If $u \in \mathscr S'(\mathbb{R}^n)$ and $v \in \mathscr S'(\mathbb{R}^m)$ are tempered distributions, then we can identify them with elements of $\mathscr D'(\mathbb{R}^n)$ and $\mathscr D'(\mathbb{R}^m)$ respectively. This allows us to form the tensor product $u \otimes v$, which is then well-defined as an element of $\mathscr D'(\mathbb{R}^n \times \mathbb{R}^m)$.
Question: Is $u \otimes v$ a tempered distribution? More precisely, does it extend to a sequentially continuous linear form on $\mathscr S(\mathbb{R}^n \times \mathbb{R}^m)$?
This extension will certainly then be unique by density of $\mathscr C_c^\infty$ in $\mathscr S$, given that it exists.
Context: I was wondering if one can extend the Fourier transform identity $\mathscr F(u\otimes v) = \mathscr F u \otimes \mathscr Fv$, which holds for Schwartz functions, to the space of tempered distributions. However, for this identity to even make sense the tensor product should first be shown to be a tempered distribution.