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.
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机译：令$ G / H $为半简单对称空间。然后，可以将空间$ L ^ 2（G / H）$分解为由$ G $的抛物子群引起的级数表示的有限总和。 $ L ^ 2（G / H）$谱中最连续的部分是由最小可能的抛物线亚组诱发的部分。在本文中，我们针对一类非紧因果对称空间引入了与谱的这部分正则相关的Hardy空间。 Hardy空间正在产生全纯函数的Hilbert空间，该空间位于管型有界对称域上，包含$ G / H $作为边界分量。构造了一个边界值图，我们证明了它在$ L ^ 2（G的最连续部分$ L _ {\ rmmc} ^ 2（G / H）$的全连续谱子自由度子空间上诱导$ G $同构。 / H）$。我们还将Hardy空间与tube域上的classicHardy空间相关联。
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