Geodesic
Continuity of the fundamental operations on distributions having a specified wave front set (with a counterexample by Semyon Alesker)
Christian Brouder 1   ; Nguyen Viet Dang 2   ; Frédéric Hélein 3
1 Sorbonne Universités, UPMC Univ. Paris 06 CNRS UMR 7590 and Muséum National d’Histoire Naturelle, IRD UMR 206
2 Institut Camille Jordan Université Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918 69622 Villeurbanne Cedex, France
3 Institut de Mathématiques de Jussieu Paris Rive Gauche Université Denis Diderot Paris 7 Bâtiment Sophie Germain 75205 Paris Cedex 13, France
Studia Mathematica, Tome 232 (2016) no. 3, pp. 201-226 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/sm8316-3-2016
Keywords: pull back push forward multiplication smooth functions extended distributions their wave front sets satisfy conditions natural investigate topological properties these operations between spaces mathcal vargamma distributions having wave front set included given closed cone vargamma cotangent space discovered alesker pull back continuous usual topology mathcal vargamma tensor product separately continuous paper topology defined which pull back push forward continuous tensor convolution products multiplication distributions hypocontinuous

Christian Brouder  1   ; Nguyen Viet Dang  2   ; Frédéric Hélein  3

1 Sorbonne Universités, UPMC Univ. Paris 06 CNRS UMR 7590 and Muséum National d’Histoire Naturelle, IRD UMR 206
2 Institut Camille Jordan Université Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918 69622 Villeurbanne Cedex, France
3 Institut de Mathématiques de Jussieu Paris Rive Gauche Université Denis Diderot Paris 7 Bâtiment Sophie Germain 75205 Paris Cedex 13, France 
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Christian Brouder; Nguyen Viet Dang; Frédéric Hélein. Continuity of the fundamental operations on distributions having a specified wave front set (with a counterexample by Semyon Alesker). Studia Mathematica, Tome 232 (2016) no. 3, pp. 201-226. doi: 10.4064/sm8316-3-2016

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