Norm of the Hilbert matrix operator between some spaces of analytic functions

TL;DR

This paper precisely calculates the norm of the Hilbert matrix operator across various spaces of analytic functions, providing exact values and bounds for different mappings.

Contribution

It offers new exact norm values and bounds for the Hilbert matrix operator between several analytic function spaces, extending previous understanding.

Findings

Exact norm of $\

Bounds for the norm on $eta$-Bloch space

Abstract

In this paper, we calculate the exact value of the norm of the Hilbert matrix operator $\mathcal{H}$ from the logarithmically weighted Korenblum space $H^\infty_{\alpha,\log}$ into Korenblum space $H^\infty_\alpha$, and from the Hardy space $H^\infty$ to the classical Bloch space $\mathcal{B}$. Furthermore, we compute the precise value of the norm on the logarithmically weighted Korenblum space $H^\infty_{\alpha,\log}$, and obtain both the lower and upper bounds of the norm on $\alpha$-Bloch space $\mathcal{B}^{\alpha}$. Finally, in the context of mapping from the Korenblum space $H^\infty_\alpha$ to the $(\alpha+1)$-Bloch space $\mathcal{B}^{\alpha+1}$, we establish the norm of $\mathcal{H}$.