A Derivative-Hilbert operator acting on BMOA space

Abstract

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

Theorems & Definitions (11)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Theorem 3.1
• Theorem 3.2