Best bounds for the Hilbert transform on \(L^ p(\mathbb{R}^ 1)\)
type: Item
instance of: scholarly article
title: Best bounds for the Hilbert transform on \(L^ p(\mathbb{R}^ 1)\)
zbMATH Open document ID: 0883.42007
author: Loukas Grafakos
published in: Mathematical Research Letters
publication date: 1997-11-04T00:00:00Z
review text: In this note the author gives a short proof (which avoids the use of conjugate function on the circle) of the following known best bounds result: The Hilbert transform \(H\), \[ Hf(x)=\text{p.v. }\Biggl({1\over\pi}\Biggr) \int_\mathbb{R} {f(x-t)\over t} dt, \] maps \(L^p(\mathbb{R})\) into itself with bound \(\tan(\pi/2p)\) for \(1<p\leq 2\) and \(\text{cot}(\pi/2p)\) for \(2\leq p<\infty\).
Mathematics Subject Classification ID: 42A50
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 1082974
zbMATH Keywords: \(L^ p\) spaces
zbMATH Keywords: subharmonic function
zbMATH Keywords: Hilbert transform
reviewed by: B. P. Duggal
MaRDI profile type: Publication
full work available at URL: mrl.1997.v4.n4.a3
OpenAlex ID: W2335044279
DOI: 10.4310/MRL.1997.V4.N4.A3
description: scientific article; zbMATH DE number 1082974