Volterra operators on Hardy spaces of Dirichlet series

Abstract

For a Dirichlet series symbol g.s/ D P n 1 bnn s , the associated Volterra operator Tg acting on a Dirichlet series f .s/ D P n 1 ann s is defined by the integral f 7! Z C1 s f .w/g0 .w/ dw: We show that Tg is a bounded operator on the Hardy space Hp of Dirichlet series with 0 < p < 1 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.cjgj/ is integrable (on the infinite polytorus) for some c > 0 whenever Tg is bounded. In particular, such g belong to Hp for every p < 1. 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