VECTOR-VALUED LEBESGUE AND HARDY SPACES FOR SYMMETRIC NORMS OVER COMPACT GROUPS

In this paper, we study gauge norms on probability spaces and their associated Lebesgue spaces, both in the scalar and vector-valued cases. We consider norms that are symmetric with respect to groups of measure-preserving transformations, and we show that such a group is ergodic if and only if every symmetric gauge norm dominates parallel to parallel to(1). When the probability space is a compact group G with Haar measure mu, we study convolution of Banach algebra-valued functions in Lebesgue spaces. When G is Abelian and its dual group is linearly ordered, we study the associated Hardy spaces. When G = T, we characterize the closed densely defined operators on H-alpha (T) affiliated with H-infinity (T).