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Continuity of the fundamental operations on distributions having a specified wave front set (with a counterexample by Semyon Alesker)

Volume 232 / 2016

Christian Brouder, Nguyen Viet Dang, Frédéric Hélein Studia Mathematica 232 (2016), 201-226 MSC: Primary 46F10; Secondary 35A18. DOI: 10.4064/sm8316-3-2016 Published online: 2 May 2016


The pull-back, push-forward and multiplication of smooth functions can be extended to distributions if their wave front sets satisfy some conditions. Thus, it is natural to investigate the topological properties of these operations between spaces ${\mathcal {D}}’_\varGamma $ of distributions having a wave front set included in a given closed cone $\varGamma $ of the cotangent space. As discovered by S. Alesker, the pull-back is not continuous for the usual topology on ${\mathcal {D}}’_\varGamma $, and the tensor product is not separately continuous. In this paper, a new topology is defined for which the pull-back and push-forward are continuous, and the tensor and convolution products and multiplication of distributions are hypocontinuous.


  • Christian BrouderSorbonne Universités, UPMC Univ. Paris 06
    CNRS UMR 7590
    Muséum National d’Histoire Naturelle, IRD UMR 206
  • Nguyen Viet DangInstitut Camille Jordan
    Université Claude Bernard Lyon 1
    43 boulevard du 11 novembre 1918
    69622 Villeurbanne Cedex, France
  • Frédéric HéleinInstitut de Mathématiques de Jussieu Paris Rive Gauche
    Université Denis Diderot Paris 7
    Bâtiment Sophie Germain
    75205 Paris Cedex 13, France

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