{"title":"平面域的伯格曼数","authors":"Christina Karafyllia","doi":"10.1215/00192082-10678837","DOIUrl":null,"url":null,"abstract":"Let $D$ be a domain in the complex plane $\\mathbb{C}$. The Hardy number of $D$, which first introduced by Hansen, is the maximal number $h(D)$ in $[0,+\\infty]$ such that $f$ belongs to the classical Hardy space $H^p (\\mathbb{D})$ whenever $0<p<h(D)$ and $f$ is holomorphic on the unit disk $\\mathbb{D}$ with values in $D$. As an analogue notion to the Hardy number of a domain $D$ in $\\mathbb{C}$, we introduce the Bergman number of $D$ and we denote it by $b(D)$. Our main result is that, if $D$ is regular, then $h(D)=b(D)$. This generalizes earlier work by the author and Karamanlis for simply connected domains. The Bergman number $b(D)$ is the maximal number in $[0,+\\infty]$ such that $f$ belongs to the weighted Bergman space $A^p_{\\alpha} (\\mathbb{D})$ whenever $p>0$ and $\\alpha>-1$ satisfy $0<\\frac{p}{\\alpha+2}<b(D)$ and $f$ is holomorphic on $\\mathbb{D}$ with values in $D$. We also establish several results about Hardy spaces and weighted Bergman spaces and we give a new characterization of the Hardy number and thus of the Bergman number of a regular domain with respect to the harmonic measure.","PeriodicalId":56298,"journal":{"name":"Illinois Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"The Bergman number of a plane domain\",\"authors\":\"Christina Karafyllia\",\"doi\":\"10.1215/00192082-10678837\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $D$ be a domain in the complex plane $\\\\mathbb{C}$. The Hardy number of $D$, which first introduced by Hansen, is the maximal number $h(D)$ in $[0,+\\\\infty]$ such that $f$ belongs to the classical Hardy space $H^p (\\\\mathbb{D})$ whenever $0<p<h(D)$ and $f$ is holomorphic on the unit disk $\\\\mathbb{D}$ with values in $D$. As an analogue notion to the Hardy number of a domain $D$ in $\\\\mathbb{C}$, we introduce the Bergman number of $D$ and we denote it by $b(D)$. Our main result is that, if $D$ is regular, then $h(D)=b(D)$. This generalizes earlier work by the author and Karamanlis for simply connected domains. The Bergman number $b(D)$ is the maximal number in $[0,+\\\\infty]$ such that $f$ belongs to the weighted Bergman space $A^p_{\\\\alpha} (\\\\mathbb{D})$ whenever $p>0$ and $\\\\alpha>-1$ satisfy $0<\\\\frac{p}{\\\\alpha+2}<b(D)$ and $f$ is holomorphic on $\\\\mathbb{D}$ with values in $D$. We also establish several results about Hardy spaces and weighted Bergman spaces and we give a new characterization of the Hardy number and thus of the Bergman number of a regular domain with respect to the harmonic measure.\",\"PeriodicalId\":56298,\"journal\":{\"name\":\"Illinois Journal of Mathematics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2022-10-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Illinois Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1215/00192082-10678837\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Illinois Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1215/00192082-10678837","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let $D$ be a domain in the complex plane $\mathbb{C}$. The Hardy number of $D$, which first introduced by Hansen, is the maximal number $h(D)$ in $[0,+\infty]$ such that $f$ belongs to the classical Hardy space $H^p (\mathbb{D})$ whenever $00$ and $\alpha>-1$ satisfy $0<\frac{p}{\alpha+2}
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