Decomposition of analytic measures on groups and measure spaces

Nakhlé Asmar and Stephen Montgomery-Smith
Department of Mathematics
University of Missouri
Columbia, MO 65211
nakhle@math.missouri.edu
stephen@math.missouri.edu

Dedicated to the memory of Edwin Hewitt

Abstract:

In this paper, we consider an arbitrary locally compact abelian group $G$, with an ordered dual group $\Gamma$, acting on a space of measures. Under suitable conditions, we define the notion of analytic measures using the representation of $G$ and the order on $\Gamma$. Our goal is to study analytic measures by applying a new transference principle for subspaces of measures, along with results from probability and Littlewood-Paley theory. As a consequence, we will derive new properties of analytic measures as well as extensions of previous work of Helson and Lowdenslager, de Leeuw and Glicksberg, and Forelli.

A.M.S. Subject Classification: 43A17, 43A32,

Keywords: orders, transference, measure space, sup path attaining, F.&M. Riesz Theorem