Norm of the Hilbert matrix operator on logarithmically weighted Bloch and Hardy spaces

TL;DR

This paper precisely calculates the norms of the Hilbert matrix operator between various classical and weighted analytic function spaces, revealing exact values and bounds for these operator norms.

Contribution

It provides exact norm values for the Hilbert matrix operator on specific weighted Bloch and Hardy spaces, and establishes bounds for more general cases.

Findings

Norm from $ ext{Bloch}$ to $ ext{logarithmic Bloch}$ is 3/2

Norm from $H^0$ to logarithmic Hardy space is 1

Bounds for $ ext{alpha-Bloch}$ and Hardy spaces with weights are established

Abstract

In this paper, we compute the exact value of the norm of the Hilbert matrix operator $\mathcal{H}$ acting from the classical Bloch space $\mathcal{B}$ into the logarithmically weighted Bloch space $\mathcal{B}_{\log}$, and show that it equals $\frac{3}{2}$; we also find that the norm from the space of bounded analytic functions $H^\infty$ into the logarithmically weighted Hardy space $H^{\infty}_{\log}$ is $1$. Furthermore, we establish both lower and upper bounds for the norm of $\mathcal{H}$ when it maps from the $\alpha$-Bloch space $\mathcal{B}^\alpha$ into the logarithmically weighted $\mathcal{B}^\alpha_{\log}$ with $1 <\alpha < 2$, and from the Hardy space $H^{1}$ into the logarithmically weighted Hardy space $H^{1}_{\log}$.