nLab Schwartz kernel

Definition

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

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

by the “integral transform”:

$$\mathcal{K}(\phi) \;\colon\; x_1 \mapsto \int_{X_2} K(x_1, x_2) \phi(x_2) \, dvol(x_2)$$

More generally, this expression makes sense for

$$K \;\in\; \mathcal{D}'(X_1 \times X_2)$$

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

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

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

Properties

Schwartz kernel theorem

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

$$\array{ \mathcal{D}'(X_1 \times X_2) &\overset{\simeq}{\longrightarrow}& \mathcal{D}'( X_2, \mathcal{D}'(X_1) ) \\ K &\mapsto& \mathcal{K} }$$

$\,$

Proposition

(partial product of distributions of several variables)

Let

$$K_1 \in \mathcal{D}'(X \times Y) \phantom{AAA} K_2 \in \mathcal{D}'(Y \times Z)$$

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

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

$$(K_1 \circ K_2)(-,-) \;\coloneqq\; \underset{Y}{\int} K_1(-,y) K_2(y,-) dvol_Y(y) \;\in\; \mathcal{D}'(X \times Z)$$

has wave front set constrained by

$$WF(K_1 \circ K_2) \;\subset\; WF(K_1) \circ WF(K_2) \;\cup\; (X \times \{0\}) \times WF(K_2) \;\cup\; WF(K_1) \times (Z \times \{0\}) \,,$$

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

(Hörmander 90, theorem 8.2.14)

Related concepts

• tensor product of distributions

• polynomial observable, microcausal observable

• correlators as differential forms on configuration spaces of points

References

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

See also

• Wikipedia, Schwartz kernel theorem