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tensor product of distributions

Contents

Idea

The tensor product of distributions is the generalization to distributions of the tensor product of smooth functions, hence it defines for two distributions uβˆˆπ’Ÿβ€²(X)u \in \mathcal{D}'(X) and vβˆˆπ’Ÿβ€²(Y)v \in \mathcal{D}'(Y) a new distribution uβŠ—vβˆˆπ’Ÿβ€²(XΓ—Y)u \otimes v \in \mathcal{D}'(X \times Y) on the Cartesian product space which, as a generalized function behaves like (uβŠ—v)(x,y)=u(x)β‹…v(y)(u \otimes v)(x,y) = u(x) \cdot v(y).

Definition

Definition

(tensor product of smooth functions)

For X 1,X 2X_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:

C ∞(X 1)βŠ— ℝC ∞(X 2) β†ͺ C ∞(X 1Γ—X 2) (f 1,f 2) ↦ f 1βŠ—f 2 \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 1βŠ—f 2)(x 1,x 2)≔f 1(x 1)β‹…f 2(x 2). (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 1βˆˆπ’Ÿβ€²(X 1)u_1 \in \mathcal{D}'(X_1) and u 2βˆˆπ’Ÿβ€²(X 2)u_2 \in \mathcal{D}'(X_2) be distributions. Then there is a unique distribution of two variables u 1βŠ—u 2βˆˆπ’Ÿβ€²(X 1Γ—X 2)u_1 \otimes u_2 \in \mathcal{D}'(X_1 \times X_2) such that for all pairs of bump functions b 1∈C c ∞(x)b_1 \in C_c^\infty(x) and b 2∈C ∞(X 2)b_2 \in C^\infty(X_2) its value on their tensor product according to def. is

u(b 1βŠ—b 2)=u 1(b 1)β‹…u 2(b 2). u(b_1 \otimes b_2) = u_1(b_1) \cdot u_2(b_2) \,.

This u 1βŠ—u 2u_1 \otimes u_2 is called the tensor product of u 1u_1 with u 2u_2

(HΓΆrmander 90, theorem 5.1.1)

Properties

Example

(wave front set of tensor product distribution)

Let uβˆˆπ’Ÿβ€²(X)u \in \mathcal{D}'(X) and vβˆˆπ’Ÿβ€²(Y)v \in \mathcal{D}'(Y) be two distributions. then the wave front set of their tensor product distribution uβŠ—vβˆˆπ’Ÿβ€²(XΓ—Y)u \otimes v \in \mathcal{D}'(X \times Y) (def. ) satisfies

WF(uβŠ—v)βŠ‚(WF(u)Γ—WF(v))βˆͺ((supp(u)Γ—{0})Γ—WF(v))βˆͺ(WF(u)Γ—(supp(v)Γ—{0})), 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(βˆ’)supp(-) denotes the support of a distribution.

(HΓΆrmander 90, theorem 8.2.9)

References

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

Last revised on October 24, 2017 at 08:57:54. See the history of this page for a list of all contributions to it.