The WAT conjecture on the torus
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- by Bassam Shayya PDF
- Proc. Amer. Math. Soc. 139 (2011), 3633-3643 Request permission
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 $\| \phi \|_{L^\infty (\mathbb U)} \leq 1$ 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.References
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Additional Information
- Bassam Shayya
- Affiliation: Department of Mathematics, American University of Beirut, Beirut, Lebanon
- Email: bshayya@aub.edu.lb
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 3633-3643
- MSC (2010): Primary 42B05; Secondary 47B33
- DOI: https://doi.org/10.1090/S0002-9939-2011-10867-0
- MathSciNet review: 2813393