{"title":"On Hollenbeck-Verbitsky conjecture for $4/3 < p < 2$","authors":"Vladan Jaguzović","doi":"arxiv-2408.17093","DOIUrl":null,"url":null,"abstract":"Let \\(P_+\\) be the Riesz's projection operator and let \\(P_-=I-P_+.\\) We find\nbest estimates of the expression \\(\\left\\lVert \\left( \\left\\lvert P_+f\n\\right\\rvert ^s + \\left\\lvert P_-f \\right\\rvert ^s \\right) ^{1/s} \\right\\rVert\n_p \\) in terms of Lebesgue p-norm of the function \\(f \\in L^p(\\mathbf{T})\\) for\n\\(p \\in (4/3,2)\\) and \\(0 < s \\leq \\frac{p}{p-1},\\) thus extending results from\n\\cite{Melentijevic_2022} and \\cite{Melentijevic_2023}, where the mentioned\nrange is not considered.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"2 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-30","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-2408.17093","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
Let \(P_+\) be the Riesz's projection operator and let \(P_-=I-P_+.\) We find
best estimates of the expression \(\left\lVert \left( \left\lvert P_+f
\right\rvert ^s + \left\lvert P_-f \right\rvert ^s \right) ^{1/s} \right\rVert
_p \) 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.