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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Cauchy transform on bounded domains


Authors: J. M. Anderson and A. Hinkkanen
Journal: Proc. Amer. Math. Soc. 107 (1989), 179-185
MSC: Primary 30E20; Secondary 47G05
MathSciNet review: 972226
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Abstract: Suppose that $ f$ is in $ {L^2}(\Delta )$ where $ \Delta $ is the unit disk, and that $ f = 0$ outside $ \Delta $. We show that then the Cauchy transform $ \mathcal{C}\,f$ of $ f$, when restricted to $ \Delta $, satisfies $ \vert\vert\mathcal{C}\,f\vert{\vert _2} \leq (2/\alpha )\vert\vert f\vert{\vert _2}$, where $ \alpha \approx 2.4048$ is the smallest positive zero of the Bessel function $ {J_0}$. This inequality is sharp.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0972226-5
PII: S 0002-9939(1989)0972226-5
Article copyright: © Copyright 1989 American Mathematical Society