Volterra operators on Hardy spaces of Dirichlet series

TL;DR

This paper characterizes the boundedness of Volterra operators on Hardy spaces of Dirichlet series via Carleson measure conditions, linking them to BMO spaces and exploring their properties through function, operator, and number theory techniques.

Contribution

It provides a comprehensive characterization of bounded Volterra operators on Hardy spaces of Dirichlet series, connecting them to Carleson measures, BMO spaces, and multiplicative coefficient structures.

Findings

Boundedness characterized by Carleson measure condition

Relation to BMOA and dual spaces of Hardy spaces

Coefficient estimates for homogeneous symbols

Abstract

For a Dirichlet series symbol $g(s) = \sum_{n \geq 1} b_n n^{-s}$, 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 $f\mapsto -\int_{s}^{+\infty} f(w)g'(w)\,dw$. We show that $\mathbf{T}_g$ is a bounded operator on the Hardy space $\mathcal{H}^p$ of Dirichlet series with $0 < p < \infty$ if and only if the symbol $g$ satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of ${\operatorname{BMOA}}(\mathbb{D})$. A further analogy with classical ${\operatorname{BMO}}$ is that $\exp(c|g|)$ 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 $p < \infty$. We relate the boundedness of $\mathbf{T}_g$ to several other ${\operatorname{BMO}}$ type spaces: ${\operatorname{BMOA}}$ 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 $g$.