The norm of the Hilbert matrix operator on Bergman spaces

TL;DR

This paper proves a conjecture about the norm of the Hilbert matrix operator on Bergman spaces, extending known results to a broader range of parameters and providing explicit formulas.

Contribution

It confirms Karapetrović's conjecture for a specific range of alpha, improving previous bounds and expanding understanding of the operator's norm on Bergman spaces.

Findings

Confirmed the conjectured norm formula for certain alpha values

Extended the range of alpha where the conjecture holds

Improved previous bounds on the operator norm

Abstract

Karapetrovi\'c conjectured that the norm of the Hilbert matrix operator on the Bergman space $A^p_\alpha$ is equal to $\pi/\sin((2+\alpha)\pi/p)$ when $-1<\alpha<p-2$. In this paper, we provide a proof of this conjecture for $0\leq \alpha \leq \frac{6p^3-29p^2+17p-2+2p\sqrt{6p^2-11p+4}}{(3p-1)^2}$, and this range of $\alpha$ improves the best known result when $\alpha>\frac{1}{47}$ and $\alpha \not=1$.