Norms and essential norms of the singular integral operator with Cauchy kernel on weighted Lebesgue spaces
type: Item
instance of: scholarly article
title: Norms and essential norms of the singular integral operator with Cauchy kernel on weighted Lebesgue spaces
zbMATH Open document ID: 1211.45010
author: Takahiko Nakazi
author: Takanori Yamamoto
published in: Integral Equations and Operator Theory
publication date: 2010-10-29T00:00:00Z
review text: Let \(\mathbb T\) be the unit circle, \(W\) some weight function on \(\mathbb T\), \(L^2(W)\) the Lebesgue space on \(\mathbb T\) with norm \(\left\| f\right\|_{W}^2= \int_{\mathbb T}| f|^2W\,dm\), and \[ Sf(\zeta)=\frac1{\pi i}\int_{\mathbb T}\frac{f(\eta)}{\eta-\zeta}d\eta \] be the singular integral operator on \(L^2(W)\), understood in the sense of Cauchy's principal value. The paper deals with calculation/estimates for the norm and essential norm of the operator \(S_{\alpha,\beta}f=\alpha Pf+\beta Qf\) in \(L^2(W)\) where \(P=(I+S)/2\) and \(Q=(I-S)/2\) are the analytic and co-analytic projections, respectively, and \(\alpha,\beta\in L^\infty\). Recall that the essential norm of \(S_{\alpha,\beta}\) is the infimum of the norms of all compact (linear) perturbations of \(S_{\alpha,\beta}\). For the case that \(\alpha\) and \(\beta\) are constant, explicit formulas for the norm and essential norm of \(S_{\alpha,\beta}\) had been obtained by \textit{I. Gohberg} and \textit{N. Krupnik} [One-dimensional linear singular integral equations. II. Basel: Birkhäuser (1992; Zbl 0781.47038)]. Morever, for the case \(W=1\), explicit formulas for the norm and essential norm of \(S_{\alpha,\beta}\) had been obtained by \textit{T. Nakazi} and \textit{T. Yamamoto} [J. Oper. Theory 40, 185--207 (1998; Zbl 0992.45009)] and by \textit{T. Nakazi} [Arch. Math. 73, 439--441 (1999; Zbl 0948.45006)], respectively. It is shown that -- up to at most constant factors for which explicit estimates are given -- the norm and essential norm of \(S_{\alpha,\beta}\) in \(L^2(W)\) relates to the norm and essential norm of \(S_{\varphi\alpha,\beta}\) in \(L^2(1)\), respectively, if \(W=| h|^2\) and \(\varphi=\overline h/h\) with an outer function \(h\).
Mathematics Subject Classification ID: 45P05
Mathematics Subject Classification ID: 45E05
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 47A30
zbMATH DE Number: 5808419
zbMATH Keywords: singular integral operator
zbMATH Keywords: weighted Lebesgue space
zbMATH Keywords: operator norm
zbMATH Keywords: essential norm
zbMATH Keywords: \(A_2\)-weight
MaRDI profile type: Publication
full work available at URL: s00020-010-1792-9
OpenAlex ID: W2083728386
cites work: Q3894416
cites work: Q4023123
cites work: Weighted Norm Inequalities for the Conjugate Function and Hilbert Transform
cites work: Essential norms of some singular integral operators
cites work: Q4529318
cites work: Q4814898
DOI: 10.1007/S00020-010-1792-9
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description: scientific article; zbMATH DE number 5808419