Lp-norms of Fourier coefficients of powers of a Blaschke factor

Abstract

We determine the asymptotic behavior of the lp-norms of the sequence of Taylor coefficients of bn, where \(\begin{array}{*{20}{c}} {b(z) = \frac{{z - \lambda }}{{1 - \lambda z}},}&{\left| \lambda \right| < 1,} \end{array}\) is an automorphism of the unit disk, p ∈ [1,∞], and n is large. It is known that in the parameter range p ∈ [1, 2] a sharp upper bound \({\left\| {{b^n}} \right\|_{l_p^A}} \leqslant {c_p}{n^{\tfrac{{2 - p}}{{2p}}}}\) holds. In this article we find that this estimate is valid even when p ∈ [1, 4). We prove that \({\left\| {{b^n}} \right\|_{l_4^A}} \leqslant {c_4}{\left( {\frac{{\log n}}{n}} \right)^{\tfrac{1}{4}}}\) and for p ∈ (4,∞] that \({\left\| {{b^n}} \right\|_{l_p^A}} \leqslant {c_p}{n^{\tfrac{{1 - p}}{{3p}}}}.\) The upper bounds are shown to be asymptotically sharp as n tends to ∞. As a direct application we prove the sharpness of existing upper estimates on analytic capacities in Beurling–Sobolev spaces. Our investigation is also motivated by a question of J. J. Schäffer about optimal estimates for norms of inverse matrices.