On Korenblum's density condition for the zero sequences of \(A^{-\alpha}\)
This paper discusses some fine points concerning a notion of density introduced by \textit{B. Korenblum} [Acta Math. 135, 187-219 (1975; Zbl 0323.30030)], which in a certain sense generalizes the classical Blaschke condition. It was introduced to obtain precise geometric information about the zeros of functions in \(A^{-\alpha}\), \(\alpha> 0\), i.e., the set of functions \(f\) analytic in the unit disk \(\mathbb{D}= \{z\in \mathbb{C}: |z|\leq 1\}\) satisfying \(\sup_{z\in \mathbb{D}} |f(z)|(1- |z|)^\alpha< \infty\). In a recent paper [Ark. Mat. 32, 237-243 (1994; Zbl 0812.30003)], the author showed that Korenblum's construction provides an expression for the infimum of those \(\alpha\) for which a given sequence constitutes the zeros of a function in \(A^{- \alpha}\). In the present paper, a more precise result is proved, revealing the right order of magnitude of the ``gap'' between the optimal necessary and sufficient conditions. Roughly, it is as follows. A potential zero sequence \(\Lambda\) is given. Take a finite set \(F\) on the unit circle, and compute the Blaschke sum of those points from \(\Lambda\) which lie inside all the Stoltz angles associated with the points of \(F\). A sharp sufficient condition is that this Blaschke sum be bounded by \(\alpha\) times the so-called Carleson characteristic of \(F\), plus a constant independent of \(F\). In the sharp necessary condition, one has to add a constant times the log of the Carleson characteristic; here the best constant is not known. An important auxiliary result is the construction of a large class of analytic functions with growth prescribed by certain well-behaved subharmonic functions; this construction is based on a decomposition of the Riesz measure of the subharmonic function and builds on a technique developed by M. Sodin and Yu. I. Lyubarskii. This auxiliary result is also used to show some delicate relations between the zero sequences of various weighted Bergman spaces, improving in an essential way results by \textit{Ch. Horowitz} [Duke Math. J. 41, 693-710 (1974; Zbl 0293.30035)].
scientific article; zbMATH DE number 871567
scientific article