Best constants for the Riesz projection
type: Item
instance of: scholarly article
title: Best constants for the Riesz projection
zbMATH Open document ID: 0963.42006
author: Brian Hollenbeck
author: Igor E. Verbitsky
published in: Journal of Functional Analysis
publication date: 2001-06-22T00:00:00Z
review text: Recall that for \(f\in L^{p}(T)\) with Fourier expansion \(f(\zeta)=\sum_{k} \widehat f(k)\zeta^{k}\) the Riesz projection onto the Hardy space \(H^{p}(T)\) is defined by \(P_{+}f(\zeta)=\sum_{k\geq 0} \widehat f(k)\zeta^{k}\), and the co-analytic projection by \(P_{-}=I-P_{+}\). The estimate \(\|\max(|P_{+}f|,|P_{-}f|)\|_{p}\leq c_{p}\|f\|_{p}\) \((1<p\leq 2)\) is proved where \(c_{p}=(\sin(\pi/p))^{-1}\). This implies \(\|P_{+}\|= c_{p}\) \((1<p<\infty)\). Analogous results for the analytic projection on the real line (half-line Fourier multiplier) and in Euclidean space and for weighted \(L^{p}\) spaces with weight function \(|\zeta-\zeta_0|^\alpha\) are obtained. For the proof, the plurisubharmonic estimate \(\max(|w|^{p}, |z|^{p})\leq c_{p}^{p} |w+\overline z|^{p}-b_{p} \text{Re}[(wz)^{p/2}]\) is shown.
Mathematics Subject Classification ID: 42A45
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42A50
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 31C10
zbMATH DE Number: 1525188
zbMATH Keywords: Riesz projection
zbMATH Keywords: operator norm
zbMATH Keywords: best constant
zbMATH Keywords: plurisubharmonic function
zbMATH Keywords: Fourier multiplier
MaRDI profile type: Publication
full work available at URL: jfan.2000.3616
OpenAlex ID: W1972753531
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DOI: 10.1006/JFAN.2000.3616
description: scientific article; zbMATH DE number 1525188