Imported homogeneous and retarded exterior smoothing

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 the obstacle is non-trapping in the complete-ray sense: for every complete unit-speed admissible generalized reflected or glancing ray $x:\mathbb{R}\to \bar{\mathrm{\Omega }}$ and every compact $K\subset {\mathbb{R}}^n$, there is $t\ge 0$ with $x(t)\mathrm{\notin }K$. Let ${\mathrm{\Delta }}_D$ be the Dirichlet Laplacian on $L^2(\mathrm{\Omega })$, let $H_D^s(\mathrm{\Omega })$ be the spectral scale of $I-{\mathrm{\Delta }}_D$, and let $\chi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$. Here $C_c^{\mathrm{\infty }}$ means smooth functions with compact support, restricted to $\mathrm{\Omega }$ when multiplying functions on the exterior.

For every $T>0$, every $s\in [-1,1]$, and every $f\in L^2((0,T);H_D^s(\mathrm{\Omega }))$, define $u(t)={\int }_0^t{e}^{i(t-\tau ){\mathrm{\Delta }}_D}\chi f(\tau )d\tau$. Then $\mathrm{\parallel }\chi u{\mathrm{\parallel }}_{L^2((0,T);H_D^{s+1}(\mathrm{\Omega }))}\le C\mathrm{\parallel }\chi f{\mathrm{\parallel }}_{L^2((0,T);H_D^s(\mathrm{\Omega }))}$.

For every $s\in [0,1]$ and $v_0\in H_D^s(\mathrm{\Omega })$, put $v(t)=e^{it{\mathrm{\Delta }}_D}{v}_0$. Then $\mathrm{\parallel }\chi v{\mathrm{\parallel }}_{L^2((0,T);H_D^{s+1\mathrm{/}2}(\mathrm{\Omega }))}\le C\mathrm{\parallel }{v}_0{\mathrm{\parallel }}_{H_D^s(\mathrm{\Omega })}$. In each estimate $C$ may depend on the obstacle, cutoff and order, but is independent of $T$ and the data. No compact-support assumption is imposed on $v_0$.

Exterior and truncated obstacle domains

Non-trapping obstacle

Dirichlet spectral Sobolev scale and duality

The Axiom of Choice

The two estimates and their independence from $T$ are imported from Burq–Gérard–Tzvetkov, Proposition 2.7, equations (2.14) and (2.15), and Remark 2.8. The source treats the nonempty nontrapping smooth obstacle in dimension at least two and uses the Dirichlet spectral scale. Its retarded estimate applies for $-1\le s\le 1$; its homogeneous estimate applies for $0\le s\le 1$. The proposition and remark were read for these assertions; their proof is not reconstructed here.

This local argument supplies the constructions used in the Statement. The two smoothing estimates and their time-uniform constants remain the external input recorded above; their proof is not claimed here.

Given: The geometry, Choice assumption, cutoff, scale, time and data in the Statement.

Goal: Justify cutoff multiplication, the time spaces, and the retarded integral without using either smoothing estimate as a premise.

1.1 Fix $s\in [-1,1]$ and set $E=H_D^s(\mathrm{\Omega })$, a complex Hilbert space by the scale construction. The restricted cutoff has bounded derivatives through order two, so [F5] defines a bounded multiplier $M_{\chi }:E\to E$, including its specified antidual extension when $s<0$. If simple functions $f_j$ converge almost everywhere in $E$ to a representative of $f$, then $M_{\chi }{f}_j$ are simple and converge there to $g=M_{\chi }f$. Hence $g$ is strongly measurable and belongs to $L^2((0,T);E)$ with norm at most $\mathrm{\parallel }{M}_{\chi }\mathrm{\parallel }\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2((0,T);E)}$. By [F7] it belongs to $L^1((0,T);E)$. The other cutoff expressions in the Statement use the same compatible multiplier at orders within $[-2,2]$. [F3, F4, F5, F7, given]

1.2 By [F6], $W$ is a strongly continuous unitary group on $E$, with uniform bound $M=1$ on $[-T,T]$. Choice supplies the Countable Choice assumption of [F8]. Apply [F8] with base time $0$ and forcing $g$ from step 1.1: the integrand is strongly measurable and its norm is integrable, and $u(t)={\int }_0^{t}W(t-\tau )g(\tau )d\tau$ is a well-defined Bochner integral in $E$, independent of the representative of $f$, and continuous in $t$. Its integral is the complex Bochner integral of [F4] by the identical simple-sum construction in [F8]. The homogeneous orbit is continuous by [F6], so its cutoff is also strongly measurable at its initial scale level. The higher output regularities and estimates are precisely the imported assertions, rather than premises of this construction. [F3, F4, F5, F6, F8, step 1.1] ∎