Norm estimates of weighted composition operators pertaining to the Hilbert Matrix

TL;DR

This paper provides a simplified proof of the Hilbert matrix operator's norm on Bergman spaces and calculates its exact value on Korenblum spaces, advancing understanding of weighted composition operators.

Contribution

It offers a new, simplified proof of the Hilbert matrix norm on Bergman spaces and determines its exact value on Korenblum spaces, extending previous results.

Findings

Norm of Hilbert matrix on Bergman space is rac{}{(rac{2\u007f}{p})} for 2 < p < 4

Exact norm of  on Korenblum spaces for 0 <    2/3

Upper bound for the norm on the scale 2/3 <   < 1

Abstract

Very recently, Bo\v{z}in and Karapetrovi\'c solved a conjecture by proving that the norm of the Hilbert matrix operator $\mathcal{H}$ on the Bergman space $A^p$ is equal to $\frac{\pi}{\sin(\frac{2\pi}{p})}$ for $2 < p < 4.$ In this article we present a partly new and simplified proof of this result. Moreover, we calculate the exact value of the norm of $\mathcal{H}$ defined on the Korenblum spaces $H^\infty_\alpha$ for $0 < \alpha \le 2/3$ and an upper bound for the norm on the scale $2/3 < \alpha < 1$.