On the norms of the multiplication operators between weighted Bergman spaces

TL;DR

This paper investigates the norms of multiplication operators between weighted Bergman spaces, providing new sharp estimates and exploring links to the Brennan conjecture and Schwarzian derivatives.

Contribution

It offers a novel sharp norm estimate for specific multiplication operators and connects these results to longstanding conjectures in complex analysis.

Findings

Established a sharp norm estimate for certain multiplication operators

Provided a proof for a previously announced norm estimate

Discussed connections between the Brennan conjecture and multiplier norms

Abstract

In this paper, we study the norms of multiplication operators acting between weighted Bergman spaces. First, we provide a proof for a norm estimate previously announced in our recent paper \cite{Jin-c}. Second, we establish a sharp norm estimate for certain special multiplication operators between weighted Bergman spaces, a result that is novel to the literature. Finally, we also discuss the connections between the Brennan conjecture and related multiplier norms induced by the Schwarzian derivative of univalent functions.