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


The WAT conjecture on the torus

Author: Bassam Shayya
Journal: Proc. Amer. Math. Soc. 139 (2011), 3633-3643
MSC (2010): Primary 42B05; Secondary 47B33
Published electronically: February 24, 2011
MathSciNet review: 2813393
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Abstract: Let $ \phi$ be a bounded holomorphic function on the unit disc $ \mathbb{U} \subset \mathbb{C}$, let $ k$ be an integer, and define the sequence $ \{ c_n \}$ by $ c_n= \int_{\partial \mathbb{U}} \phi(z)^n z^{k-n} dz$. Nazarov and Shapiro conjectured that if $ \Vert \phi \Vert _{L^\infty(\mathbb{U})} \leq 1$ and $ \phi$ is not a rotation, then $ \lim_{n \to \infty} c_n = 0$. A consequence of this conjecture would be that any composition operator is weakly asymptotically Toeplitz on the Hardy space $ H^2(\mathbb{U})$. We formulate a higher-dimensional version of this conjecture and use Fourier analytic techniques to obtain results that improve on what is currently known in dimension one.

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Bassam Shayya
Affiliation: Department of Mathematics, American University of Beirut, Beirut, Lebanon

PII: S 0002-9939(2011)10867-0
Received by editor(s): June 21, 2010
Received by editor(s) in revised form: September 1, 2010
Published electronically: February 24, 2011
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.