SEMINARIO DE ANÁLISIS COMPLEJO (COMPLEX ANALYSIS SEMINAR)

On the Korenblum-Hayman domination principle for
weighted Bergman spaces

Dragan VUKOTI´C
UAM

Wednesday, February 26, 2025 at 15:00
Room 520, Module 17, Department of Mathematics,
Universidad Autónoma de Madrid

Abstract:
The domination (or maximum) principle for Bergman spaces states that there exists
a fixed annulus {z : R < |z| < 1} such that, whenever an analytic function in a Bergman
space Ap has larger modulus than another such function on that annulus, its norm in
the space is larger than that of the second function. This principle was conjectured in
the late 1980s by Korenblum and was proved in 1999 by Hayman for A2 and, shortly
thereafter, by Hinkkanen for 1 ≤ p < ∞. In 2018, Boˇzin and Karapetrovi´c showed
that Korenblum’s principle fails for all p with 0 < p < 1 by exhibiting a universal
counterexample.
In this joint work with Iason Efraimidis and Adri´an Llinares, we show that the
domination principle continues to be valid for all weighted Bergman spaces Ap
w with
arbitrary (non-negative and integrable) radial weights w in the case 1 ≤ p <∞. Under
the mild additional assumption liminfr→0+ w(r ) > 0, we show that the principle fails
whenever 0 < p < 1.
Some basic details of the theory will be reviewed and several open problems will be
stated in the talk.