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Clark measures on the torus


Author: Evgueni Doubtsov
Journal: Proc. Amer. Math. Soc. 148 (2020), 2009-2017
MSC (2010): Primary 30J05, 32A35; Secondary 31C10, 46E27, 46J15
DOI: https://doi.org/10.1090/proc/14846
Published electronically: December 30, 2019
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Abstract: Let $ \mathbb{D}$ denote the unit disc of $ \mathbb{C}$ and let $ \mathbb{T}= \partial \mathbb{D}$. Given a holomorphic function $ \varphi : \mathbb{D}^n \to \mathbb{D}$, $ n\ge 2$, we study the corresponding family $ \sigma _\alpha [\varphi ]$, $ \alpha \in \mathbb{T}$, of Clark measures on the torus $ \mathbb{T}^n$. If $ \varphi $ is an inner function, then we introduce and investigate related isometric operators $ T_\alpha $ mapping analogs of model spaces into $ L^2(\sigma _\alpha )$, $ \alpha \in \mathbb{T}$.


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Additional Information

Evgueni Doubtsov
Affiliation: Department of Mathematics and Computer Science, St. Petersburg State University, Line 14th (Vasilyevsky Island), 29, St. Petersburg 199178, Russia; and St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
Email: dubtsov@pdmi.ras.ru

DOI: https://doi.org/10.1090/proc/14846
Keywords: Hardy space, polydisc, inner function, model space, pluriharmonic measure
Received by editor(s): June 15, 2019
Received by editor(s) in revised form: September 4, 2019
Published electronically: December 30, 2019
Additional Notes: This research was supported by the Russian Science Foundation (grant No. 19-11-00058).
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2019 American Mathematical Society