# Ph.D. Oral Defense: "Carlson's Theorem for Different Measures"

Meredith Sargent, Washington University in Saint Louis

Abstract: Hedenmalm, Lindqvist, and Seip in 1997 revitalized the modern study of Dirichlet series by defining the space $\mathcal{H}^2$ and considering it as isometrically isomorphic to the Hardy space of the infinite polytorus $H^2(\mathbb{T}^\infty)$. This allowed a new viewpoint to be applied to classical theorems, including Carlson's theorem about the integral in the mean of a Dirichlet series. Carlson's theorem holds only for vertical lines in the right half plane, and cannot be extended to the boundary in full generality (as shown by Saksman and Seip in 2009.) However, Carlson's theorem can be shown to hold on the imaginary axis for a more restrictive class of Dirichlet series, and we shall do so.

The main result contained in this talk is a generalized version of Carlson's theorem: given a Borel probability measure on the polytorus, a measure is constructed on the imaginary axis so that the integral in the mean is equal to the integral on the polytorus.

Host: John McCarthy