Area operators on Hardy spaces of Dirichlet series

We introduce area operators $\mathbb A_{\mu ,l}$ in the Dirichlet series setting for $l \gt 0$ and positive Borel measures $\mu $ on the right half-plane $\mathbb C_0$. It is proved that if $\mu $ is a Carleson measure on $\mathbb C_0$, then for $0 \lt p \lt \infty $, the area operator $\mathbb A_{\mu ,l}$ is bounded from the Hardy space $\mathscr H^p_0$ of Dirichlet series vanishing at $+\infty $ to some $L^p$-space. We also give an application of our methods to Volterra operators.