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

Abstract

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\_$\Gamma$ of distributions having a wave front set included in a given closed cone $\Gamma$ of the cotangent space. As discovered by S. Alesker, the pull-back is not continuous for the usual topology on D\_$\Gamma$ , 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.