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

Sharp results on the integrability of the derivative of an interpolating Blaschke product The Schwarz-Pick lemma readily implies that the derivative of any Blaschke product belongs to all the Bergman spaces A p 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 A 1 . However, the question of whether there exists an interpolating Blaschke product B with B ′ ∉ A 1 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 ′ ∈ H p if and only if (1 – | a k |) 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 – | a k |) 1/2 < ∞ but B ′ ∉ H 1/2 , and (b) an interpolating Blaschke products B whose sequence of zeros of satisfies (1 – | a k |) 1– p < ∞, for all p ∈ (0, 1/2), whose derivative B ′ does not belong to the Nevanlinna class. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

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

Peláez, José Ángel
Forum Mathematicum , Volume 20 (6) – Nov 1, 2008
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References (25)

Publisher
de Gruyter
Copyright
© de Gruyter 2008
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/FORUM.2008.046
Publisher site
See Article on Publisher Site

Abstract

The Schwarz-Pick lemma readily implies that the derivative of any Blaschke product belongs to all the Bergman spaces A p 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 A 1 . However, the question of whether there exists an interpolating Blaschke product B with B ′ ∉ A 1 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 ′ ∈ H p if and only if (1 – | a k |) 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 – | a k |) 1/2 < ∞ but B ′ ∉ H 1/2 , and (b) an interpolating Blaschke products B whose sequence of zeros of satisfies (1 – | a k |) 1– p < ∞, for all p ∈ (0, 1/2), whose derivative B ′ does not belong to the Nevanlinna class.

Journal

Forum Mathematicum – de Gruyter

Published: Nov 1, 2008

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