Integral means and boundary limits of Dirichlet series

TL;DR

This paper investigates the boundary behavior of Dirichlet series within Hardy spaces, revealing limitations of classical theorems and exploring boundary limits and embeddings in infinite-dimensional contexts.

Contribution

It demonstrates that Carlson's integral mean theorem does not extend to the boundary for HD^, and discusses boundary limits and embedding problems for these Hardy spaces.

Findings

Classical Carlson theorem fails on the imaginary axis for HD^.

Boundary limits analogous to Fatou's theorem are established for HD^p.

The embedding problem for HD^p remains open except when p is an even integer.

Abstract

We study the boundary behavior of functions in the Hardy spaces HD^p for ordinary Dirichlet series. Our main result, answering a question of H. Hedenmalm, shows that the classical F. Carlson theorem on integral means does not extend to the imaginary axis for functions in HD^\infty, i.e., for ordinary Dirichlet series in H^\infty of the right half-plane. We discuss an important embedding problem for HD^p, the solution of which is only known when p is an even integer. Viewing HD^p as Hardy spaces of the infinite-dimensional polydisc, we also present analogues of Fatou's theorem.