A necessary condition for Calderón-Zygmund singular integral operators
type: Item
instance of: scholarly article
title: A necessary condition for Calderón-Zygmund singular integral operators
zbMATH Open document ID: 0979.42004
DOI: 10.1007/BF01259372
author: James E. Daly
published in: The Journal of Fourier Analysis and Applications
publication date: 2002-02-05T00:00:00Z
full work available at URL: https://eudml.org/doc/59607
review text: The author treats the following Calderón-Zygmund singular integral operator \[ T_\Omega (f)(x)=\text{p.v. }\int_{\mathbb R^n}f(x-y)\Omega(y/|y|)|y|^{-n} dy, \] where \(\Omega\) is an integrable function with mean value zero on the unit sphere \(S^{n-1}\). In 1977, W. Connett showed that if \(\Omega\) is in the Hardy space \(H^1(S^{n-1})\), then \(T_\Omega\) is \(L^p(\mathbb R^n)\) bounded for \(1<p<\infty\). The author gives the following partial reverse of this: if \(T_\Omega\) is bounded on the Hardy space \(H^1(\mathbb R^n)\), then the restriction of \(\Omega\) to \(S^{n-1}\) is a function in \(H^1(S^{n-1})\). To prove this, he uses the F. and M. Riesz theorem on \(S^{n-1}\) and spherical harmonics expansion.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 1340857
zbMATH Keywords: Calderón-Zygmund singular integral
zbMATH Keywords: Hardy spaces
zbMATH Keywords: multipliers
reviewed by: Kôzô Yabuta
MaRDI profile type: Publication
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OpenAlex ID: W2086758415
description: scientific article; zbMATH DE number 1340857