Sharp results on the integrability of the derivative of an interpolating Blaschke product

Abstract

The Schwarz-Pick lemma readily implies that the derivative of any Blaschke product belongs to all the Bergman spaces Ap with 0 < p < 1. It is also well known that this result is sharp: there exist a Blaschke product whose derivative does not belong to A1. However, the question of whether there exists an interpolating Blaschke product B with B′ ∉ A1 remained open. In this paper we give an explicit construction of such an interpolating Blaschke product B.