A Derivative-Hilbert operator acting on BMOA space

Abstract

Let $\hat{I}¼$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix ${\mathcal{H}}_{\hat{I}}¼=(\hat{I}{¼}_{n,k}{)}_{n,k\ge 0}$ with entries $\hat{I}{¼}_{n,k}=\hat{I}{¼}_{n+k}$, where $\hat{I}{¼}_n={\int }_{[0,1)}{t}^{n}d\hat{I}¼(t)$, induces, formally, the Derivative-Hilbert operator $${\mathcal{D}\mathcal{H}}_{\hat{I}}¼(f)(z)=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\infty }\hat{I}{¼}_{n,k}{a}_k\right)(n+1)z^n,z\in \mathbb{D},$$ where $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ is an analytic function in $\mathbb{D}$. We characterize the measures $\hat{I}¼$ for which ${\mathcal{D}\mathcal{H}}_{\hat{I}}¼$ is a bounded operator on $BMOA$ space. We also study the analogous problem from the $\hat{I}\pm$-Bloch space ${\mathcal{B}}_{\hat{I}}\pm (\hat{I}\pm >0)$ into the $BMOA$ space.

arXiv admin note: text overlap with arXiv:2410.20435