Theorem: Fixed-kernel Dini criterion

Status: Conditional; implication proved for displayed mathematical data.

Let $u_{\delta }\in {\mathcal{D}}^{\prime }(X;E)$ have scalar scale regularity. Suppose, on every compact support $H$,

Then $u_{\delta }$ has a strong distributional limit $u_0$, with

Proof. Scalar absolute continuity gives

The bound makes every pairing Cauchy. The anchor $u_{{\delta }_{\ast }}$ has a finite-order compact-local estimate; adding the derivative integral gives such an estimate for the limit. It is therefore a distribution. The tail estimate is uniform on bounded test sets and proves strong convergence.