Theorem: Microlocal upgrade

Let $v_{\delta }=t_{\delta }-S_{\delta }\to T$ strongly, and let $\Gamma \subset T^{\ast }U\setminus 0$ be closed and conic. Assume $v_{\delta }\in {{\mathcal{D}}^{\prime }}_{\Gamma }(U;E_{\mathrm{out}})$. For every $\zeta \in {C_c}^{\infty }(U)$, closed frequency cone $V_0\subset {\mathbb{R}}^{16}\setminus 0$ with

and integer $r\ge 0$, suppose

uniformly for small scales, componentwise.

Then

and $v_{\delta }\to T$ in the normal topology associated with $\Gamma$.

Proof. Strong convergence gives convergence of each localized Fourier value. Passing to the limit in the uniform rapid-decay bounds proves $\mathrm{WF}(T)\subset \Gamma$.

For the order-$r$ Fourier seminorm of the difference, the uniform order-$r+1$ bounds make the tail above radius $R$ bounded by $C/(1+R)$, for both $v_{\delta }$ and $T$. Within that radius, the localized exponential tests form a bounded test set, so convergence is uniform. First choose $R$ large, then let $\delta \downarrow 0$. Together with the strong seminorms, this proves normal convergence.