Abstract/Summary

For a Dirichlet series symbol g(s)=∑n≥1bnn−s, the associated Volterra operator Tg acting on a Dirichlet series f(s)=∑n≥1ann−s is defined by the integral f↦−∫+∞sf(w)g′(w)dw. We show that Tg is a bounded operator on the Hardy space Hp 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 Hp for every p<∞. We relate the boundedness of Tg to several other BMO-type spaces: BMOA in half-planes, the dual of H1, 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 H2. 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.