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Proposition 2.2: retarded local smoothing

PDF p. 7: Proposition 2.2; (2.2)

Statement

Assume the Axiom of Choice. Let $n \geq 2$ and let $Ω = R^{n} ∖ Θ$ be the connected exterior of a nonempty compact obstacle with smooth boundary. Assume that $Θ$ is non-trapping in the complete-ray sense: for every complete unit-speed admissible generalized reflected or glancing ray $x : R \to \overline{Ω}$ and every compact $K \subset R^{n}$ , there is $t \geq 0$ with $x (t) \notin K$ . Let $Δ_{D}$ be the Dirichlet Laplacian, $H_{D}^{s} (Ω)$ the spectral scale of $I - Δ_{D}$ , and $W (t) = e^{i t Δ_{D}}$ .

For every $χ \in C_{c}^{\infty} (R^{n})$ there is a constant $C > 0$ such that, for every $T > 0$ , $s \in [- 1, 1]$ , and $f \in L^{2} (0, T; H_{D}^{s} (Ω))$ , the retarded evolution $u (t) = \int_{0}^{t} W (t - r) χ f (r) d r$ satisfies $∥ χ u ∥_{L^{2} (0, T; H_{D}^{s + 1} (Ω))} \leq C ∥ χ f ∥_{L^{2} (0, T; H_{D}^{s} (Ω))} .$ The cutoff acts by multiplication after restriction to $Ω$ . The constant may depend on the obstacle and cutoff, but is independent of $T$ , $s$ , and $f$ . The time spaces are Bochner spaces and the integral is a Bochner integral in $H_{D}^{s}$ .

Facts & Assumptions

[F1]

The smooth exterior geometry and generalized-ray non-trapping convention are specified by Exterior and truncated obstacle domains and Non-trapping obstacle.

[F2]

The Dirichlet spectral scale, including its negative-order pivot-antidual interpretation, is specified by Dirichlet spectral Sobolev scale and duality. Its flow $W (t)$ is a strongly continuous unitary group on every scale level Scale isometries and free Schrödinger evolution. The spaces $L^{2} (0, T; H_{D}^{s})$ and their Bochner integrals use Continuous and mixed time-space function spaces.

[F3]

For each smooth compactly supported cutoff $η$ , Imported homogeneous and retarded exterior smoothing gives the causal estimate with input $η h$ , output cutoff $η$ , and orders $- 1 \leq s \leq 1$ ; its constant is independent of $T$ and the data, and may depend on the order.

[F4]

Multiplication by a smooth compactly supported function is bounded on $H_{D}^{q}$ for $- 2 \leq q \leq 2$ ; the negative-order action is the compatible pivot-dual extension Endpoint multiplier bounds extended to negative spectral orders.

[F5]

A smooth compactly supported function equal to one on a prescribed sufficiently large closed ball exists by A smooth bump between concentric Euclidean balls.

[F6]

The lifted spectral-scale interpolation theorem gives the exact Bochner-space interpolation identity and the geometric-mean endpoint operator bound for maps whose source and target are such time-space scales Lifted spectral measure and isometric Bochner scale interpolation. The assumptions of the cited scale and interpolation results include the Axiom of Choice The Axiom of Choice.

[F7]

Scalar integral Cauchy–Schwarz gives $\int_{0}^{t} a (r) d r \leq t^{1 / 2} ∥ a ∥_{L^{2} (0, t)}$ for each nonnegative $a \in L^{2} (0, t)$ , by pairing $a$ with the constant function $1$ in Cauchy-Schwarz inequality for L².

Proof

[F1] The smooth exterior geometry and generalized-ray non-trapping convention are specified by Exterior and truncated obstacle domains and Non-trapping obstacle.

[F2] The Dirichlet spectral scale, including its negative-order pivot-antidual interpretation, is specified by Dirichlet spectral Sobolev scale and duality. Its flow $W (t)$ is a strongly continuous unitary group on every scale level Scale isometries and free Schrödinger evolution. The spaces $L^{2} (0, T; H_{D}^{s})$ and their Bochner integrals use Continuous and mixed time-space function spaces.

[F3] For each smooth compactly supported cutoff $η$ , Imported homogeneous and retarded exterior smoothing gives the causal estimate with input $η h$ , output cutoff $η$ , and orders $- 1 \leq s \leq 1$ ; its constant is independent of $T$ and the data, and may depend on the order.

