For a Dirichlet series symbol , the associated Volterra operator acting on a Dirichlet series is defined by the integral
We show that is a bounded operator on the Hardy space of Dirichlet series with 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 . A further analogy with classical is that is integrable (on the infinite polytorus) for some whenever is bounded. In particular, such g belong to for every . We relate the boundedness of to several other -type spaces: in half-planes, the dual of , 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 on reproducing kernels for appropriate sequences of subspaces of . 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.
Funding statement: The first and third author are supported by Grant 227768 of the Research Council of Norway.
The authors are grateful to Alexandru Aleman and Frédéric Bayart for helpful discussions and remarks. They would also like to express their gratitude to the anonymous referee for a very careful review of the paper.
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