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

 

On unboundedness of maximal operators for directional Hilbert transforms


Author: G. A. Karagulyan
Journal: Proc. Amer. Math. Soc. 135 (2007), 3133-3141
MSC (2000): Primary 42B25, 42B20
Published electronically: June 19, 2007
MathSciNet review: 2322743
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that for any infinite set of unit vectors $ U$ in $ \mathbb{R}^2$ the maximal operator defined by

$\displaystyle H_Uf(x)=\sup_{u\in U}\bigg\vert{pv}\int_{-\infty }^\infty \frac{f(x-tu)}{t}dt\bigg\vert,\quad x\in \mathbb{R}^2, $

is not bounded in $ L^2(\mathbb{R}^2)$.


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Additional Information

G. A. Karagulyan
Affiliation: Institute of Mathematics, Armenian National Academy of Sciences, Marshal Baghramian ave. 24b, Yerevan, 375019, Armenia
Address at time of publication: Department of Applied Mathematics, Yerevan State University, Yerevan, Armenia
Email: karagul@instmath.sci.am

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08731-X
PII: S 0002-9939(07)08731-X
Keywords: Hilbert transform, maximal function
Received by editor(s): February 21, 2006
Published electronically: June 19, 2007
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.