Is there some regularizing version of Schwartz kernel theorem for topological spaces, i.e., in the form of

Every continuous linear map $A\colon C\prime(X_2) \to C(X_1)$ is given by a kernel $k \in C(X_1 \times X_2)$?

Here $C(\cdot)$ is the space of continuous functions, equipped with a suitable topology, and $C\prime(\cdot)$ its dual space. The topological spaces $X_1$ and $X_2$ may be as nice as you want (e.g., locally compact Hausdorff).

Note that I need the kernel to be continuous (this is the reason why the operator $A$ is per assumption regularizing).

Or maybe I have to use other spaces like $C_c(\cdot)$ (compactly supported continuous functions) in the formulation of the theorem? Because I'm not even sure about the right formulation of such a theorem.

Thanks in advance, Alex