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

TL;DR

This paper studies the topological properties of fundamental operations on distributions with specified wave front sets, introducing a new topology that ensures continuity and hypocontinuity of these operations, addressing previous discontinuity issues.

Contribution

A new topology on distribution spaces with constrained wave front sets is proposed, ensuring continuity of pull-back, push-forward, and other operations, with a counterexample illustrating prior limitations.

Findings

Pull-back is not continuous under the usual topology.

A new topology makes pull-back and push-forward continuous.

Tensor and convolution products become hypocontinuous under the new topology.

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.