{"title":"Cauchy transforms and Szegő projections in dual Hardy spaces: inequalities and Möbius invariance","authors":"David E. Barrett, Luke D. Edholm","doi":"arxiv-2407.13033","DOIUrl":null,"url":null,"abstract":"Dual pairs of interior and exterior Hardy spaces associated to a simple\nclosed Lipschitz planar curve are considered, leading to a M\\\"obius invariant\nfunction bounding the norm of the Cauchy transform $\\bf{C}$ from below. This\nfunction is shown to satisfy strong rigidity properties and is closely\nconnected via the Berezin transform to the square of the Kerzman-Stein\noperator. Explicit example calculations are presented. For ellipses, a new\nasymptotically sharp lower bound on the norm of $\\bf{C}$ is produced.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"78 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.13033","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Dual pairs of interior and exterior Hardy spaces associated to a simple
closed Lipschitz planar curve are considered, leading to a M\"obius invariant
function bounding the norm of the Cauchy transform $\bf{C}$ from below. This
function is shown to satisfy strong rigidity properties and is closely
connected via the Berezin transform to the square of the Kerzman-Stein
operator. Explicit example calculations are presented. For ellipses, a new
asymptotically sharp lower bound on the norm of $\bf{C}$ is produced.