Hilbert matrix operator acting between conformally invariant spaces

TL;DR

The paper analyzes the Hilbert matrix operator $\mathcal H$ acting from $H^\infty$ into conformally invariant Banach spaces, focusing on the sharp norm into $\text{BMOA}$ and on boundedness criteria into $M(\mathcal{D}_\mu)$. It establishes the exact norm $\|\mathcal H\|_{H^\infty\to\text{BMOA}}=1+\dfrac{\pi}{\sqrt{2}}$ and provides a set of equivalent conditions linking boundedness to $\log(1-z)\in M(\mathcal{D}_\mu)$ and to $V_\mu$, along with explicit norm formulas for radial $\mu$ such as those associated with the $Q_p$ spaces. The work further derives mean Lipschitz and Dirichlet-space results, including a precise norm formula $\|\mathcal H\|_{H^\infty\to M(\mathcal{D}_\mu)}=1+\left(\int_{\mathbb{D}} \frac{4}{|1-z^{2}|^{2}} U_\mu(z) dA(z)\right)^{1/2}$ for radial $\mu$, and connects these with the analogous Cesàro operator. By providing concrete norms and criteria, the paper advances understanding of Hilbert-type operators on conformally invariant spaces and sets the stage for further extensions to non-radial measures and broader Dirichlet-type frameworks.

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 $μ$ such that $\mathcal H$ is bounded from $H^\infty$ into the conformally invariant Dirichlet space $M(\mathcal{D}_μ)$. For particular measures $μ$, we also provide the norm of $\mathcal{H}$ from $H^\infty$ into $M(\mathcal{D}_μ)$.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• proof : Proof of Theorem \ref{['T:norm h h to bmoa']}
• Theorem 4.1
• proof
• Proposition 4.2
• proof
• proof : Proof of Theorem \ref{['h from h to mdm']}