Weighted norm inequalities for some singular integral operators
type: Item
instance of: scholarly article
title: Weighted norm inequalities for some singular integral operators
zbMATH Open document ID: 0905.47039
author: Takahiko Nakazi
author: Takanori Yamamoto
published in: Journal of Functional Analysis
publication date: 1997-10-01T00:00:00Z
full work available at URL: http://hdl.handle.net/2115/69097
review text: Let \[ Sf(\zeta)= {1\over\pi i} \int_T {f(z)\over z-\zeta} dz \] be the singular integral operator on the unit circle \(T\). For functions \(\alpha\) and \(\beta\) in \(L^\infty\) set \[ S_{\alpha,\beta}= {\alpha- \beta\over 2} S+ {\alpha+ \beta\over 2} I, \] where \(I\) is the identity operator. For a nonnegative function \(W\) in \(L^1\) let \[ \| f\|_W= \Biggl\{ \int_T| f|^2 W du\Biggr\}^{1/2}. \] The authors obtain conditions in order that the estimates \[ \| S_{\alpha, \beta}f\|_W\leq C\| f\|_W,\quad \| S_{\alpha, \beta}f\|_W\geq \delta\| f\|_W, \] hold for \(f\in A+\overline A_0\), where \(A\) is the disc algebra, that is, the algebra of all continuous functions in \(T\) whose negative Fourier coefficients vanish, and \(\overline A_0\) is the set of all complex conjugates of functions in \(A_0\), the subspace of all functions in \(A\) whose mean value is zero. For that purpose they introduce a class of weights generalizing that by Helson and Szegö. The authors also obtain a necessary and sufficient condition for the existence of a weight \(U\) such that the inequality \[ \| S_{\alpha, \beta}f\|_U\leq C\| f\|_W,\quad\text{or}\quad \| S_{\alpha, \beta}f\|_W\geq \delta\| f\|_U, \] holds for all \(f\in A+\overline A_0\).
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 1069295
zbMATH Keywords: singular integral operators
zbMATH Keywords: weighted norm inequality
zbMATH Keywords: unit circle
zbMATH Keywords: Helson-Szegö weights
zbMATH Keywords: inner function
zbMATH Keywords: disc algebra
reviewed by: Hiroki Tanabe
MaRDI profile type: Publication
OpenAlex ID: W2079620051
cites work: Q3894416
cites work: Q3922907
cites work: Q4023123
cites work: A problem in prediction theory
cites work: Weighted Norm Inequalities for the Conjugate Function and Hilbert Transform
cites work: Q3901955
cites work: Q4286345
cites work: Some singular integral operators and Helson-Szegö measures
cites work: Shorter Notes: Positive H 1/2 Functions are Constants
cites work: Q4150744
cites work: Invertibility of some singular integral operators in a lifting theorem
cites work: Q4865571
DOI: 10.1006/JFAN.1997.3100
description: scientific article; zbMATH DE number 1069295