Distributions on product spaces

2011-06-30T10:59:50Z

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.

Answer by Andrey Rekalo for Distributions on product spaces

2011-06-30T11:49:02Z

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".)

Distributions on product spaces - MathOverflow

Distributions on product spaces

Answer by Andrey Rekalo for Distributions on product spaces