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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the smoothness of eigenfunctions of hyponormal singular integral operators


Author: Kevin Clancey
Journal: Proc. Amer. Math. Soc. 31 (1972), 475-479
DOI: https://doi.org/10.1090/S0002-9939-1972-0284873-4
MathSciNet review: 0284873
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Abstract: Fix $ \varphi \in {L^\infty }(E)$ and let $ E \subset R$ be bounded and measurable; for $ 1 < p < \infty $ consider the bounded linear operator

$\displaystyle Tf(s) = sf(s) + \frac{{\varphi (s)}}{\pi }\int _E^\ast\frac{{\bar... ...rphi (t)f(t)}}{{t - s}}dt\;{\mkern 1mu} {\text{a}}{\text{.e}}{\text{. }}s \in E$

where $ f \in {L^p}(E)$. If $ \nu = \lambda + i\mu \in C$ then there are no nonzero $ {L^p}(E)$ solutions of $ Tf = \nu f$ for $ p > 2$ in case $ \lambda $ is a point of positive Lebesgue density in the complement of $ E$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0284873-4
Keywords: Singular integral, hyponormal operator
Article copyright: © Copyright 1972 American Mathematical Society