广义Hörmander分布空间的泛函性质I：对偶理论、完备化和拓扑
Functional properties of Generalized Hörmander spaces of distributions I : Duality theory, completions and bornologifications
摘要：
具有其$C^\infty$波前设置在圆锥$Λ$中的分布的空间$D'_Λ$在物理学中变得重要，因为它在弯曲时空中的量子场论的表述中起作用。它也是微观局部分析的基本对象，但从功能分析的角度没有得到很好的研究。为了计算它在开锥情形下的完备化，我们引入了广义空间$D'_{γ,Λ}$，其中我们也控制了包含在$Λ$中的第二锥$γ$中的H^s波前集的并集.我们可以计算这类空间的自然拓扑的拓扑学和拓扑学性质、完备化和拓扑化。我们所有的拓扑结构都是核的，当它们是准LB时，我们可以描述它们。我们也给具体的微局部表示的有界和等度连续集在这些空间和工作的一般支持条件，包括未来的紧或空间紧的支持条件，全球双曲流形，物理应用的动机，将在第二篇论文中开发。
英文摘要：
The space $D'_Λ$ of distributions having their $C^\infty$ wavefront set in a cone $Λ$ has become important in physics because of its role in the formulation of quantum field theory in curved spacetime. It is also a basic object in microlocal analysis, but not well studied from a functional analytic viewpoint. In order to compute its completion in the open cone case, we introduce generalized spaces $D'_{γ,Λ}$ where we also control the union of H^s-wave front sets in a second cone $γ$ contained in $Λ$. We can compute bornological and topological duals, completions and bornologifications of natural topologies for spaces in this class. All our topologies are nuclear, ultrabornological when bornological and we can describe when they are quasi-LB. We also give concrete microlocal representations of bounded and equicontinuous sets in those spaces and work with general support conditions including future compact or space compact support conditions on globally hyperbolic manifolds, as motivated by physics applications to be developed in a second paper.
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