Theorem: Fixed-kernel distributional criterion

Fix $K_{\ast }\in {\mathcal{K}}_{\mathrm{int}}$. Suppose every test pairing of $t_{\delta }={\mathsf{F}}_{\delta }(K_{\ast })$ has a limit, and for every compact $L⋐U$ there are $C_L,m_L$ such that

uniformly in $\delta$. Then $t_{\delta }$ converges strongly to a distribution $t$.

This conclusion concerns $K_{\ast }$ only, not a continuous map on all of ${\mathcal{K}}_{\mathrm{int}}$.

Proof. The limit inherits the local finite-order estimates, so is a distribution. On a bounded test set supported in $L$, the tests are totally bounded in the $C^{m_L}$ norm. Use a finite net, the uniform estimate, and convergence on its finitely many tests to obtain uniform convergence on the bounded set.