Causal compactification and Hardy spaces

Abstract

Let M = G / H \mathcal {M}=G/H be a irreducible symmetric space of Cayley type. Then M \mathcal {M} is diffeomorphic to an open and dense G G -orbit in the Shilov boundary of G / K × G / K G/K\times G/K . This compactification of M \mathcal {M} is causal and can be used to give answers to questions in harmonic analysis on M \mathcal {M} . In particular we relate the Hardy space of M \mathcal {M} to the classical Hardy space on the bounded symmetric domain G / K × G / K G/K\times G/K . This gives a new formula for the Cauchy-Szegö kernel for M \mathcal {M} .