{"title":"Estimates of Partial Derivatives for Harmonic Functions on the Unit Disc","authors":"","doi":"10.1007/s40315-023-00514-3","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>Let <span> <span>\\(f = P[F]\\)</span> </span> denote the Poisson integral of <em>F</em> in the unit disc <span> <span>\\({{\\mathbb {D}}}\\)</span> </span> with <em>F</em> absolutely continuous on the unit circle <span> <span>\\({{\\mathbb {T}}}\\)</span> </span> and <span> <span>\\(\\dot{F}\\in L^p({{\\mathbb {T}}})\\)</span> </span>, where <span> <span>\\(\\dot{F}(e^{it}) = \\frac{d}{dt} F(e^{it})\\)</span> </span>. We show that for <span> <span>\\(p\\in (1,\\infty )\\)</span> </span>, the partial derivatives <span> <span>\\(f_z\\)</span> </span> and <span> <span>\\(\\overline{f_{\\bar{z}}}\\)</span> </span> belong to the holomorphic Hardy space <span> <span>\\(H^p({{\\mathbb {D}}})\\)</span> </span>. In addition, for <span> <span>\\(p=1\\)</span> </span> or <span> <span>\\(p=\\infty \\)</span> </span>, <span> <span>\\(f_z\\)</span> </span> and <span> <span>\\(\\overline{f_{\\bar{z}}}\\in H^p({{\\mathbb {D}}})\\)</span> </span> if and only if <span> <span>\\(H(\\dot{F})\\in L^p({{\\mathbb {T}}})\\)</span> </span>, the Hilbert transform of <span> <span>\\(\\dot{F}\\)</span> </span> and in that case, we have <span> <span>\\(2izf_z=P[\\dot{F}+iH(\\dot{F})]\\)</span> </span>. Our main tools are integral representations of <span> <span>\\(f_z\\)</span> </span> and <span> <span>\\(f_{\\overline{z}}\\)</span> </span> in terms of <span> <span>\\(\\dot{F}\\)</span> </span> and M. Riesz’s theorem on conjugate functions. This simplifies and extends the recent results of Chen et al. [<span>1</span>] (J. Geom. Anal., 2021) and Zhu [<span>17</span>] (J. Geom. Anal., 2021).</p>","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"55 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods and Function Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40315-023-00514-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let \(f = P[F]\) denote the Poisson integral of F in the unit disc \({{\mathbb {D}}}\) with F absolutely continuous on the unit circle \({{\mathbb {T}}}\) and \(\dot{F}\in L^p({{\mathbb {T}}})\), where \(\dot{F}(e^{it}) = \frac{d}{dt} F(e^{it})\). We show that for \(p\in (1,\infty )\), the partial derivatives \(f_z\) and \(\overline{f_{\bar{z}}}\) belong to the holomorphic Hardy space \(H^p({{\mathbb {D}}})\). In addition, for \(p=1\) or \(p=\infty \), \(f_z\) and \(\overline{f_{\bar{z}}}\in H^p({{\mathbb {D}}})\) if and only if \(H(\dot{F})\in L^p({{\mathbb {T}}})\), the Hilbert transform of \(\dot{F}\) and in that case, we have \(2izf_z=P[\dot{F}+iH(\dot{F})]\). Our main tools are integral representations of \(f_z\) and \(f_{\overline{z}}\) in terms of \(\dot{F}\) and M. Riesz’s theorem on conjugate functions. This simplifies and extends the recent results of Chen et al. [1] (J. Geom. Anal., 2021) and Zhu [17] (J. Geom. Anal., 2021).
Abstract Let \(f = P[F]\) denote the Poisson integral of F in the unit disc \({{\mathbb {D}}}) with F absolutely continuous on the unit circle \({{\mathbb {T}}}\) and\(\dot{F}\in L^p({{\mathbb {T}}})\.其中 \(\dot{F}(e^{it}) = \frac{d}{dt}F(e^{it})\) 。我们证明对于 \(p\in (1,\infty )\)的偏导数 \(f_z\) 和 \(\overline{f_{bar\{z}}) 属于全形哈代空间 \(H^p({{mathbb {D}})\) 。此外，对于（p=1）或（p=infty），当且仅当（H(dot{F})\in L^p({{\mathbb {T}}})）时，（f_z）和（overline{f_{bar{z}}}\in H^p({{\mathbb {D}}})）。在这种情况下，我们就有\(2izf_z=P[\dot{F}+iH(\dot{F})]\) 。我们的主要工具是 \(f_z\) 和 \(f_{\overline{z}}\) 在 \(\dot{F}\) 和 M. Riesz 的共轭函数定理方面的积分表示。这简化并扩展了 Chen 等人 [1] (J. Geom. Anal., 2021) 和 Zhu [17] (J. Geom. Anal., 2021) 的最新成果。
期刊介绍：
CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.
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