[F4] Multiplication by a smooth compactly supported function is bounded on $H_{D}^{q}$ for $- 2 \leq q \leq 2$ ; the negative-order action is the compatible pivot-dual extension Endpoint multiplier bounds extended to negative spectral orders.

[F5] A smooth compactly supported function equal to one on a prescribed sufficiently large closed ball exists by A smooth bump between concentric Euclidean balls.

[F6] The lifted spectral-scale interpolation theorem gives the exact Bochner-space interpolation identity and the geometric-mean endpoint operator bound for maps whose source and target are such time-space scales Lifted spectral measure and isometric Bochner scale interpolation. The assumptions of the cited scale and interpolation results include the Axiom of Choice The Axiom of Choice.

[F7] Scalar integral Cauchy–Schwarz gives $\int_{0}^{t} a (r) d r \leq t^{1 / 2} ∥ a ∥_{L^{2} (0, t)}$ for each nonnegative $a \in L^{2} (0, t)$ , by pairing $a$ with the constant function $1$ in Cauchy-Schwarz inequality for L².

Given: The obstacle, cutoff $χ$ , time $T$ , order $s$ , and function $f$ in the statement.

Goal: Prove the displayed retarded estimate with one constant independent of $T$ and $s$ .

1.1F2F4F5F7given

Choose a real $η \in C_{c}^{\infty} (R^{n})$ equal to one on a neighborhood of $supp χ$ , using a ball containing that support and [F5]. Thus $χ η = χ$ . For each $T$ , define $S_{T} g (t) = χ \int_{0}^{t} W (t - r) η g (r) d r$ . Strong continuity of $W$ on the relevant scale and bounded cutoff multiplication give strongly measurable integrands for $g$ in either endpoint time-space. Applying [F7] to $a (r) = ∥ η g (r) ∥_{H_{D}^{q}}$ , with $q = - 1, 1$ , gives $\int_{0}^{t} ∥ W (t - r) η g (r) ∥_{H_{D}^{q}} d r \leq t^{1 / 2} ∥ η g ∥_{L^{2} (0, t; H_{D}^{q})} < \infty$ by unitarity, so the stated Bochner integrals exist.

2.1F3F4step 1.1

Apply [F3] with cutoff $η$ and input $g$ at each endpoint $q = - 1, 1$ . Since $χ η = χ$ , multiplying its output by $χ$ and using [F4] gives $∥ S_{T} g ∥_{L^{2} H_{D}^{q + 1}} \leq M_{q} ∥ g ∥_{L^{2} H_{D}^{q}}$ , where $M_{q}$ is independent of $T$ . The constants include the bounded multiplication norms of $χ$ on $H_{D}^{q + 1}$ and $η$ on $H_{D}^{q}$ . The endpoint definitions agree on their common domain because they use the same integral and compatible scale actions.

3.1F6step 2.1

For $- 1 < s < 1$ , set $θ = (s + 1) / 2$ . Apply [F6] to the consistent endpoint map $S_{T}$ , with source orders $- 1, 1$ and target orders $0, 2$ . It gives $∥ S_{T} g ∥_{L^{2} H_{D}^{s + 1}} \leq M_{- 1}^{1 - θ} M_{1}^{θ} ∥ g ∥_{L^{2} H_{D}^{s}}$ . This bound and the endpoint bounds are all at most $\max (M_{- 1}, M_{1})$ , independently of $s$ and $T$ .

4.1F4step 1.1step 2.1step 3.1given∎

Put $g = χ f$ . By [F4], $g$ belongs to $L^{2} (0, T; H_{D}^{s})$ and $η g = g$ : the identity holds on the common spectral core and extends by the compatible bounded multiplier actions, including at negative orders. Hence $S_{T} g = χ u$ . Substitution into the bound from step 3.1, with the endpoint cases from step 2.1, proves the claimed estimate with $C = \max (M_{- 1}, M_{1})$ .

Source notes

The original article is cited for attribution of Proposition 2.2 and its equation (2.2). The causal smoothing input is the independent Burq–Gérard–Tzvetkov result recorded in Imported homogeneous and retarded exterior smoothing; the uniformity in the order range is obtained by the endpoint interpolation above. No Christ–Kiselev argument is used.

Depends on

Used by · 1

Lemma 4.1: retarded smoothing down to negative order two

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2