$l_{p}$-norms of Fourier coefficients of powers of a Blaschke factor

Abstract

We determine the asymptotic behavior of the $l_{p}$-norms of the sequence of Taylor coefficients of $b^{n}$, where $b=\frac{z-\lambda}{1-\bar{\lambda}z}$ is an automorphism of the unit disk, $p\in[1,\infty]$, and $n$ is large. It is known that in the parameter range $p\in[1,2]$ a sharp upper bound \begin{align*} \left|\!\left|b^{n}\right|\!\right|_{l_{p}^A}\leq C_{p}n^{\frac{2-p}{2p}} \end{align*} holds. In this article we find that this estimate is valid even when $p\in[1,4)$. We prove that \begin{align*} \left|\!\left|b^{n}\right|\!\right|_{l_{4}^A}\leq C_{4}\left(\frac{\log n}{n}\right)^{\frac{1}{4}} \end{align*} and for $p\in(4,\infty]$ that \begin{align*} \left|\!\left|b^{n}\right|\!\right|_{l_{p}^A}\leq C_{p}n^{\frac{1-p}{3p}} & . \end{align*} We prove that our upper bounds are sharp as $n$ tends to $\infty$ i.e. they have the correct asymptotic $n$ dependence.