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Let $G/H$ be a semisimple symmetric space. Then the space $L^2(G/H)$ can bedecomposed into a finite sum of series representations induced from parabolicsubgroups of $G$. The most continuous part of the spectrum of $L^2(G/H)$ is thepart induced from the smallest possible parabolic subgroup. In this paper weintroduce Hardy spaces canonically related to this part of the spectrum for aclass of non-compactly causal symmetric spaces. The Hardy space is areproducing Hilbert space of holomorphic functions living on a tube typebounded symmetric domain, containing $G/H$ as a boundary component. A boundaryvalue map is constructed and we show that it induces an $G$-isomorphism onto amultiplicity free subspace of full spectrum in the most continuous part $L_{\rmmc}^2(G/H)$ of $L^2(G/H)$. We also relate our Hardy space with the classicalHardy space on the tube domain.

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