Condition d'existence du produit de deux distributions. (Existence condition for the product of two distributions) (Q1821309)

For u,v\(\in {\mathcal D}'({\mathbb{R}}^ n)\) the product \(u\cdot v\) is said to exist if \(\lim_{\epsilon \to 0}u_{\epsilon}\cdot v_{\epsilon}\) exists in \({\mathcal D}'({\mathbb{R}}^ n)\) for each mollifier \(\omega\) (where \(u_{\epsilon}:=u*\omega_{\epsilon}\) and \(\omega_{\epsilon}(x):=\frac{1}{\epsilon^ n}\omega (\frac{x}{\epsilon}))\) and if the limit does not depend on \(\omega\). The author derives a sufficient condition for the existence of \(u\cdot v\) formulated in terms of wave front sets which are defined with respect to certain linear subspaces of \({\mathcal D}'({\mathbb{R}}^ n)\). In doing this he extends a previous result of \textit{L. Hörmander} [Acta Math. 127, 79-183 (1971; Zbl 0212.466)].