I'm having a bit of problem filling in the gap for Theorem 5.2.6. in Hormander's first volume on linear PDE. It says that if $\kappa \in \mathcal{C}^{\infty}(X_1 \times X_2)$ is a smooth function then there exists a continuous operator $K: \mathcal{E}'(X_2) \rightarrow \mathcal{C}^{\infty}(X_1)$ with Schwartz kernel $\kappa$. Now this operator is constructed by the formula $Ku(x_1) = \langle u, \kappa(x_1, \cdot) \rangle$ and it is clear for me that this is a map $ \mathcal{E}'(X_2) \rightarrow \mathcal{C}^{\infty}(X_1)$. But I cannot get the continuity statement. From my understanding here I need to show that if $u_j \rightarrow u$ in $\mathcal{E}'(X_2)$ with the weak star topology then we must have $$\sup_{x_1 \in K} |\langle u - u_j, \partial^{\alpha}_{x_1} \kappa(x_1,\cdot) \rangle| \rightarrow 0$$ for all compact subsets $K \subset X_1$ and multiindex $\alpha$.
I know that perhaps one possible way to get this is to show that $\{ \partial^{\alpha}_{x_1}\kappa(x_1,\cdot) \ / \ x_1 \in K \} $ is compact, then a theorem from Reed and Simons says that the uniform convergence holds. However Hormander seems to hint that this is elementary. So I must be missing something quiet easy here. I would appreciate if somebody can point out how do I obtain this.
Many thanks!
Let $f_j^{\alpha}(x \in K)=\langle u-u_j,\, \partial_{x_1}^{\alpha}\kappa(x_1,\cdot)\rangle$.
It’s not hard to see that $f_j^{\alpha}$ is continuous, bounded, and actually differentiable wrt each coordinate, with derivative wrt the $k$-th coordinate being $f_j^{\alpha+e_k}$.
The last part is because, for $x \in K$, in $C^{\infty}(L)$ (where $L$ is a compact subset the interior of which contains the supports of $u$ and $u_j$), $\frac{\partial_{x_1}^{\alpha}\kappa(x+he_k,\cdot)-\partial_{x_1}^{\alpha}\kappa(x,\cdot)}{h} \rightarrow \partial_{x_1}^{\alpha+e_k}\kappa(x,\cdot)$. This converges itself follows from easy Taylor estimates on the derivatives of $\kappa$.
Therefore, the $f_j$ converge pointwise to zero and are equi-Lipschitz continuous on $K$ compact. It’s then an elementary topology exercise to show from scratch that $f_j$ converges uniformly. You can also, of course, use Ascoli or Dini (the latter on the $\sup_{l \geq j}\,|f_l^{\alpha}|$ which are continuous).