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Functional Properties of Hörmander’s Space of Distributions Having a Specified Wavefront Set

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Abstract

The space \({\mathcal{D}_\Gamma^\prime}\) of distributions having their wavefront sets in a closed cone \({\Gamma}\) has become important in physics because of its role in the formulation of quantum field theory in curved spacetime. In this paper, the topological and bornological properties of \({\mathcal{D}_\Gamma^\prime}\) and its dual \({\mathcal{E}_\Lambda^\prime}\) are investigated. It is found that \({\mathcal{D}_\Gamma^\prime}\) is a nuclear, semi-reflexive and semi-Montel complete normal space of distributions. Its strong dual \({\mathcal{E}_\Lambda^\prime}\) is a nuclear, barrelled and (ultra)bornological normal space of distributions which, however, is not even sequentially complete. Concrete rules are given to determine whether a distribution belongs to \({\mathcal{D}_\Gamma^\prime}\) , whether a sequence converges in \({\mathcal{D}_\Gamma^\prime}\) and whether a set of distributions is bounded in \({\mathcal{D}_\Gamma^\prime}\) .

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Authors and Affiliations

  1. Institut Camille Jordan UMR 5208, Université de Lyon, Université Lyon 1, 43 bd. du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Yoann Dabrowski

  2. Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, Sorbonne Universités, UMR CNRS 7590, UPMC Univ. Paris 06, Muséum National d’Histoire Naturelle, IRD UMR 206, 4 place Jussieu, 75005, Paris, France

    Christian Brouder

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  1. Yoann Dabrowski
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Communicated by M. Salmhofer

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Dabrowski, Y., Brouder, C. Functional Properties of Hörmander’s Space of Distributions Having a Specified Wavefront Set. Commun. Math. Phys. 332, 1345–1380 (2014). https://doi.org/10.1007/s00220-014-2156-0

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