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Gaussian analytic functions of bounded mean oscillation

Alon Nishry and Elliot Paquette

Vol. 16 (2023), No. 1, 89–117
Abstract

We consider random analytic functions given by a Taylor series with independent, centered complex Gaussian coefficients. We give a new sufficient condition for such a function to have bounded mean oscillation. Under a mild regularity assumption this condition is optimal. We give as a corollary a new bound for the norm of a random Gaussian Hankel matrix. Finally, we construct some exceptional Gaussian analytic functions which in particular disprove the conjecture that a random analytic function with bounded mean oscillation always has vanishing mean oscillation.

Keywords
function theory on the disc, bounded mean oscillation, Gaussian analytic functions, Bloch, probability
Mathematical Subject Classification 2010
Primary: 30B20
Secondary: 30H35, 47B80
Milestones
Received: 8 February 2020
Revised: 25 February 2021
Accepted: 4 May 2021
Published: 14 April 2023
Authors
Alon Nishry
School of Mathematical Sciences
Department of Pure Mathematics
Tel Aviv University
Tel Aviv
Israel
Elliot Paquette
Department of Mathematics
McGill University
Montreal
Canada

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