Continuity of the fundamental operations on distributions having a specified wave front set (with a counterexample by Semyon Alesker) (Q2809347)

The authors consider the spaces \({\mathcal D}'_\Gamma\) of distributions \(u\in {\mathcal D}'({\mathbb R}^n)\) whose wave front set is contained in the closed cone \(\Gamma\) of the cotangent space. It is well known that several operations involving distributions are sequentially continuous between appropriate spaces of type \({\mathcal D}'_\Gamma\) endowed with Hörmander's topology, which is the locally convex topology defined in terms of all the seminorms on \({\mathcal D}'({\mathbb R}^n)\) for the weak topology and the seminorms of the form NEWLINE\[NEWLINE \|u\|_{N,V,\chi} = \sup_{\xi\in V}\left(1+|\xi|\right)^N\left|\widehat{u\chi}(\xi)\right|, NEWLINE\]NEWLINE where \(N \geq 0\), \(\chi\in {\mathcal D}({\mathbb R}^n)\) and \(V\subset {\mathbb R}^n\) is a closed cone with NEWLINE\[NEWLINE\left(\text{supp}\;\chi \times V\right)\cap \Gamma = \emptyset.NEWLINE\]NEWLINE Section 2 of the paper describes a counterexample due to \textit{S. Alesker} [Geom. Funct. Anal. 20, No. 5, 1073--1143 (2010; Zbl 1213.52013)] that shows that the pull-back is not continuous for Hörmander's topology. Section 3 describes the normal topology on \({\mathcal D}'_\Gamma,\) which is obtained from Hörmander's topology after replacing the seminorms of the weak topology of \({\mathcal D}'({\mathbb R}^n)\) by those of the strong topology. Section 4 shows that the tensor product is hypocontinuous for the normal topology, while Sections 5--6 prove the continuity of the pull-back and push-forward operations. Section 6 finishes with a coordinate invariant definition of the wave front set. Some technical results are included in the final Section 7.