Schwartz kernel




Let X 1,X 2X_1, X_2 be two open subsets of Cartesian spaces. Then a smooth function KC (X 1×X 2)K \in C^\infty(X_1 \times X_2) on the Cartesian product of these two manifolds defines a linear function

C cp (X 2) C (X 1) ϕ 𝒦(ϕ), \array{ C^\infty_{cp}(X_2) &\overset{}{\longrightarrow}& C^\infty(X_1) \\ \phi &\mapsto& \mathcal{K}(\phi) } \,,

by the “integral transform”:

𝒦(ϕ):x 1 X 2K(x 1,x 2)ϕ(x 2)dvol(x 2) \mathcal{K}(\phi) \;\colon\; x_1 \mapsto \int_{X_2} K(x_1, x_2) \phi(x_2) \, dvol(x_2)

More generally, this expression makes sense for

K𝒟(X 1×X 2) K \;\in\; \mathcal{D}'(X_1 \times X_2)

a distribution on X 1×X 2X_1 \times X_2 (a “distribution of two variables”), which makes the result itself in general be a distribution

C cp (X 2) 𝒟(X 1) ϕ 𝒦(ϕ). \array{ C^\infty_{cp}(X_2) &\overset{}{\longrightarrow}& \mathcal{D}'(X_1) \\ \phi &\mapsto& \mathcal{K}(\phi) } \,.

Here KK is called the integral kernel and 𝒦(ϕ)\mathcal{K}(\phi) the corresponding integral transform.


Schwartz kernel theorem

The Schwarz kernel theorem states that this construction constitutes a linear isomorphism between Schwartz integral kernels and “distribution-valued distributions”

𝒟(X 1×X 2) 𝒟(X 2,𝒟(X 1)) K 𝒦 \array{ \mathcal{D}'(X_1 \times X_2) &\overset{\simeq}{\longrightarrow}& \mathcal{D}'( X_2, \mathcal{D}'(X_1) ) \\ K &\mapsto& \mathcal{K} }



(partial product of distributions of several variables)


K 1𝒟(X×Y)AAAK 2𝒟(Y×Z) K_1 \in \mathcal{D}'(X \times Y) \phantom{AAA} K_2 \in \mathcal{D}'(Y \times Z)

be two distributions of two variables. For their product of distributions to be defined over YY, Hörmander's criterion on the pair of wave front sets WF(K 1),WF(K 2)WF(K_1), WF(K_2) needs to hold for the wave front wave vectors along XX and YY taken to be zero.

If this is satisfied, then composition of integral kernels (if it exists)

(K 1K 2)(,)YK 1(,y)K 2(y,)dvol Y(y)𝒟(X×Z) (K_1 \circ K_2)(-,-) \;\coloneqq\; \underset{Y}{\int} K_1(-,y) K_2(y,-) dvol_Y(y) \;\in\; \mathcal{D}'(X \times Z)

has wave front set constrained by

WF(K 1K 2)WF(K 1)WF(K 2)(X×{0})×WF(K 2)WF(K 1)×(Z×{0}), WF(K_1 \circ K_2) \;\subset\; WF(K_1) \circ WF(K_2) \;\cup\; (X \times \{0\}) \times WF(K_2) \;\cup\; WF(K_1) \times (Z \times \{0\}) \,,

where on the left the composition symbol means composition of relations of wave vectors over points in YY.

(Hörmander 90, theorem 8.2.14)


  • Lars Hörmander, section 5.2 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990 (pdf)

See also

Last revised on November 9, 2018 at 06:14:25. See the history of this page for a list of all contributions to it.