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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 the sweeping out property for convolution operators of discrete measures


Author: G. A. Karagulyan
Journal: Proc. Amer. Math. Soc. 139 (2011), 2543-2552
MSC (2000): Primary 42B25
Published electronically: December 22, 2010
MathSciNet review: 2784819
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Abstract: Let $ \mu_n$ be a sequence of discrete measures on the unit circle $ \mathbb{T}=\mathbb{R}/\mathbb{Z}$ with $ \mu_n(0)=0$, and $ \mu_n((-\delta,\delta))\to 1$, as $ n\to\infty$. We prove that the sequence of convolution operators $ (f\ast\mu_n)(x)$ is strong sweeping out; i.e., there exists a set $ E\subset\mathbb{T}$ such that

$\displaystyle \limsup\limits_{n\to\infty} (\mathbb{I}_E\ast\mu_n)(x)= 1,\quad \liminf\limits_{n\to\infty}(\mathbb{I}_E\ast\mu_n)(x)= 0, $

almost everywhere on $ \mathbb{T}$.


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Additional Information

G. A. Karagulyan
Affiliation: Institute of Mathematics of Armenian National Academy of Sciences, Baghramian Avenue 24b, 0019, Yerevan, Armenia
Email: g.karagulyan@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10829-8
PII: S 0002-9939(2010)10829-8
Keywords: Discrete measures, bounded entropy theorem, sweeping out property, Bellow problem
Received by editor(s): July 11, 2010
Published electronically: December 22, 2010
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.