Hilbert matrix operator acting between conformally invariant spaces

TL;DR

This paper investigates the Hilbert matrix operator's behavior between bounded analytic functions and conformally invariant spaces, providing norm estimates and characterizations of boundedness for specific measures.

Contribution

It characterizes the norm of the Hilbert matrix operator from $H^{infty}$ to $ ext{BMOA}$ and describes conditions for boundedness into conformally invariant Dirichlet spaces.

Findings

Norm of $ ext{H}$ from $H^{infty}$ to $ ext{BMOA}$ is determined.

Conditions for boundedness of $ ext{H}$ into $M( ext{D}_{ extmu})$ are characterized.

Explicit norms are provided for particular measures.

Abstract

In this article we study the action of the the Hilbert matrix operator $\mathcal H$ from the space of bounded analytic functions into conformally invariant Banach spaces. In particular, we describe the norm of $\mathcal{H}$ from $H^\infty$ into $\text{BMOA}$ and we characterize the positive Borel measures $\mu$ such that $\mathcal H$ is bounded from $H^\infty$ into the conformally invariant Dirichlet space $M(\mathcal{D}_\mu )$. For particular measures $\mu$, we also provide the norm of $\mathcal{H}$ from $H^\infty$ into $M(\mathcal{D}_\mu )$.