Lemma 4.1: retarded smoothing down to negative order two

PDF p. 17: Lemma 4.1; (4.1)

Assume the Axiom of Choice. Let $n\ge 2$ and let $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$ be the connected exterior of a nonempty compact obstacle with smooth boundary. Assume that $\mathrm{\Theta }$ is nontrapping: every generalized reflected ray leaves each compact set in finite forward time. Let ${\mathrm{\Delta }}_D$ be the Dirichlet Laplacian, $H_D^s(\mathrm{\Omega })$ the spectral scale of $I-{\mathrm{\Delta }}_D$, with negative orders understood by pivot antiduality, and $W(t)=e^{it{\mathrm{\Delta }}_D}$.

For every $T>0$, real $\eta \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ and $s\in [-2,1]$, there is $C>0$ such that, whenever $g=\eta f\in L^2(0,T;H_D^s(\mathrm{\Omega }))$, the retarded integral $u(t)={\int }_0^{t}W(t-r)g(r)dr$ satisfies $\mathrm{\parallel }\eta u{\mathrm{\parallel }}_{L^2(0,T;H_D^{s+1}(\mathrm{\Omega }))}\le C\mathrm{\parallel }g{\mathrm{\parallel }}_{L^2(0,T;H_D^s(\mathrm{\Omega }))}\mathrm{.}$ The time spaces and integral are Bochner spaces and a Bochner integral. Cutoffs act by their bounded extensions on the indicated scale spaces. The constant may depend on $T$, $s$, the obstacle and the cutoff, but not on $f$.

The original article is cited for attribution of Lemma 4.1 and equation (4.1). The proof above uses the two stated suppliers; their independent inputs and source reading limits are recorded in those items. The explicit Choice assumption records the hypotheses of those suppliers and is not attributed to the article.