Volterra operators on Hardy spaces of Dirichlet series

For a Dirichlet series symbol $g(s)={\sum }_{n\ge 1}{b}_n{n}^{-s}$, the associated Volterra operator ${\mathbf{T}}_g$ acting on a Dirichlet series $f(s)={\sum }_{n\ge 1}{a}_n{n}^{-s}$ is defined by the integral $f\mapsto -\int_{s}^{+\infty} f(w)g’(w),dw$\mathrm{.}Weshowthat$$\mathbf{T}$_g$ is a bounded operator on the Hardy space ${\mathcal{H}}^p$ of Dirichlet series with $0 < p < \infty$ifandonlyifthesymbol$g$ satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of $\mathrm{BMOA}(\mathbb{D})$. A further analogy with classical $\mathrm{BMO}$ is that $\exp (c\mathrm{\mid }g\mathrm{\mid })$ is integrable (on the infinite polytorus) for some $c>0$ whenever ${\mathbf{T}}_g$ is bounded. In particular, such $g$ belong to ${\mathcal{H}}^p$ for every $p<\mathrm{\infty }$. We relate the boundedness of ${\mathbf{T}}_g$ to several other $\mathrm{BMO}$ type spaces: $\mathrm{BMOA}$ in half-planes, the dual of ${\mathcal{H}}^1$, and the space of symbols of bounded Hankel forms. Moreover, we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for $m$-homogeneous symbols as well as for general symbols. Finally, we consider the action of ${\mathbf{T}}_g$ on reproducing kernels for appropriate sequences of subspaces of ${\mathcal{H}}^2$. Our proofs employ function and operator theoretic techniques in one and several variables; a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbols $g$.