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On unboundedness of maximal operators for directional Hilbert transforms

Author(s): G. A. Karagulyan
Journal: Proc. Amer. Math. Soc. 135 (2007), 3133-3141.
MSC (2000): Primary 42B25, 42B20
Posted: June 19, 2007
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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: 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
Posted: June 19, 2007
Communicated by: Michael T. Lacey
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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