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Bulletin of the London Mathematical Society Advance Access originally published online on March 11, 2009
Bulletin of the London Mathematical Society 2009 41(3):411-422; doi:10.1112/blms/bdp004
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© 2009 London Mathematical Society

Integral means and boundary limits of Dirichlet series

Eero Saksman

Department of Mathematics and Statistics
University of Helsinki
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FI-00014 Helsinki
Finland
eero.saksman@helsinki.fi

Kristian Seip

Department of Mathematical Sciences
Norwegian University of Science and Technology
NO-7491 Trondheim
Norway

Received 5 December 2007. Revision received 23 November 2008.

This paper deals with the boundary behaviour of functions in the Hardy spaces Formula 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 Formula {infty}, that is, for the ordinary Dirichlet series in H{infty} of the right half-plane. We discuss an important embedding problem for Formula p, the solution of which is only known when p is an even integer. Viewing Formula p as Hardy spaces of the infinite-dimensional polydisc, we also present analogues of Fatou's theorem.

2000 Mathematics Subject Classification 30B50, 42B30 (primary), 46E15, 46J15 (secondary).

The first author is supported by the Academy of Finland, projects no. 113826 and 118765. The second author is supported by the Research Council of Norway grant 185359/V30. This research is part of the European Science Foundation Networking Programme Harmonic and Complex Analysis and its Applications (HCAA).


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