A Derivative Hilbert operator acting from Bergman spaces to Hardy spaces

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix ${\mathcal{H}}_{\mu }=({\mu }_{n,k}{)}_{n,k\ge 0}$ with entries ${\mu }_{n,k}={\mu }_{n+k}$, where ${\mu }_n={\int }_{[0,1)}{t}^{n}d\mu (t)$, formally induces the operator as follows:

$${\mathcal{D}\mathcal{H}}_{\mu }(f)(z)=\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^{\mathrm{\infty }}{\mu }_{n,k}{a}_k\right)(n+1)z^n,z\in \mathbb{D},$$

where $f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ is an analytic function in $\mathbb{D}$. In this article, we characterize those positive Borel measures on $[0,1)$ such that ${\mathcal{D}\mathcal{H}}_{\mu }$ is bounded (resp., compact) from Bergman spaces ${\mathcal{A}}^p$ into Hardy spaces $H^q$, where $0<p,q<\mathrm{\infty }$.