nLab tensor product of distributions

Idea

The tensor product of distributions is the generalization to distributions of the tensor product of smooth functions, hence it defines for two distributions $u \in \mathcal{D}'(X)$ and $v \in \mathcal{D}'(Y)$ a new distribution $u \otimes v \in \mathcal{D}'(X \times Y)$ on the Cartesian product space which, as a generalized function behaves like $(u \otimes v)(x,y) = u(x) \cdot v(y)$.

Definition

Definition

(tensor product of smooth functions)

For $X_1, X_2$ two open subsets of some Cartesian space, there is an injection from the tensor product of the real vector spaces of smooth functions on the separate spaces to that on the Cartesian product space:

$$\array{ C^\infty(X_1) \otimes_{\mathbb{R}} C^\infty(X_2) &\overset{}{\hookrightarrow}& C^\infty(X_1 \times X_2) \\ (f_1, f_2) &\mapsto& f_1 \otimes f_2 }$$

with

$$(f_1 \otimes f_2)(x_1, x_2) \coloneqq f_1(x_1) \cdot f_2(x_2) \,.$$

Proposition

(tensor product of distributions)

Let $u_1 \in \mathcal{D}'(X_1)$ and $u_2 \in \mathcal{D}'(X_2)$ be distributions. Then there is a unique distribution of two variables $u_1 \otimes u_2 \in \mathcal{D}'(X_1 \times X_2)$ such that for all pairs of bump functions $b_1 \in C_c^\infty(x)$ and $b_2 \in C^\infty(X_2)$ its value on their tensor product according to def. is

$$u(b_1 \otimes b_2) = u_1(b_1) \cdot u_2(b_2) \,.$$

This $u_1 \otimes u_2$ is called the tensor product of $u_1$ with $u_2$

(Hörmander 90, theorem 5.1.1)

Properties

Example

(wave front set of tensor product distribution)

Let $u \in \mathcal{D}'(X)$ and $v \in \mathcal{D}'(Y)$ be two distributions. then the wave front set of their tensor product distribution $u \otimes v \in \mathcal{D}'(X \times Y)$ (def. ) satisfies

$$WF(u \otimes v) \;\subset\; \left( WF(u) \times WF(v) \right) \cup \left( \left( supp(u) \times \{0\} \right) \times WF(v) \right) \cup \left( WF(u) \times \left( supp(v) \times \{0\} \right) \right) \,,$$

where $supp(-)$ denotes the support of a distribution.

(Hörmander 90, theorem 8.2.9)

Related concepts

• Schwartz kernel theorem

References

• Lars Hörmander, section 5.1 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990 (pdf)