Korenblum's principle for Bergman spaces with radial weights

TL;DR

The paper addresses the Korenblum domination principle for Bergman spaces with radial weights by extending known results from unweighted spaces to weighted spaces $A^p_w$ with nonnegative radial $w$. For $p\ge 1$, the principle holds in $A^p_w$, and the corresponding Korenblum radius satisfies $c(p)<1$, with the supremum of admissible radii strictly less than 1. Under the additional assumption $\liminf_{r\to0^+} w(r)>0$, the principle fails for $0<p<1$, while without this assumption the behavior can vary (including potential hold under special constructions). The approach leverages classical integral-mean estimates and subharmonicity arguments, mirroring and generalizing prior work by Hayman, Hinkkanen, Schuster, BK, and Ka to the radial-weighted setting, and highlighting the robustness of the Korenblum phenomenon beyond standard unweighted spaces.

Abstract

We show that the Korenblum maximum (domination) principle is valid for weighted Bergman spaces $A^p_w$ with arbitrary (non-negative and integrable) radial weights $w$ in the case $1\le p<\infty$. We also notice that in every weighted Bergman space the supremum of all radii for which the principle holds is strictly smaller than one. Under the mild additional assumption $\liminf_{r\to 0^+} w(r)>0$, we show that the principle fails whenever $0<p<1$.

Theorems & Definitions (3)

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