Let $U \subset \mathbb{R}^n$ be open, and let all test functions and distributions be complex-valued. Let $R: C_c^\infty(U) \to \mathcal{D}'(U)$ be a continuous linear operator. Let $K_R \in \mathcal{D}'(U \times U)$ denote its Schwartz kernel, with the convention that
for all $\phi, \psi \in C_c^\infty(U)$. Assume that $R$ is properly supported, meaning that if $\Gamma := \operatorname{supp} K_R \subset U \times U$, then both coordinate projections $\Gamma \to U$ are proper.
paragraph
admin
Then $K_R$ is represented by a function $K \in C^\infty(U \times U)$ if and only if there exists a unique continuous [linear map](/page/Linear%20Map)
from $\mathcal{D}'(U)$ with its strong dual topology to $C^\infty(U)$ with its usual Fréchet topology such that, for every $\phi \in C_c^\infty(U)$, the [regular distribution](/page/Regular%20Distribution) associated to $\widetilde R(T_\phi)$ equals $R\phi$, where $T_\phi \in \mathcal{D}'(U)$ is defined by