Fractional Volterra-type operators on Hardy spaces

A family of non-trivial paraproducts acting on $H(\mathbb {D})$, the space of analytic functions over the open unit disk in the complex plane, are introduced and studied. We first establish their existence and the associated reconstruction formulae. A natural consequence is that they induce Volterra-type operators $\mathfrak {V}_{\alpha ,\beta }^{\varphi }$, with symbol $\varphi \in H(\mathbb {D})$, for parameters $$ (\alpha ,\beta )\in \mathbb {K}:=(((\mathbb {C} \setminus \mathbb {Z})\cup \{0\}) \times \mathbb {C}) \cup (\mathbb {N} \times \mathbb {Z}).$$ If $\alpha =\beta =1$, then $\mathfrak {V}_{\alpha ,\beta }^{\varphi }$ reduces to the standard Volterra integration operator. Moreover, our construction improves the definition of a fractional Volterra operator of M. Pavlović in 2019. Then we characterize those $\varphi $ such that $\mathfrak {V}_{\alpha ,\beta }^{\varphi }$ is bounded from $H^p$ to $H^q$, where $$0 \lt p,q \lt \infty , \quad \alpha \gt 0 \quad \text {and} \quad \alpha \gt \beta .$$ Several open problems, in connection with the Hardy–Stein formula, g-functions, and Luecking’s characterization of Carleson measures, arise naturally for the range $\alpha \le 0$ or $ \alpha \le \beta $.