Volterra operators between Hardy spaces of vector-valued Dirichlet series

Let $2\le p<\mathrm{\infty }$ and $X$ be a complex infinite-dimensional Banach space. It is proved that if $X$ is $p$-uniformly PL-convex, then there is no nontrivial bounded Volterra operator from the weak Hardy space ${\mathcal{H}}_p^{\text{weak}}(X)$ to the Hardy space ${\mathcal{H}}_p^{+}(X)$ of vector-valued Dirichlet series. To obtain this, a Littlewood–Paley inequality for Dirichlet series is established.