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Invertibility Threshold for Nevanlinna Quotient Algebras

Part of: Spaces and algebras of analytic functions Function theory on the disc

Published online by Cambridge University Press:  10 September 2021

Artur Nicolau  and
Pascal J. Thomas
Artur Nicolau*
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain
Pascal J. Thomas
Affiliation:
Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France e-mail: pascal.thomas@math.univ-toulouse.fr
*
e-mail: artur@mat.uab.cat
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Abstract

Let $\mathcal {N}$ be the Nevanlinna class, and let B be a Blaschke product. It is shown that the natural invertibility criterion in the quotient algebra $\mathcal {N} / B \mathcal {N}$ , that is, $|f| \ge e^{-H} $ on the set $B^{-1}\{0\}$ for some positive harmonic function H, holds if and only if the function $- \log |B|$ has a harmonic majorant on the set $\{z\in \mathbb {D}:\rho (z,\Lambda )\geq e^{-H(z)}\}$ , at least for large enough functions H. We also study the corresponding class of positive harmonic functions H on the unit disc such that the latter condition holds. We also discuss the analogous invertibility problem in quotients of the Smirnov class.

MSC classification

Primary: 30H15: Nevanlinna class and Smirnov class
Secondary: 30H80: Corona theorems 30J10: Blaschke products
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 1 , February 2023 , pp. 225 - 244
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

First author is supported by the Generalitat de Catalunya (grant 2017 SGR 395) and the Spanish Ministerio de Ciencia e Innovación (project MTM2017-85666-P).

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