Korenblum's principle for Bergman spaces with radial weights

TL;DR

This paper extends the Korenblum maximum principle to weighted Bergman spaces with radial weights for p≥1, and identifies conditions under which it fails for p<1, revealing limitations based on weight behavior.

Contribution

It demonstrates the validity of the Korenblum principle for weighted Bergman spaces with arbitrary radial weights when p≥1 and explores its failure for p<1 under certain conditions.

Findings

Korenblum principle holds for p≥1 in weighted Bergman spaces with radial weights.

Supremum radius for the principle is always less than one.

The principle fails for 0<p<1 if the weight satisfies a mild positivity condition.

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 $0<p<1$.