On Taylor coefficients and $L_p$-continuity moduli of Blaschke products

I. É. Verbitskii

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part X, Tome 107 (1982), pp. 27-35

Résumé

Let $$ B=\prod_{k\geq1}b_{z_k},\quad b_{z_k}\overset{\text{def}}=\frac{|z_k|}{z_k} \frac{z_k-z}{1-\bar z_k z},\quad |z_k|<1, $$ be a Blaschke product, let $\widehat{B}(z)$ denote its $k$-th Taylor coefficient. Suppose $\{z_k\}$ splits into a finite union of sequences $\{\xi_k\}$ satisfying $$ \sup_{k\geq1}\frac{1-|\xi_{k+1}|}{1-|\xi_{k}|}<1. $$ The following assertions are proved to be equivalent: 1. $\{z_k\}_{k\geq1}\in(\omega N)$; 2. $\widehat{B}(k)=O(1/k)$, $k\to\infty$; 3. $\sum_{k\geq n}|\widehat{B}|^2=O(n^{-1})$; 4. $B\in\operatorname{Lip}(1/p,L^p)$ for some $p$, $1<p<\infty$.