Hilbert type operators acting from the Bloch space into Bergman spaces

Abstract Let µ be a finite positive Borelmeasure on $[0,1)$ and $\alpha \gt -1$ . The generalized integral operator of Hilbert type $\mathcal {I}_{\mu_{\alpha+1}}$ is defined on the spaces $H(\mathbb{D})$ of analytic functions in the unit disc $\mathbb{D}$ as follows: \begin{equation*}\mathcal {I}_{\mu_{\alpha+1}}(f)(z)=\int_{0}^{1} \frac{f(t)}{(1-tz)^{\alpha+1}}d\mu(t),\ \ f\in H(\mathbb{D}),\ \ z\in \mathbb{D} .\end{equation*} In this paper, we give a unified characterization of the measures µ for which the operator $\mathcal {I}_{\mu_{\alpha+1}}$ is bounded from the Bloch space to a Bergman space for all $\alpha \gt -1$ . Additionally, we also investigate the action of $\mathcal {I}_{\mu_{\alpha+1}}$ from the Bloch space to the Hardy spaces and the Besov spaces.