Sharp results on the integrability of the derivative of an interpolating Blaschke product
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José Ángel Peláez
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.
A result of W. S. Cohn asserts that if 
and B is an interpolating Blaschke product with sequence of zeros of 
, then B′ ∈ Hp if and only if 
(1 – |ak|)1–p < ∞. We prove that Cohn's result is no longer true for 
. Indeed, we construct: (a) an interpolating Blaschke product B whose sequence of zeros of satisfies (1 – |ak|)1/2 < ∞ but B′ ∉ H1/2, and (b) an interpolating Blaschke products B whose sequence of zeros of satisfies (1 – |ak|)1–p < ∞, for all p ∈ (0, 1/2), whose derivative B′ does not belong to the Nevanlinna class.
© de Gruyter 2008
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- Inequalities for Euler's gamma function
- Integers without divisors from a fixed arithmetic progression
- Sharp results on the integrability of the derivative of an interpolating Blaschke product
- The Gauss map of pseudo-algebraic minimal surfaces
- Gödel incompleteness in AF C*-algebras
- Quasi-regular Dirichlet forms on Riemannian path and loop spaces
- Divided differences and generalized Taylor series
- A regularity result for a class of degenerate Yang-Mills connections in critical dimensions
Articles in the same Issue
- Inequalities for Euler's gamma function
- Integers without divisors from a fixed arithmetic progression
- Sharp results on the integrability of the derivative of an interpolating Blaschke product
- The Gauss map of pseudo-algebraic minimal surfaces
- Gödel incompleteness in AF C*-algebras
- Quasi-regular Dirichlet forms on Riemannian path and loop spaces
- Divided differences and generalized Taylor series
- A regularity result for a class of degenerate Yang-Mills connections in critical dimensions
