Pedro Barbosa, Arturo Fernández-Pérez, Víctor León
We introduce the notion of the textit{Bruce-Roberts number} for holomorphic 1-forms relative to complex analytic varieties. Our main result shows that the Bruce-Roberts number of a 1-form $omega$ with respect to a complex analytic hypersurface $X$ with an isolated singularity can be expressed in terms of the textit{Ebeling--Gusein-Zade index} of $omega$ along $X$, the textit{Milnor number} of $omega$ and the textit{Tjurina number} of $X$. This result allows us to recover known formulas for the Bruce-Roberts number of a holomorphic function along $X$ and to establish connections between this number, the radial index, and the local Euler obstruction of $omega$ along $X$. Moreover, we present applications to both global and local holomorphic foliations in complex dimension two.
{"title":"The Bruce-Roberts number of holomorphic 1-forms along complex analytic varieties","authors":"Pedro Barbosa, Arturo Fernández-Pérez, Víctor León","doi":"arxiv-2409.01237","DOIUrl":"https://doi.org/arxiv-2409.01237","url":null,"abstract":"We introduce the notion of the textit{Bruce-Roberts number} for holomorphic\u00001-forms relative to complex analytic varieties. Our main result shows that the\u0000Bruce-Roberts number of a 1-form $omega$ with respect to a complex analytic\u0000hypersurface $X$ with an isolated singularity can be expressed in terms of the\u0000textit{Ebeling--Gusein-Zade index} of $omega$ along $X$, the textit{Milnor\u0000number} of $omega$ and the textit{Tjurina number} of $X$. This result allows\u0000us to recover known formulas for the Bruce-Roberts number of a holomorphic\u0000function along $X$ and to establish connections between this number, the radial\u0000index, and the local Euler obstruction of $omega$ along $X$. Moreover, we\u0000present applications to both global and local holomorphic foliations in complex\u0000dimension two.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201824","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 Carleson measures on NTA and ADP domains in the Heisenberg groups $mathbb{H}^n$ and provide two characterizations of such measures: (1) in terms of the level sets of subelliptic harmonic functions and (2) via the $1$-quasiconformal family of mappings on the Kor'anyi--Reimann unit ball. Moreover, we establish the $L^2$-bounds for the square function $S_{alpha}$ of a subelliptic harmonic function and the Carleson measure estimates for the BMO boundary data, both on NTA domains in $mathbb{H}^n$. Finally, we prove a Fatou-type theorem on $(epsilon, delta)$-domains in $mathbb{H}^n$.
{"title":"Carleson measures on domains in Heisenberg groups","authors":"Tomasz Adamowicz, Marcin Gryszówka","doi":"arxiv-2409.01096","DOIUrl":"https://doi.org/arxiv-2409.01096","url":null,"abstract":"We study the Carleson measures on NTA and ADP domains in the Heisenberg\u0000groups $mathbb{H}^n$ and provide two characterizations of such measures: (1)\u0000in terms of the level sets of subelliptic harmonic functions and (2) via the\u0000$1$-quasiconformal family of mappings on the Kor'anyi--Reimann unit ball.\u0000Moreover, we establish the $L^2$-bounds for the square function $S_{alpha}$ of\u0000a subelliptic harmonic function and the Carleson measure estimates for the BMO\u0000boundary data, both on NTA domains in $mathbb{H}^n$. Finally, we prove a\u0000Fatou-type theorem on $(epsilon, delta)$-domains in $mathbb{H}^n$.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201852","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 show that certain moduli spaces of vector bundles over blown-up primary Hopf surfaces admit no compact components. These are the moduli spaces used by Andrei Teleman in his work on the classification of class $VII$ surfaces.
{"title":"Vector bundles on blown-up Hopf surfaces","authors":"Matei Toma","doi":"arxiv-2408.17330","DOIUrl":"https://doi.org/arxiv-2408.17330","url":null,"abstract":"We show that certain moduli spaces of vector bundles over blown-up primary\u0000Hopf surfaces admit no compact components. These are the moduli spaces used by\u0000Andrei Teleman in his work on the classification of class $VII$ surfaces.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"89 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201854","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 $D$ be a bounded domain in the complex plane with Lipschitz boundary. In the paper, we construct an integral solution operator $T[f]$ for any $overline{partial}$ closed $(0,1)$-form $fin L^p_{(0,1)}(D^n)$ solving the Cauchy-Riemain equation $overline{partial} u=f$ on the product domains $D^n$ and obtain the $L^p$-estimates for all $1
{"title":"$overline{partial}$-Estimates on the product of bounded Lipschitz domain","authors":"Song-Ying Li, Sujuan Long, Jie Lao","doi":"arxiv-2409.00293","DOIUrl":"https://doi.org/arxiv-2409.00293","url":null,"abstract":"Let $D$ be a bounded domain in the complex plane with Lipschitz boundary. In\u0000the paper, we construct an integral solution operator $T[f]$ for any\u0000$overline{partial}$ closed $(0,1)$-form $fin L^p_{(0,1)}(D^n)$ solving the\u0000Cauchy-Riemain equation $overline{partial} u=f$ on the product domains $D^n$\u0000and obtain the $L^p$-estimates for all $1<ple infty$.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"178 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201860","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 (P_+) be the Riesz's projection operator and let (P_-=I-P_+.) We find best estimates of the expression (leftlVert left( leftlvert P_+f rightrvert ^s + leftlvert P_-f rightrvert ^s right) ^{1/s} rightrVert _p ) in terms of Lebesgue p-norm of the function (f in L^p(mathbf{T})) for (p in (4/3,2)) and (0 < s leq frac{p}{p-1},) thus extending results from cite{Melentijevic_2022} and cite{Melentijevic_2023}, where the mentioned range is not considered.
