On the bilinear Hilbert transform
type: Item
instance of: scholarly article
title: On the bilinear Hilbert transform
zbMATH Open document ID: 0963.42007
author: Michael T. Lacey
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/228249
full work available at URL: Lacey.MAN.html
review text: The author and \textit{C. Thiele} in their highly acclaimed paper ``On Calderón's conjecture'' [Ann. Math. (2) 149, No. 2, 475-496 (1999; Zbl 0934.42012)] have proved that the bilinear Hilbert transform \[ Hfg(x):= \lim_{\varepsilon \rightarrow 0}\int_{|y|>\varepsilon}{f(x+y)g(x-y) {\frac{dy}{y}}} \] extends to a bounded operator on \(L^p\times L^p\) into \(L^r\) if \(\frac{1}{p}+\frac{1}{q}=\frac{1}{r}\), \(1<p,q\leq{\infty}\) and \(\frac{2}{3}<r< \infty\). This proves, in particular, a conjecture of Calderón from 1964 that \(H\) maps \(L^2 \times L^2\) into \(L^1\). The current highly readable article illustrates the method of proof of the general result by giving a complete proof of the weaker result that the bilinear Hilbert transform maps \(L^2 \times L^2\) into weak \(L^1\).
Mathematics Subject Classification ID: 42A50
zbMATH DE Number: 1184299
zbMATH Keywords: Hilbert transform
zbMATH Keywords: time-frequency analysis
zbMATH Keywords: Calderón conjecture
reviewed by: B. P. Duggal
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1184299