Assume $u, v \in \mathcal{D}'(\mathbb{R}^n)$ are distributions with compact support. Denote by $\operatorname{WF}(\bullet) \subset T^*\mathbb{R}^n \setminus 0$ the wavefront set of a distribution $\bullet$. If $\operatorname{WF}(u) \cap \operatorname{WF}(v) = \emptyset$, then their product $uv$ is well defined. If $uv = 0$, does this imply that at an open and dense set of points one of the two distributions vanish, that is, $\operatorname{supp}(u)^c \cup \operatorname{supp}(v)^c \subset \mathbb{R}^n$ is open and dense?

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For the background on the wavefront set and distribution theory, see Chapter 8 of Hörmander's book [The Analysis of Linear Partial Differential Operators I](https://doi.org/10.1007/978-3-642-61497-2). For products under the wavefront set condition, see Theorem 8.2.10 of the same book. I've also asked this question on [Mathematics StackExchange][2].

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As noted below by Vinicius in the comments, if $u, v \in C^\infty_c(\mathbb{R})$ with $\operatorname{supp}(u) = [-1, 0]$ and $\operatorname{supp}(v) = [0, 1]$, then $uv = 0$ but $\operatorname{supp}(u)^c \cup \operatorname{supp}(v)^c = \mathbb{R}\setminus 0$, so open and dense is the most we can hope for.


  [1]: https://link.springer.com/book/10.1007/978-3-642-61497-2
  [2]: https://math.stackexchange.com/questions/4479812/product-of-distributions-under-wavefront-set-condition-is-zero