让（P_+）成为里兹投影算子，让（P_-=I-P_+.）成为里兹投影算子。 我们可以找到表达式 (leftlVert left( leftlvert P_+frightrvert ^s + leftlvert P_-f rightrvert ^s right) ^{1/s} 的最佳估计值。rightr Vert_p) in terms of Lebesgue p-norm of the function (f in L^p(mathbf{T})) for(p in (4/3,2)) and (0 < s leq frac{p}{p-1}、)从而扩展了来自cite{Melentijevic_2022}和cite{Melentijevic_2023}的结果，在这两个结果中没有考虑提到的范围。
{"title":"On Hollenbeck-Verbitsky conjecture for $4/3 < p < 2$","authors":"Vladan Jaguzović","doi":"arxiv-2408.17093","DOIUrl":"https://doi.org/arxiv-2408.17093","url":null,"abstract":"Let (P_+) be the Riesz's projection operator and let (P_-=I-P_+.) We find\u0000best estimates of the expression (leftlVert left( leftlvert P_+f\u0000rightrvert ^s + leftlvert P_-f rightrvert ^s right) ^{1/s} rightrVert\u0000_p ) in terms of Lebesgue p-norm of the function (f in L^p(mathbf{T})) for\u0000(p in (4/3,2)) and (0 < s leq frac{p}{p-1},) thus extending results from\u0000cite{Melentijevic_2022} and cite{Melentijevic_2023}, where the mentioned\u0000range is not considered.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201853","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 derive a useful result about the zeros of the $k$-polar polynomials on the unit circle; in particular we obtain a ring shaped region containing all the zeros of these polynomials. Some examples are presented.
{"title":"Localization of zeros of polar polynomials on the unit disc","authors":"Roberto S. Costas-Santos, Abdelhamid Rehouma","doi":"arxiv-2409.00156","DOIUrl":"https://doi.org/arxiv-2409.00156","url":null,"abstract":"We derive a useful result about the zeros of the $k$-polar polynomials on the\u0000unit circle; in particular we obtain a ring shaped region containing all the\u0000zeros of these polynomials. Some examples are presented.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201850","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}
Fabrizio CataneseBayreuth and KIAS Seoul, Davide FrapportiPolitecnico Milano, Christian GleissnerBayreuth, Wenfei LiuXiamen, Matthias SuchüttHannover
In this first part we describe the group $Aut_{mathbb{Z}}(S)$ of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface $S$ with Kodaira dimension $kappa(S)=1$), in the initial case $ chi(mathcal{O}_S) =0$. In particular, in the case where $Aut_{mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $Aut_{mathbb{Z}}(S)$ is nontrivial, it is one of the following groups: $mathbb{Z}/2, mathbb{Z}/3, (mathbb{Z}/2)^2$. We also show with easy examples that the groups $mathbb{Z}/2, mathbb{Z}/3$ do effectively occur. Respectively, in the case where $Aut_{mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components.
{"title":"On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$","authors":"Fabrizio CataneseBayreuth and KIAS Seoul, Davide FrapportiPolitecnico Milano, Christian GleissnerBayreuth, Wenfei LiuXiamen, Matthias SuchüttHannover","doi":"arxiv-2408.16936","DOIUrl":"https://doi.org/arxiv-2408.16936","url":null,"abstract":"In this first part we describe the group $Aut_{mathbb{Z}}(S)$ of\u0000cohomologically trivial automorphisms of a properly elliptic surface (a minimal\u0000surface $S$ with Kodaira dimension $kappa(S)=1$), in the initial case $\u0000chi(mathcal{O}_S) =0$. In particular, in the case where $Aut_{mathbb{Z}}(S)$ is finite, we give the\u0000upper bound 4 for its cardinality, showing more precisely that if\u0000$Aut_{mathbb{Z}}(S)$ is nontrivial, it is one of the following groups:\u0000$mathbb{Z}/2, mathbb{Z}/3, (mathbb{Z}/2)^2$. We also show with easy examples\u0000that the groups $mathbb{Z}/2, mathbb{Z}/3$ do effectively occur. Respectively, in the case where $Aut_{mathbb{Z}}(S)$ is infinite, we give\u0000the sharp upper bound 2 for the number of its connected components.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201855","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 note, we prove that one can use the generalized Bergman kernels to approximate the minimal $L^2$ integrals with respect to ideals of the ring of germs of holomorhpic functions.
