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Proposition 2.1: cut-off resolvent estimate

PDF p. 6: Proposition 2.1; (2.1)

Statement

Assume the Axiom of Choice. Let $n \geq 2$ and let $Θ \subset R^{n}$ be a nonempty compact set with smooth boundary and connected exterior $Ω = R^{n} ∖ Θ$ . Suppose 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 on $Ω$ , with domain $H^{2} (Ω) \cap H_{0}^{1} (Ω)$ , and let $H_{D}^{s} (Ω)$ be the spectral scale of $I - Δ_{D}$ .

For every real-valued $χ \in C_{c}^{\infty} (R^{n})$ and every $s \in [- 1, 1]$ , there are $C > 0$ and $ε_{0} > 0$ such that, for every $κ \in R$ and $0 < ε \leq ε_{0}$ , $∥ χ (- Δ_{D} - (κ \pm i ε)^{2})^{- 1} χ ∥_{H_{D}^{s} (Ω) \to H_{D}^{s + 1} (Ω)} \leq C .$ The cutoffs act by multiplication after restriction to $Ω$ ; the operator norm refers to the bounded extension between the indicated scale spaces. The constants may depend on $Θ$ , $χ$ and $s$ , but not on $κ$ or $ε$ .

Statement references

Exterior and truncated obstacle domains

Non-trapping obstacle

Dirichlet spectral Sobolev scale and duality

Nontrapping cutoff resolvent estimates

The Axiom of Choice

Facts & Assumptions

[F1]

The exterior geometry, smooth boundary convention and generalized-ray non-trapping condition are those of Exterior and truncated obstacle domains and Non-trapping obstacle.

[F2]

The Dirichlet operator and its scale, including the pivot-antidual interpretation at negative orders, are those of Dirichlet spectral Sobolev scale and duality. This scale and the following suppliers retain the Axiom of Choice The Axiom of Choice.

[F3]

For every real nonnegative $ρ \in C_{c}^{\infty} (R^{n})$ , the non-trapping cutoff resolvent input Nontrapping cutoff resolvent estimates gives the displayed uniform $H_{D}^{s} \to H_{D}^{s + 1}$ estimate for $- 1 \leq s \leq 1$ , with constants independent of $κ$ and sufficiently small positive $ε$ .

[F4]

Multiplication by a smooth compactly supported function is bounded on $H_{D}^{t} (Ω)$ for $- 2 \leq t \leq 2$ , with the negative-order action defined by pivot duality and agreeing with ordinary multiplication on $L^{2} (Ω)$ Endpoint multiplier bounds extended to negative spectral orders.

[F5]

For $0 < r < R$ , there is a smooth $[0, 1]$ -valued function on $R^{n}$ equal to one on ${\overline{B}}_{r} (0)$ and supported in $B_{R} (0)$ A smooth bump between concentric Euclidean balls.

Proof

[F1] The exterior geometry, smooth boundary convention and generalized-ray non-trapping condition are those of Exterior and truncated obstacle domains and Non-trapping obstacle.

[F2] The Dirichlet operator and its scale, including the pivot-antidual interpretation at negative orders, are those of Dirichlet spectral Sobolev scale and duality. This scale and the following suppliers retain the Axiom of Choice The Axiom of Choice.

[F3] For every real nonnegative $ρ \in C_{c}^{\infty} (R^{n})$ , the non-trapping cutoff resolvent input Nontrapping cutoff resolvent estimates gives the displayed uniform $H_{D}^{s} \to H_{D}^{s + 1}$ estimate for $- 1 \leq s \leq 1$ , with constants independent of $κ$ and sufficiently small positive $ε$ .

[F4] Multiplication by a smooth compactly supported function is bounded on $H_{D}^{t} (Ω)$ for $- 2 \leq t \leq 2$ , with the negative-order action defined by pivot duality and agreeing with ordinary multiplication on $L^{2} (Ω)$ Endpoint multiplier bounds extended to negative spectral orders.

[F5] For $0 < r < R$ , there is a smooth $[0, 1]$ -valued function on $R^{n}$ equal to one on ${\overline{B}}_{r} (0)$ and supported in $B_{R} (0)$ A smooth bump between concentric Euclidean balls.

Given: The obstacle, cutoff $χ$ , and order $s$ in the Statement.

Goal: Establish the stated uniform cutoff-resolvent bound.

1.1F5givenchoose

Since $supp χ$ is compact, choose $0 < r < R$ with $supp χ \subset B_{r} (0)$ . By [F5], choose a real $ρ \in C_{c}^{\infty} (R^{n})$ with $0 \leq ρ \leq 1$ and $ρ = 1$ on $supp χ$ . Thus $χ ρ = ρ χ = χ$ .

2.1F1F2F3step 1.1given

The hypotheses of [F3] hold for $ρ$ by [F1] and step 1.1. Hence $T_{κ, ε}^{\pm} = ρ (- Δ_{D} - (κ \pm i ε)^{2})^{- 1} ρ$ has norm at most a constant $C_{ρ}$ from $H_{D}^{s}$ to $H_{D}^{s + 1}$ uniformly for $κ \in R$ and $0 < ε \leq ε_{0}$ .

3.1F2F4step 1.1step 2.1given∎

Both $s$ and $s + 1$ lie in $[- 2, 2]$ , so [F4] bounds multiplication $M_{χ}$ on both $H_{D}^{s}$ and $H_{D}^{s + 1}$ . On $L^{2} (Ω)$ the identity $M_{χ} T_{κ, ε}^{\pm} M_{χ} = χ (- Δ_{D} - (κ \pm i ε)^{2})^{- 1} χ$ follows from $χ ρ = χ$ . The bounded composition consequently supplies the stated extension, since the common spectral core is dense in $H_{D}^{s}$ . Its norm is at most $∥ M_{χ} ∥_{H_{D}^{s + 1} \to H_{D}^{s + 1}} C_{ρ} ∥ M_{χ} ∥_{H_{D}^{s} \to H_{D}^{s}}$ , independent of $κ$ and $ε$ . This proves the estimate.

Source notes

The original article is cited for the statement and attribution to its reference [9]; it is not used as proof of the estimate. The resolvent estimate for a nonnegative cutoff is the independent imported input recorded in Nontrapping cutoff resolvent estimates. The multiplier composition above is the local argument extending that input to the article's real-valued cutoffs.

Depends on

Used by · 2

Dependency tree · 2 levels

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2