Lemma: Leading boundary-layer estimates

For $1\le m=|\gamma |\le 5$, uniformly on $|d(S)|\le \Lambda \delta$ in the fixed non-tip domain,

The scale derivative here is at fixed $S$. Also $J_{\delta }$ and its scale derivative are bounded by $C\Vert w{\Vert }_0$.

For $d(S)>4\delta$, propagation derivatives and their scale derivatives are bounded by $C_m\Vert w{\Vert }_{2m}$. For $d(S)<-4\delta$, they vanish.

Proof. Differentiate $J=\Theta (d)\Gamma$. The unique highest boundary term of order $m$ is

The other boundary terms contain ${{\delta }_{\mathrm{D}}}^{(j)}(d)$, $j\le m-2$, and the interior term is bounded by Lemma lem:regularity.

Put $H=P+Q$, and solve the translated boundary in $H^0$:

Temporal coarea contributes ${\delta }^{-m}$ for the highest term. The profile density gives $H_{\ast }^{(m-1)}$, the normal gives $k(S{)}^{\gamma }$, and the coefficient gives ${\mathcal{G}}_{\overline{S}}[w]$. Their changes are $O(\delta )$ in the required temporal derivatives. Lemma lem:regularity, together with the shifted-weight Taylor estimate, bounds these changes by $C\delta \Vert w{\Vert }_{2m}$. Lower boundary terms are at most $O({\delta }^{1-m})$; interior terms are bounded.

For the scale derivative, replace $\rho$ by

Its normal marginal is

and

This is the fixed-$S$ scale derivative of the leading term. The same estimates apply to every lower boundary and interior term for this compact signed profile.

Finally,

on the profile support. For $d>4\delta$, only smooth timelike coefficient derivatives occur; for $d<-4\delta$, all translated arguments are outside causal support.