Continuity of the fundamental operations on distributions having a specified wave front set (with a counterexample by Semyon Alesker)
Volume 232 / 2016
Studia Mathematica 232 (2016), 201-226
MSC: Primary 46F10; Secondary 35A18.
DOI: 10.4064/sm8316-3-2016
Published online: 2 May 2016
Abstract
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.
