2 added 33 characters in body

In general you can only say that $$WF(uv) \subseteq \Bigg[ WF(u) \cup WF(v) \cup \Big[WF(u)+WF(v)\Big]\Bigg]$$ where the set $$WF(u) + WF(v) = \left\lbrace (x,\xi +\eta)| (x,\xi)\in WF(u), (x,\eta)\in WF(v) \right\rbrace$$ (note that a sufficient criterion for the product is to be well-defined is precisely when the above set contains no points of the form $(x,0)$).

See, e.g. chapter 11 of Friedlander and Joshi Introduction to The Theory of Distributions.

But quite obviously the $\subseteq$ is not always an equality: just take $u,v$ two compactly supported distributions with distinct supports. Note that given $WF(u)$ and $WF(v)$ you only know the singular support of $u$ and $v$ and not their actual supports.

1

In general you can only say that $$WF(uv) \subseteq \Bigg[ WF(u) \cup WF(v) \cup \Big[WF(u)+WF(v)\Big]\Bigg]$$ where the set $$WF(u) + WF(v) = \left\lbrace (x,\xi +\eta)| (x,\xi)\in WF(u), (x,\eta)\in WF(v) \right\rbrace$$ (note that the product is well-defined precisely when the above set contains no points of the form $(x,0)$).

See, e.g. chapter 11 of Friedlander and Joshi Introduction to The Theory of Distributions.

But quite obviously the $\subseteq$ is not always an equality: just take $u,v$ two compactly supported distributions with distinct supports. Note that given $WF(u)$ and $WF(v)$ you only know the singular support of $u$ and $v$ and not their actual supports.