The bilinear Hilbert transform is pointwise finite
type: Item
instance of: scholarly article
title: The bilinear Hilbert transform is pointwise finite
zbMATH Open document ID: 0894.42005
DOI: 10.4171/RMI/227
author: Michael T. Lacey
published in: Revista Matemática Iberoamericana
publication date: 1998-08-19T00:00:00Z
full work available at URL: https://eudml.org/doc/39536
review text: For the bilinear Hilbert transform \[ H(fg)(x)=\text{p.v. }\int^\infty_{-\infty} f(x+ y) g(x- y){dy\over y}, \] the author proved that if \(f\in L^\infty\) and \(g\in L^2\) are supported on \([0,1]\), then \[ \| H(fg)\|_{L^1(\mathbb{R})}\leq C\| f\|_{L^\infty(\mathbb{R})}\| g\|_{L^2(\mathbb{R})}. \] As a consequence, the bilinear Hilbert transform is pointwise finite for \(f\in L^\infty\) and \(g\in L^2\). The paper gives some concrete indication that the conjecture, by A. Calderón, that \(H\) maps \(L^2\times L^\infty\) into \(L^2\) could be true. A further development of the paper can be found in [\textit{M. Lacey} and \textit{C. Thiele}, Proc. Natl. Acad. Sci. USA 94, No. 1, 33-35 (1997)].
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 1136589
zbMATH Keywords: Calderón conjecture
zbMATH Keywords: Carleson measure
zbMATH Keywords: bilinear Hilbert transform
reviewed by: Dashan Fan
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1136589