Integral means and boundary limits of Dirichlet series

Abstract

This paper deals with the boundary behaviour of functions in the Hardy spaces ℋ p for ordinary Dirichlet series. The main result, answering a question of Hedenmalm, shows that the classical Carlson theorem on integral means does not extend to the imaginary axis for functions in ℋ ∞ , that is, for the ordinary Dirichlet series in H∞ of the right half-plane. We discuss an important embedding problem for ℋ p , the solution of which is only known when p is an even integer. Viewing ℋ p as Hardy spaces of the infinite-dimensional polydisc, we also present analogues of Fatou's theorem.