Pluriharmonic Clark measures and analogs of model spaces

Aleksandrov, Aleksei B.; Doubtsov, Evgueni

doi:10.1016/j.crma.2018.11.013

Complex analysis/Functional analysis

Pluriharmonic Clark measures and analogs of model spaces
[Mesures de Clark pluriharmoniques et analogues des espaces modèles]

Présenté par : the Editorial Board

Aleksandrov, Aleksei B.^1,² ; Doubtsov, Evgueni ¹

¹ St. Petersburg Department of V.A. Steklov Institute of Mathematics, Fontanka 27, St. Petersburg 191023, Russia
² Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr. 28, St. Petersburg, 198504, Russia

Comptes Rendus. Mathématique, Tome 357 (2019) no. 1, pp. 7-12.

Résumé
Abstract

Soit $B_{d}$ la boule unité de $C^{d}$ , $d \geq 1$ . Étant donnée une fonction intérieure $I : B_{d} \to B_{1}$ , nous étudions la famille correspondante $σ_{α} [I]$ , $α \in \partial B_{1}$ , de mesures de Clark pluriharmoniques sur la sphère complexe. Nous introduisons et étudions les opérateurs unitaires $U_{α}$ entre des analogues des espaces modèles et $L^{2} (σ_{α})$ , $α \in \partial B_{1}$ . En particulier, nous caractérisons explicitement l'ensemble des $U_{α}^{⁎} f$ telles que $f σ_{α}$ soit une mesure pluriharmonique.

Let $B_{d}$ denote the unit ball of $C^{d}$ , $d \geq 1$ . Given an inner function $I : B_{d} \to B_{1}$ , we study the corresponding family $σ_{α} [I]$ , $α \in \partial B_{1}$ , of pluriharmonic Clark measures on the complex sphere. We introduce and investigate related unitary operators $U_{α}$ mapping analogs of model spaces onto $L^{2} (σ_{α})$ , $α \in \partial B_{1}$ . In particular, we explicitly characterize the set of $U_{α}^{⁎} f$ such that $f σ_{α}$ is a pluriharmonic measure.

Reçu le : 2018-04-25
Accepté le : 2018-11-23
Publié le : 2018-12-14

DOI : 10.1016/j.crma.2018.11.013

Affiliations des auteurs :

Aleksandrov, Aleksei B. ^{1, 2} ; Doubtsov, Evgueni ¹

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Aleksandrov, Aleksei B.; Doubtsov, Evgueni. Pluriharmonic Clark measures and analogs of model spaces. Comptes Rendus. Mathématique, Tome 357 (2019) no. 1, pp. 7-12. doi : 10.1016/j.crma.2018.11.013. http://www.numdam.org/articles/10.1016/j.crma.2018.11.013/

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Cité par Sources :

^☆ This research was supported by the Russian Science Foundation (grant No. 18-11-00053).