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

By Christian Brouder, Nguyen Viet Dang and Frédéric Hélein


29 pages, 1 figure.The pull-back, push-forward and multiplication of smooth functions can be extended to distributions if their wave front set satisfies some conditions. Thus, it is natural to investigate the topological properties of these operations between spaces D_Γ of distributions having a wave front set included in a given closed cone Γ of the cotangent space. As discovered by S. Alesker, the pull-back is not continuous for the usual topology on D_Γ , 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, the tensor and convolution products and the multiplication of distributions are hypocontinuous

Topics: microlocal analysis, functional analysis, mathematical physics, renormalization, [ MATH.MATH-FA ] Mathematics [math]/Functional Analysis [math.FA], [ PHYS.MPHY ] Physics [physics]/Mathematical Physics [math-ph]
Year: 2016
DOI identifier: 10.4064/sm8316-3-2016
OAI identifier: oai:HAL:hal-01069072v2
Provided by: Hal-Diderot

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