Theorem: Quantitative scale-integrability criterion

Let $t_{\delta }$ be a strongly smooth distribution-valued family on $0<\delta \le {\delta }_0$, with $D_{\delta }=\delta {\partial }_{\delta }{t}_{\delta }$. Suppose that for every $L⋐U$, there are $m_L,C_L$ and a nonnegative measurable $a_L$ such that

and

Then $t_{\delta }$ converges strongly to a distribution $t_0$, with

If $a_L(s)=s^{\alpha }$, $\alpha >0$, then

and, for dyadic scales within the parameter interval,

The same integral controls parameters between adjacent dyadic scales.

Proof. For $0<{\delta }_1<{\delta }_2\le {\delta }_0$,

The hypothesis gives a Cauchy pairing for each test. The limit is a distribution: locally its value is bounded by the finite-order estimate for $t_{{\delta }_0}$ plus the finite increment bound, increasing the test order if necessary. Taking ${\delta }_1\downarrow 0$ gives (eq:Dini-remainder) uniformly on bounded test sets. The positive-power estimates follow by integration.