Functional properties of Hörmander's space of distributions having a specified wavefront set (Q462895)

The topological and bornological properties of the space \(\mathcal{D}'_\Gamma\) of distributions having wavefront set in a closed cone \(\Gamma\) and its dual \(\mathcal{E}'_\Lambda\) are studied exhaustively. These spaces play an important role in perturbative algebraic quantum field theory (pAQFT) [\textit{R. Brunetti} et al., Adv. Theor. Math. Phys. 13, No. 5, 1541--1599 (2009; Zbl 1201.81090)]. In addition to the so-called Hörmander topology typically used in the literature, also the normal topology on \(\mathcal{D}'_\Gamma\) is investigated. In particular, it is shown that in this topology \(\mathcal{D}'_\Gamma\) is a normal space of distributions which is Hausdorff, nuclear and semi-reflexive. \(\mathcal{E}'_\Lambda\) equipped with the strong topology is Hausdorff, nuclear, and barrelled, but not even sequentially complete. In view of applications in pAQFT, also criteria for boundedness and convergence in \(\mathcal{D}'_\Gamma\) are given.