在本论文中，我们证明了可以利用广义伯格曼核来近似求全偶函数环的理想的最小 $L^2$ 积分。
{"title":"A remark on Bergman kernels and minimal $L^2$ integrals","authors":"Shijie Bao, Qi'an Guan","doi":"arxiv-2408.16372","DOIUrl":"https://doi.org/arxiv-2408.16372","url":null,"abstract":"In this note, we prove that one can use the generalized Bergman kernels to\u0000approximate the minimal $L^2$ integrals with respect to ideals of the ring of\u0000germs of holomorhpic functions.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"93 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201859","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 $mathcal{A}$ be the class of all analytic functions $f$ defined on the open unit disk $mathbb{D}$ with the normalization $f(0)=0=f^{prime}(0)-1$. This paper examines the radius of concavity for various subclasses of $mathcal{A}$, namely $mathcal{S}_0^{(n)}$, $mathcal{K(alpha,beta)}$, $mathcal{tilde{S^*}(beta)}$, and $mathcal{S}^*(alpha)$. It also presents results for various classes of analytic functions on the unit disk. All the radii are best possible.
{"title":"Sharp radius of concavity for certain classes of analytic functions","authors":"Molla Basir Ahamed, Rajesh Hossain","doi":"arxiv-2408.15544","DOIUrl":"https://doi.org/arxiv-2408.15544","url":null,"abstract":"Let $mathcal{A}$ be the class of all analytic functions $f$ defined on the\u0000open unit disk $mathbb{D}$ with the normalization $f(0)=0=f^{prime}(0)-1$.\u0000This paper examines the radius of concavity for various subclasses of\u0000$mathcal{A}$, namely $mathcal{S}_0^{(n)}$, $mathcal{K(alpha,beta)}$,\u0000$mathcal{tilde{S^*}(beta)}$, and $mathcal{S}^*(alpha)$. It also presents\u0000results for various classes of analytic functions on the unit disk. All the\u0000radii are best possible.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201856","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 Bohr radius for an arbitrary class $mathcal{F}$ of analytic functions of the form $f(z)=sum_{n=0}^{infty}a_nz^n$ on the unit disk $mathbb{D}={zinmathbb{C} : |z|<1}$ is the largest radius $R_{mathcal{F}}$ such that every function $finmathcal{F}$ satisfies the inequality begin{align*} dleft(sum_{n=0}^{infty}|a_nz^n|, |f(0)|right)=sum_{n=1}^{infty}|a_nz^n|leq d(f(0), partial f(mathbb{D})), end{align*} for all $|z|=rleq R_{mathcal{F}}$ , where $d(0, partial f(mathbb{D}))$ is the Euclidean distance. In this paper, our aim is to determine the sharp improved Bohr radius for the classes of analytic functions $f$ satisfying differential subordination relation $zf^{prime}(z)/f(z)prec h(z)$ and $f(z)+beta zf^{prime}(z)+gamma z^2f^{primeprime}(z)prec h(z)$, where $h$ is the Janowski function. We show that improved Bohr radius can be obtained for Janowski functions as root of an equation involving Bessel function of first kind. Analogues results are obtained in this paper for $alpha$-convex functions and typically real functions, respectively. All obtained results in the paper are sharp and are improved version of [{Bull. Malays. Math. Sci. Soc.} (2021) 44:1771-1785].
{"title":"Sharp Bohr radius involving Schwarz functions for certain classes of analytic functions","authors":"Molla Basir Ahamed, Partha Pratim Roy","doi":"arxiv-2408.14773","DOIUrl":"https://doi.org/arxiv-2408.14773","url":null,"abstract":"The Bohr radius for an arbitrary class $mathcal{F}$ of analytic functions of\u0000the form $f(z)=sum_{n=0}^{infty}a_nz^n$ on the unit disk\u0000$mathbb{D}={zinmathbb{C} : |z|<1}$ is the largest radius $R_{mathcal{F}}$\u0000such that every function $finmathcal{F}$ satisfies the inequality\u0000begin{align*} dleft(sum_{n=0}^{infty}|a_nz^n|,\u0000|f(0)|right)=sum_{n=1}^{infty}|a_nz^n|leq d(f(0), partial f(mathbb{D})),\u0000end{align*} for all $|z|=rleq R_{mathcal{F}}$ , where $d(0, partial\u0000f(mathbb{D}))$ is the Euclidean distance. In this paper, our aim is to\u0000determine the sharp improved Bohr radius for the classes of analytic functions\u0000$f$ satisfying differential subordination relation $zf^{prime}(z)/f(z)prec\u0000h(z)$ and $f(z)+beta zf^{prime}(z)+gamma z^2f^{primeprime}(z)prec h(z)$,\u0000where $h$ is the Janowski function. We show that improved Bohr radius can be\u0000obtained for Janowski functions as root of an equation involving Bessel\u0000function of first kind. Analogues results are obtained in this paper for\u0000$alpha$-convex functions and typically real functions, respectively. All\u0000obtained results in the paper are sharp and are improved version of [{Bull.\u0000Malays. Math. Sci. Soc.} (2021) 44:1771-1785].","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"729 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201858","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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