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Gaussian analytic functions of bounded mean oscillation

Alon Nishry and Elliot Paquette

Vol. 16 (2023), No. 1, 89–117

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.

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

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