Schwartz kernel theorem - Imported Knowlpedia Content

Statement

Let $X\subseteq\mathbb R^m$ and $Y\subseteq\mathbb R^n$ be open. The Schwartz kernel theorem states that every continuous linear map

A:\mathcal D(Y)\longrightarrow\mathcal D'(X)

from the test-function space on $Y$ to the distributions on $X$ has a unique distribution $K_A\in\mathcal D'(X\times Y)$ such that

\langle A\varphi,\psi\rangle =\langle K_A,\psi\otimes\varphi\rangle

for all $\varphi\in\mathcal D(Y)$ and $\psi\in\mathcal D(X)$ . Conversely, every such $K_A$ determines a continuous map $A$ . The distribution $K_A$ is called the Schwartz kernel of $A$ .

Bilinear formulation

Equivalently, every separately continuous bilinear functional on $\mathcal D(X)\times\mathcal D(Y)$ is evaluation against a unique distribution on $X\times Y$ . The operator formulation follows by applying this statement to $(\psi,\varphi)\mapsto\langle A\varphi,\psi\rangle$ . Separate continuity is the natural hypothesis here; on test-function spaces it supplies the hypocontinuity needed in the tensor-product formulation Trèves, Chapter 51.

Role of nuclearity

The theorem reflects the nuclearity of the test-function space. Nuclearity identifies the completed tensor-product topology needed to pass from bilinear functionals on $\mathcal D(X)\times\mathcal D(Y)$ to continuous linear functionals on $\mathcal D(X\times Y)$ . This mechanism extends the theorem to several other nuclear function spaces.

Examples and scope

An integral operator with kernel $k(x,y)$ has Schwartz kernel given by the regular distribution induced by $k$ . Differential operators have kernels supported on the diagonal, typically derivatives of the delta distribution. The theorem does not assert that every kernel is a function: singular distributional kernels are essential. On manifolds or for vector bundles, one uses the corresponding test sections and density conventions.

References

Laurent Schwartz, Théorie des distributions, Hermann, 1966. Bibliographic record. Relevant: the kernel theorem for distributions.
François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. Publisher record. Relevant: Chapter 51 on the kernel theorem.