Fractional Volterra‐type operators from Bergman spaces to Hardy spaces

A new family of Volterra‐type operators Vα,βφ(·)$\mathfrak {V}_{\alpha,\beta }^{\varphi }(\cdot)$ based on bona fide fractional calculus is introduced in [12] by constructing analytic paraproducts acting on H(D)$H(\mathbb {D})$ and their boundedness between Hardy spaces is characterized for certain parameter ranges there. This paper is a natural companion to [12] in the sense that it characterizes those φ$\varphi$’s such that Vα,βφ$\mathfrak {V}_{\alpha,\beta }^{\varphi }$ is bounded from weighted Bergman spaces Lap(dAγ)$L_a^p(dA_\gamma)$ to Hardy spaces Hq$H^q$ for the range 0 −1,α>0andγ+p(α−β)>−1.$$\begin{equation*} \hspace*{40pt}0 -1, \quad \alpha >0 \quad \text{and} \quad \gamma +p(\alpha -\beta)>-1. \end{equation*}$$The case α=β=1$\alpha =\beta =1$ extends earlier results of Wu [32] and Miihkinen et al. [22]. Besides standard techniques in this area, the proof relies on certain recent results on fractional integration operators obtained in [14, 35].