the asymptotical uniform bounds of Jacobi\u0000polynomials. We also show a relation between weighted Chebyshev polynomials on\u0000the unit circle and Jacobi weighted Chebyshev polynomials on [-1,1]. This\u0000generalizes work by Lachance et al. In order to complete the picture we provide\u0000numerical experiments on the remaining cases that our proof does not cover.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225825","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 the gain in regularity of the distance to the boundary of a domain in $R^m$. In particular, we show that if the signed distance function happens to be merely differentiable in a neighborhood of a boundary point, it and the boundary have to be $mathcal C^{1,1}$ regular. Conversely, we study the regularity of the distance function under regularity hypotheses of the boundary. Along the way, we point out that any solution to the eikonal equation, differentiable everywhere in a domain of the Euclidean space, admits a gradient which is locally Lipschitz.
{"title":"Boundary regularity for the distance functions, and the eikonal equation","authors":"Nikolai Nikolov, Pascal J. Thomas","doi":"arxiv-2409.01774","DOIUrl":"https://doi.org/arxiv-2409.01774","url":null,"abstract":"We study the gain in regularity of the distance to the boundary of a domain\u0000in $R^m$. In particular, we show that if the signed distance function happens\u0000to be merely differentiable in a neighborhood of a boundary point, it and the\u0000boundary have to be $mathcal C^{1,1}$ regular. Conversely, we study the\u0000regularity of the distance function under regularity hypotheses of the\u0000boundary. Along the way, we point out that any solution to the eikonal\u0000equation, differentiable everywhere in a domain of the Euclidean space, admits\u0000a gradient which is locally Lipschitz.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201907","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}
Evgeny Sevost'yanov, Victoria Desyatka Zarina Kovba
We consider mappings that distort the modulus of families of paths in the opposite direction in the manner of Poletsky's inequality. Here we study the case when the mappings are not closed, in particular, they do not preserve the boundary of the domain under the mapping. Under certain conditions, we obtain results on the continuous boundary extension of such mappings in the sense of prime ends. In addition, we obtain corresponding results on the equicontinuity of families of such mappings in terms of prime ends.
{"title":"On the prime ends extension of unclosed inverse mappings","authors":"Evgeny Sevost'yanov, Victoria Desyatka Zarina Kovba","doi":"arxiv-2409.02956","DOIUrl":"https://doi.org/arxiv-2409.02956","url":null,"abstract":"We consider mappings that distort the modulus of families of paths in the\u0000opposite direction in the manner of Poletsky's inequality. Here we study the\u0000case when the mappings are not closed, in particular, they do not preserve the\u0000boundary of the domain under the mapping. Under certain conditions, we obtain\u0000results on the continuous boundary extension of such mappings in the sense of\u0000prime ends. In addition, we obtain corresponding results on the equicontinuity\u0000of families of such mappings in terms of prime ends.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"178 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201823","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 $Omega subsetwidehat{mathbb{C}}$ be a multiply connected domain, and let $picolon mathbb{D}toOmega$ be a universal covering map. In this paper, we analyze the boundary behaviour of $pi$, describing the interplay between radial limits and angular cluster sets, the tangential and non-tangential limit sets of the deck transformation group, and the geometry and the topology of the boundary of $Omega$. As an application, we describe accesses to the boundary of $Omega$ in terms of radial limits of points in the unit circle, establishing a correspondence in the same spirit as in the simply connected case. We also develop a theory of prime ends for multiply connected domains which behaves properly under the universal covering, providing an extension of the Carath'eodory--Torhorst Theorem to multiply connected domains.
{"title":"Boundary behaviour of universal covering maps","authors":"Gustavo R. Ferreira, Anna Jové","doi":"arxiv-2409.01070","DOIUrl":"https://doi.org/arxiv-2409.01070","url":null,"abstract":"Let $Omega subsetwidehat{mathbb{C}}$ be a multiply connected domain, and\u0000let $picolon mathbb{D}toOmega$ be a universal covering map. In this paper,\u0000we analyze the boundary behaviour of $pi$, describing the interplay between\u0000radial limits and angular cluster sets, the tangential and non-tangential limit\u0000sets of the deck transformation group, and the geometry and the topology of the\u0000boundary of $Omega$. As an application, we describe accesses to the boundary of $Omega$ in terms\u0000of radial limits of points in the unit circle, establishing a correspondence in\u0000the same spirit as in the simply connected case. We also develop a theory of\u0000prime ends for multiply connected domains which behaves properly under the\u0000universal covering, providing an extension of the Carath'eodory--Torhorst\u0000Theorem to multiply connected domains.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201851","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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