On the continuity of the product of distributions in local Sobolev spaces

We consider the space ${\mathcal{H}}_L^{s,r}(O)$ consisting of all local Sobolev distributions of order $s$ on an open set $O$ whose Sobolev wave front set of order $r$ is contained in the closed conic set $L\subseteq O\times ({\mathbb{R}}^m\mathrm{\backslash }\{0\})$. We introduce a locally convex topology on ${\mathcal{H}}_L^{s,r}(O)$ and show that the ordinary product of smooth functions uniquely extends to a continuous bilinear mapping ${\mathcal{H}}_{L_1}^{r_1,r^{\mathrm{\prime }}}(O)\times {\mathcal{H}}_{L_2}^{r_2,r^{\mathrm{\prime }\mathrm{\prime }}}(O)\to {\mathcal{H}}_L^{s,r}(O)$, for appropriate $s$ and $r$ when $L_1$ and $L_2$ are in a favorable position. The key ingredient in our proof is to employ Hörmander’s idea of considering the pullback by the diagonal map $x\mapsto (x,x)$ of the tensor product of two distributions.