## Distributions on product spaces

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.

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It's been a while but I'm pretty sure that Treves's book contains the tensorproduct construction. – Johannes Hahn Jun 30 2011 at 11:22

According to the Schwartz Kernel Theorem and its variants, there are the canonical isomorphisms $$\mathcal{D}^{\prime} \left(X\right)\tilde\otimes \mathcal{D}^{\prime} \left(Y\right)\simeq\mathcal{D}^{\prime}\left(X\times Y\right),$$ $$\mathcal{E}^{\prime} \left(X\right)\tilde\otimes \mathcal{E}^{\prime} \left(Y\right)\simeq\mathcal{E}^{\prime} \left(X\times Y\right),$$ $$\mathcal{S}^{\prime} \left(\mathbb R^n\right)\tilde\otimes \mathcal{S}^{\prime} \left(\mathbb R^m\right)\simeq\mathcal{S}^{\prime} \left(\mathbb R^{n+m}\right),$$ where $E\tilde\otimes F$ is the completion of the space $E\otimes F$.

Roughly speaking, this follows from the fact that the corresponding spaces of test functions $\mathcal{D}$, $\mathcal{C}^{\infty}$, and $\mathcal{S}$ are nuclear Fréchet spaces, and one has the canonical isomorphisms $$E^{\prime}\tilde\otimes F^{\prime}\simeq \left(E\tilde\otimes F\right)^{\prime}\simeq L(E; F'),$$ provided that $E$ and $F$ are nuclear Fréchet spaces. (Here the duals carry the strong dual topology and the space $L(E;F ')$ of continuous linear mappings is endowed with the topology of bounded convergence.)

As Johannes mentioned in his comment, a detailed presentation of the Schwartz Kernel Theorem and its versions for various spaces of distributions can be found in Topological Vector Spaces, Distributions and Kernels by Trèves. (More specifically, take a look at Chapt. 51, "Examples of Nuclear Spaces. The Kernels Theorem".)

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Interesting - this seems to be a slightly stronger Schwartz kernel theorem than the one I know (Theorem 6.1.1 in Friedlander/Joshi). So there is no way to avoid nuclear theory if I want to see distributions as coalgebra? – darij grinberg Jun 30 2011 at 13:47
Well, the topological tensor products and nuclear spaces approach to the theory of distributions goes back to Schwartz and Grothendieck, I think. The original proof of the Schwartz kernel theorem was obtained at the time when the TVS-based methodology had been 'in vogue'. Since then several more or less elementary presentations have appeared. See e.g. the note by Gask mscand.dk/article.php?id=1584 Hope, it helps. – Andrey Rekalo Jun 30 2011 at 15:10
In the same vein as the last comment by Andrey, for the case of S' the easiest proof I know is in the article by Barry Simon "Distributions and their Hermite expansions", J. Mathematical Phys. 12 1971 140–148. You can also look at the appendix to section V.3 of vol 1 of Reed & Simon. – Abdelmalek Abdesselam Jun 30 2011 at 15:41
Andrey: how exactly does the theorem in your link (which is more or less equivalent to the version of the kernel theorem I have in mind) yield the properties of tensor products you brought up in your post? – darij grinberg Jun 30 2011 at 15:52
Now, while it's true that $L(\mathcal{D} \left(X\right);\mathcal{D}^{\prime} \left(Y\right))\simeq\mathcal{D}^{\prime} \left(X\right)\tilde\otimes \mathcal{D}^{\prime} \left(Y\right)$, it's not shown in the note. I believe that the latter isomorphism can be established without invoking the theory of nuclear spaces, but I cannot give you a precise reference. That's what they call folklore, I think. – Andrey Rekalo Jul 1 2011 at 15:40