A Derivative-Hilbert operator acting on BMOA space

TL;DR

This paper characterizes measures for which a Derivative-Hilbert operator, defined via a Hankel matrix, is bounded on the BMOA space and explores related boundedness on the alpha-Bloch space.

Contribution

It provides a characterization of measures ensuring the boundedness of the Derivative-Hilbert operator on BMOA and studies its behavior from alpha-Bloch space to BMOA.

Findings

Characterization of measures for boundedness on BMOA

Boundedness conditions from alpha-Bloch space to BMOA

Analysis of the Derivative-Hilbert operator's properties

Abstract

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\geq 0}$ with entries $\mu_{n,k}=\mu_{n+k}$, where $\mu_{n}=\int_{[0,1)}t^nd\mu(t)$, induces, formally, the Derivative-Hilbert operator $$\mathcal{DH}_\mu(f)(z)=\sum_{n=0}^\infty\left(\sum_{k=0}^\infty \mu_{n,k}a_k\right)(n+1)z^n , ~z\in \mathbb{D},$$ where $f(z)=\sum_{n=0}^\infty a_nz^n$ is an analytic function in $\mathbb{D}$. We characterize the measures $\mu$ for which $\mathcal{DH}_\mu$ is a bounded operator on $BMOA$ space. We also study the analogous problem from the $\alpha$-Bloch space $\mathcal{B}_\alpha(\alpha>0)$ into the $BMOA$ space.