# Volterra operators on Hardy spaces of Dirichlet series

Ole Fredrik Brevig, Karl-Mikael Perfekt and Kristian Seip

# Abstract

For a Dirichlet series symbol g(s)=n1bnn-s, the associated Volterra operator 𝐓g acting on a Dirichlet series f(s)=n1ann-s is defined by the integral

f-s+f(w)g(w)𝑑w.

We show that 𝐓g is a bounded operator on the Hardy space p of Dirichlet series with 0<p< 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 BMOA(𝔻). A further analogy with classical BMO is that exp(c|g|) is integrable (on the infinite polytorus) for some c>0 whenever 𝐓g is bounded. In particular, such g belong to p for every p<. We relate the boundedness of 𝐓g to several other BMO-type spaces: BMOA in half-planes, the dual of 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 𝐓g on reproducing kernels for appropriate sequences of subspaces of 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.

Funding statement: The first and third author are supported by Grant 227768 of the Research Council of Norway.

# Acknowledgements

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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