The WAT conjecture on the torus

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.