On the Number of Zeros of Functions in
Analytic Quasianalytic Classes

Sasha Sodin

School of Mathematical Sciences, Queen Mary University of London, London E1 4NS,United Kingdom School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel.
E-mail: a.sodin@qmul.ac.uk

Received February 16, 2019, revised June 3, 2019.

Abstract

A space of analytic functions in the unit disc with uniformly continuous
derivatives is said to be quasianalytic if the boundary value of a non-zero
function from the class can not have a zero of infinite multiplicity. Such
classes were described in the 1950-s and 1960-s by Carleson, Rodrigues-
Salinas and Korenblum. A non-zero function from a quasianalytic space
of analytic functions can only have a finite number of zeros in the closed
disc. Recently, Borichev, Frank, and Volberg proved an explicit estimate on
the number of zeros for the case of quasianalytic Gevrey classes. Here, an
estimate of similar form for general analytic quasianalytic classes is proved
using a reduction to the classical quasianalyticity problem.

Mathematics Subject Classification 2000: 26E10, 30D60, 30H99. Key words: quasianalytic class, analytic quasianalyticity, number of zeros.