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Sharp results on the integrability of the derivative of an interpolating Blaschke product.
- Source :
-
Forum Mathematicum . 2008, Vol. 20 Issue 6, p1039-1054. 16p. - Publication Year :
- 2008
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *BLASCHKE products
*COMPLEX variables
*MATHEMATICAL sequences
*MATHEMATICS
*ALGEBRA
Subjects
Details
- Language :
- English
- ISSN :
- 09337741
- Volume :
- 20
- Issue :
- 6
- Database :
- Academic Search Index
- Journal :
- Forum Mathematicum
- Publication Type :
- Academic Journal
- Accession number :
- 35680079
- Full Text :
- https://doi.org/10.1515/FORUM.2008.046