This question comes from Grigis-Sjorstrand's "Microlocal Analysis."
Let $X,Y$ be open subets of Euclidean space, and let $K \in C^\infty(X\times Y)$, and consider the associated operator $A$, defined initially on $C^\infty_c(Y)$ by $$Au(x) = \int_Y K(x,y)u(y)\ dy.$$
Show that $A$ extends to a continuous operator $A:\mathcal E'(Y) \to C^\infty(X)$. Here $\mathcal E'(Y) = (C^\infty(Y))'$ is the space of compactly supported distributions.
This should be a straightforward application of definitions, but I'm having some trouble. $A$ clearly has the correct codomain. I just need to derive an estimate of the following form, for $u \in C_c^\infty(Y)$: for any compact subset $S \subseteq X$ and $k \geq 0$, there exists $v_1,\ldots,v_n \in C^\infty(Y)$ such that $$||Au||_{C^k(S)} \leq C\max_{1 \leq i \leq n}|\langle u,v_i\rangle|.$$
All I have though is that $$||Au||_{C^k(S)} \leq \max_{|\alpha| \leq k}\sup_{x \in S} \left|\int \partial^\alpha_x K(x,y)u(y)\ dy\right|.$$
If $K$ were a finite-rank operator of the form $K(x,y) = \sum_{i=1}^k f_i(x)g_i(y)$ then this would certainly be sufficient. I've tried using density of such functions in $C^\infty$ but I don't see how to use it.