I hope this is suitable to MO.

**Question.** Let $X$ and $Y$ be two open sets in $\mathbb R^n$ and $\mathbb R^m$, respectively. In what sense can we consider $\mathcal{D}^{\prime} \left(X\times Y\right)$ as a tensor product of $\mathcal{D}^{\prime} \left(X\right)$ and $\mathcal{D}^{\prime} \left(Y\right)$ (where $\mathcal{D}^{\prime}$ means distributions of the standard kind, i. e., those acting on $\mathcal{C}^{\infty}$ functions of compact support)? What if $\mathcal{D}^{\prime}$ is replaced by $\mathcal{E}^{\prime}$ (distributions with compact support) or $\mathcal{S}^{\prime}$ (tempered distributions)?

**Remarks.** I am trying to understand in how far distributions form a coalgebra, and what can be derived from this viewpoint. The applicability of coalgebras to distribution theory seems to be one of the selling points of coalgebra and Hopf algebra theory, but I have yet to see a place where this is actually elaborated upon and applied to yield nontrivial results. "I have yet to see" does not mean much, though, as I am a complete greenhorn at analysis, and there is not much literature avaliable on the coalgebra side.