Volterra operators on Hardy spaces of Dirichlet series

Abstract

Abstract For a Dirichlet series symbol g ⁢ ( s ) = ∑ n ≥ 1 b n ⁢ n - s {g(s)=\sum_{n\geq 1}b_{n}n^{-s}} , the associated Volterra operator 𝐓 g {\mathbf{T}_{g}} acting on a Dirichlet series f ⁢ ( s ) = ∑ n ≥ 1 a n ⁢ n - s {f(s)=\sum_{n\geq 1}a_{n}n^{-s}} is defined by the integral f ↦ - ∫ s + ∞ f ⁢ ( w ) ⁢ g ′ ⁢ ( w ) ⁢ 𝑑 w . {f\mapsto-\int_{s}^{+\infty}f(w)g^{\prime}(w)\,dw}. We show that 𝐓 g {\mathbf{T}_{g}} is a bounded operator on the Hardy space ℋ p {\mathcal{H}^{p}} of Dirichlet series with 0 < p < ∞ {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 BMOA ⁡ ( 𝔻 ) {{\operatorname{BMOA}}(\mathbb{D})} . A further analogy with classical BMO {{\operatorname{BMO}}} is that exp ⁡ ( c ⁢ | g | ) {\exp(c|g|)} is integrable (on the infinite polytorus) for some c > 0 {c>0} whenever 𝐓 g {\mathbf{T}_{g}} is bounded. In particular, such g belong to ℋ p {\mathcal{H}^{p}} for every p < ∞ {p<\infty} . We relate the boundedness of 𝐓 g {\mathbf{T}_{g}} to several other BMO {{\operatorname{BMO}}} -type spaces: BMOA {{\operatorname{BMOA}}} in half-planes, the dual of ℋ 1 {\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 𝐓 g {\mathbf{T}_{g}} on reproducing kernels for appropriate sequences of subspaces of ℋ 2 {\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.