On unboundedness of maximal operators for directional Hilbert transforms
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- by G. A. Karagulyan PDF
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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 \begin{equation*} H_Uf(x)=\sup _{u\in U}\bigg |\operatorname {pv}\int _{-\infty }^\infty \frac {f(x-tu)}{t}dt\bigg |,\quad x\in \mathbb {R}^2, \end{equation*} is not bounded in $L^2(\mathbb {R}^2)$.References
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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
- Received by editor(s): February 21, 2006
- Published electronically: June 19, 2007
- Communicated by: Michael T. Lacey
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 3133-3141
- MSC (2000): Primary 42B25, 42B20
- DOI: https://doi.org/10.1090/S0002-9939-07-08731-X
- MathSciNet review: 2322743