{"title":"On M. Riesz Conjugate Function Theorem for Harmonic Functions","authors":"David Kalaj","doi":"10.1007/s11118-024-10150-8","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\(L^p(\\textbf{T})\\)</span> be the Lesbegue space of complex-valued functions defined in the unit circle <span>\\(\\textbf{T}=\\{z: |z|=1\\}\\subseteq \\mathbb {C}\\)</span>. In this paper, we address the problem of finding the best constant in the inequality of the form: </p><span>$$ \\Vert f\\Vert _{L^p(\\textbf{T})}\\le A_{p,b} \\Vert (|P_+ f|^2+b| P_{-} f|^2)^{1/2}\\Vert _{L^p(\\textbf{T})}. $$</span><p>Here <span>\\(p\\in [1,2]\\)</span>, <span>\\(b>0\\)</span>, and by <span>\\(P_{-} f\\)</span> and <span>\\( P_+ f\\)</span> are denoted the co-analytic and analytic projections of a function <span>\\(f\\in L^p(\\textbf{T})\\)</span>. The sharpness of the constant <span>\\(A_{p,b}\\)</span> follows by taking a family quasiconformal harmonic mapping <span>\\(f_c\\)</span> and letting <span>\\(c\\rightarrow 1/p\\)</span>. The result extends a sharp version of M. Riesz conjugate function theorem of Pichorides and Verbitsky and some well-known estimates for holomorphic functions.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"50 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Potential Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11118-024-10150-8","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let \(L^p(\textbf{T})\) be the Lesbegue space of complex-valued functions defined in the unit circle \(\textbf{T}=\{z: |z|=1\}\subseteq \mathbb {C}\). In this paper, we address the problem of finding the best constant in the inequality of the form:
Here \(p\in [1,2]\), \(b>0\), and by \(P_{-} f\) and \( P_+ f\) are denoted the co-analytic and analytic projections of a function \(f\in L^p(\textbf{T})\). The sharpness of the constant \(A_{p,b}\) follows by taking a family quasiconformal harmonic mapping \(f_c\) and letting \(c\rightarrow 1/p\). The result extends a sharp version of M. Riesz conjugate function theorem of Pichorides and Verbitsky and some well-known estimates for holomorphic functions.
期刊介绍:
The journal publishes original papers dealing with potential theory and its applications, probability theory, geometry and functional analysis and in particular estimations of the solutions of elliptic and parabolic equations; analysis of semi-groups, resolvent kernels, harmonic spaces and Dirichlet forms; Markov processes, Markov kernels, stochastic differential equations, diffusion processes and Levy processes; analysis of diffusions, heat kernels and resolvent kernels on fractals; infinite dimensional analysis, Gaussian analysis, analysis of infinite particle systems, of interacting particle systems, of Gibbs measures, of path and loop spaces; connections with global geometry, linear and non-linear analysis on Riemannian manifolds, Lie groups, graphs, and other geometric structures; non-linear or semilinear generalizations of elliptic or parabolic equations and operators; harmonic analysis, ergodic theory, dynamical systems; boundary value problems, Martin boundaries, Poisson boundaries, etc.