{"title":"从对数布洛赫空间作用于伯格曼空间的导数-希尔伯特算子","authors":"Shanli Ye, Yun Xu","doi":"10.1007/s10473-024-0516-1","DOIUrl":null,"url":null,"abstract":"<p>Let <i>μ</i> be a positive Borel measure on the interval [0, 1). The Hankel matrix <span>\\(\\cal{H}_{\\mu}=(\\mu_{n,k})_{n,k\\geq 0}\\)</span> with entries <i>μ</i><sub><i>n,k</i></sub> = <i>μ</i><sub><i>n</i>+<i>k</i></sub>, where <i>μ</i><sub><i>n</i></sub> = ⨜<sub>[0,1)</sub> <i>t</i><sup><i>n</i></sup>d<i>μ</i>(<i>t</i>), induces, formally, the operator</p><span>$$\\cal{DH}_\\mu(f)(z)=\\sum\\limits_{n=0}^\\infty\\left(\\sum\\limits_{k=0}^\\infty \\mu_{n,k}a_k\\right)(n+1)z^n, ~z\\in \\mathbb{D},$$</span><p>where <span>\\(f(z)=\\sum\\limits_{n=0}^\\infty a_nz^n\\)</span> is an analytic function in ⅅ. We characterize the measures <i>μ</i> for which <span>\\(\\cal{DH}_\\mu\\)</span> is bounded (resp., compact) operator from the logarithmic Bloch space <span>\\(\\mathscr{B}_{L^{\\alpha}}\\)</span> into the Bergman space <span>\\(\\cal{A}^p\\)</span>, where 0 ≤ <i>α</i> < ∞, 0 < <i>p</i> < ∞. We also characterize the measures <i>μ</i> for which <span>\\(\\cal{DH}_\\mu\\)</span> is bounded (resp., compact) operator from the logarithmic Bloch space <span>\\(\\mathscr{B}_{L^{\\alpha}}\\)</span> into the classical Bloch space <span>\\(\\mathscr{B}\\)</span>.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"64 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A derivative-Hilbert operator acting from logarithmic Bloch spaces to Bergman spaces\",\"authors\":\"Shanli Ye, Yun Xu\",\"doi\":\"10.1007/s10473-024-0516-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <i>μ</i> be a positive Borel measure on the interval [0, 1). The Hankel matrix <span>\\\\(\\\\cal{H}_{\\\\mu}=(\\\\mu_{n,k})_{n,k\\\\geq 0}\\\\)</span> with entries <i>μ</i><sub><i>n,k</i></sub> = <i>μ</i><sub><i>n</i>+<i>k</i></sub>, where <i>μ</i><sub><i>n</i></sub> = ⨜<sub>[0,1)</sub> <i>t</i><sup><i>n</i></sup>d<i>μ</i>(<i>t</i>), induces, formally, the operator</p><span>$$\\\\cal{DH}_\\\\mu(f)(z)=\\\\sum\\\\limits_{n=0}^\\\\infty\\\\left(\\\\sum\\\\limits_{k=0}^\\\\infty \\\\mu_{n,k}a_k\\\\right)(n+1)z^n, ~z\\\\in \\\\mathbb{D},$$</span><p>where <span>\\\\(f(z)=\\\\sum\\\\limits_{n=0}^\\\\infty a_nz^n\\\\)</span> is an analytic function in ⅅ. We characterize the measures <i>μ</i> for which <span>\\\\(\\\\cal{DH}_\\\\mu\\\\)</span> is bounded (resp., compact) operator from the logarithmic Bloch space <span>\\\\(\\\\mathscr{B}_{L^{\\\\alpha}}\\\\)</span> into the Bergman space <span>\\\\(\\\\cal{A}^p\\\\)</span>, where 0 ≤ <i>α</i> < ∞, 0 < <i>p</i> < ∞. We also characterize the measures <i>μ</i> for which <span>\\\\(\\\\cal{DH}_\\\\mu\\\\)</span> is bounded (resp., compact) operator from the logarithmic Bloch space <span>\\\\(\\\\mathscr{B}_{L^{\\\\alpha}}\\\\)</span> into the classical Bloch space <span>\\\\(\\\\mathscr{B}\\\\)</span>.</p>\",\"PeriodicalId\":50998,\"journal\":{\"name\":\"Acta Mathematica Scientia\",\"volume\":\"64 1\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-08-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Scientia\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10473-024-0516-1\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Scientia","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10473-024-0516-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
A derivative-Hilbert operator acting from logarithmic Bloch spaces to Bergman spaces
Let μ be a positive Borel measure on the interval [0, 1). The Hankel matrix \(\cal{H}_{\mu}=(\mu_{n,k})_{n,k\geq 0}\) with entries μn,k = μn+k, where μn = ⨜[0,1)tndμ(t), induces, formally, the operator
where \(f(z)=\sum\limits_{n=0}^\infty a_nz^n\) is an analytic function in ⅅ. We characterize the measures μ for which \(\cal{DH}_\mu\) is bounded (resp., compact) operator from the logarithmic Bloch space \(\mathscr{B}_{L^{\alpha}}\) into the Bergman space \(\cal{A}^p\), where 0 ≤ α < ∞, 0 < p < ∞. We also characterize the measures μ for which \(\cal{DH}_\mu\) is bounded (resp., compact) operator from the logarithmic Bloch space \(\mathscr{B}_{L^{\alpha}}\) into the classical Bloch space \(\mathscr{B}\).
期刊介绍:
Acta Mathematica Scientia was founded by Prof. Li Guoping (Lee Kwok Ping) in April 1981.
The aim of Acta Mathematica Scientia is to present to the specialized readers important new achievements in the areas of mathematical sciences. The journal considers for publication of original research papers in all areas related to the frontier branches of mathematics with other science and technology.