HARDY SPACES FOR NON-COMPACTLY CAUSAL SYMMETRIC SPACES AMD THE MOST CONTINUOUS SPECTRUM

SIMON GINDIKIN BERNHARD KRÖTZ GESTUR ÓLAFSSON

Abstract. Let G/H be a semisimple symmetric space. Then the space L^{2}(G/H) can be decomposed into a finite sum of series representations induced from parabolic subgroups of G. The most continuous part of the spectrum of L^{2}(G/H) is the part induced from the smallest possible parabolic subgroup. In this paper we introduce Hardy spaces canonically related to this part of the spectrum for a class of noncompactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions living on a tube type bounded symmetric space, containing G/H as a boundary component. A boundary value map is constructed and we show that it induces an G-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part L_{mc}^{2}(G/H) of L^{2}(G/H). We also relate our Hardy space with the classical Hardy space on the tube domain.