Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum

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Let G/H be a semisimple symmetric space. Then the space L2 (G/H) can be decomposed into a finite sum of series of 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 non-compactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions on a bounded symmetric domain of tube type, containing G/H as a boundary component. A boundary value map is constructed and we show that it induces a G-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part Lmc2(G/H) of L2(G/H). We also relate our Hardy space to the classical Hardy space on the bounded symmetric domain.

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Mathematische Annalen

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