On the \(T(1)\)-theorem for the Cauchy integral
type: Item
instance of: scholarly article
title: On the \(T(1)\)-theorem for the Cauchy integral
zbMATH Open document ID: 1039.42011
DOI: 10.1007/BF02384497
author: J. Verdera
published in: Arkiv för Matematik
publication date: 2004-03-15T00:00:00Z
review text: The author defines a weaker growth condition for the measure \(\mu\) than the standard doubling condition \(\mu (2D)\leq const\times\mu (D)\) for each disc \(D\), and gives an alternative proof, with respect to two recent proofs, of the \(L^2(\mu )\)-boundedness of the Cauchy integral operator, i.e., by the integral \(\int_{C}^{}| f| ^2d\mu\) (\(T(1)-\)theorem). Then he obtains the estimate from below of the analytic capacity of a compact subset contained in the contour \(C\) in terms of total Menger curvature. He shows that if the doubling condition fails then \(L^2\)-boundedness does not imply the \(L^\infty -\text{BMO}\) estimate, \(\text{BMO}(\mu )\) being the space of locally integrable functions with respect to \(\mu\).
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 2057365
zbMATH Keywords: Cauchy integral
zbMATH Keywords: \(T(1)\) theorem
zbMATH Keywords: analytic capacity
zbMATH Keywords: \(L^2(\mu)\)-boundedness
zbMATH Keywords: doubling condition
reviewed by: Jacek Gilewicz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2057365