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Resolvent adjoint transport to order minus two

Statement

Assume the Axiom of Choice. Let $n \geq 2$ , and let $Θ \subset R^{n}$ be a nonempty compact obstacle with smooth boundary and connected exterior $Ω = R^{n} ∖ Θ$ . Assume it is nontrapping 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 and let $H_{D}^{s}$ be the spectral Hilbert scale of $A = I - Δ_{D}$ , with negative orders identified by the $L^{2} (Ω)$ pivot antidual pairing. Let $η \in C_{c}^{\infty} (R^{n})$ be real, and write $K_{z} = η (- Δ_{D} - z)^{- 1} η$ on $L^{2} (Ω)$ whenever the resolvent exists.

There are $C > 0$ and $ϵ_{0} > 0$ , depending on the obstacle and $η$ , such that for every real $λ$ , $0 < ϵ \leq ϵ_{0}$ , and either sign, $K_{(λ \pm i ϵ)^{2}}$ extends uniquely to a bounded operator $H_{D}^{- 2} \to H_{D}^{- 1}$ of norm at most $C$ . For every $z$ with $Re z < 0$ , the same extension satisfies $∥ K_{z} ∥_{H_{D}^{- 2} \to H_{D}^{- 1}} \leq C (1 + ∣ Re z ∣)^{- 1 / 2}$ . The constants are uniform in the indicated parameters.

Statement references

Resolvent and spectrum of an unbounded operator

Exterior and truncated obstacle domains

Non-trapping obstacle

Dirichlet spectral Sobolev scale and duality

The Axiom of Choice

Facts & Assumptions

[F1]

The exterior Dirichlet realization is self-adjoint, with $- Δ_{D} \geq 0$ Exterior Dirichlet realization, H² domain and form domain.

[F2]

The scale is the completion of the common spectral core; its negative spaces are isometrically the pivot antiduals of the corresponding positive spaces Dirichlet spectral Sobolev scale and duality, Positive spectral Hilbert scale: duality and interpolation.

[F3]

For a nonnegative smooth compactly supported cutoff $χ$ , the imported cutoff resolvent estimate applies from $H_{D}^{1}$ to $H_{D}^{2}$ , uniformly on the two squared-parameter families and with the stated negative-real-part decay Nontrapping cutoff resolvent estimates.

[F4]

Multiplication by a smooth compactly supported function is bounded on each $H_{D}^{s}$ for $- 2 \leq s \leq 2$ ; on negative orders it is defined by the pivot pairing with the conjugate multiplier Endpoint multiplier bounds extended to negative spectral orders.

[F5]

The self-adjoint Borel calculus identifies the adjoint of a bounded spectral multiplier with the multiplier by its complex conjugate Unbounded Borel functional calculus: domains, products, spectral mapping. The assumed Axiom of Choice supplies the choice hypotheses of the cited scale and calculus results The Axiom of Choice.

[F6]

The explicit smooth-cutoff construction gives $ρ \in C_{c}^{\infty} (R^{n})$ with $0 \leq ρ \leq 1$ , $ρ = 1$ for $∣ x ∣ \leq 1$ , and $ρ = 0$ for $∣ x ∣ \geq 2$ ; for every $R > 0$ , $ρ (x / R)$ is again smooth and compactly supported Explicit compactly supported smooth cutoffs.

[F7]

For $B = - Δ_{D}$ on the complex Hilbert space $L^{2} (Ω)$ , the resolvent set $ρ (B)$ consists of those $z \in C$ for which $z I - B : D (B) \to L^{2} (Ω)$ is bijective with bounded everywhere defined inverse $R_{B} (z) = (z I - B)^{- 1}$ ; the spectrum is $σ (B) = C ∖ ρ (B)$ Resolvent and spectrum of an unbounded operator. Since $B - z I = - (z I - B)$ , the inverse used here is $(B - z I)^{- 1} = - R_{B} (z)$ , with the same resolvent set and the same operator norm.

Proof

Given: The geometry, nontrapping assumption, scale, cutoff and parameter ranges in the Statement.

Goal: Construct the stated negative-order extensions and prove their bounds.

1.1F3F4F6given

Since $η$ has compact support, the increasing open balls $B_{m} (0)$ , $m \geq 1$ , cover $supp η$ , and a finite subcover gives $R > 0$ with $supp η \subset B_{R} (0)$ ; if the support is empty, take $R = 1$ . Take $ρ$ from [F6] and set $χ (x) = ρ (x / R)$ . Then $χ \in C_{c}^{\infty} (R^{n})$ , $0 \leq χ \leq 1$ , and $χ = 1$ on $B_{R} (0)$ , an open neighborhood of $supp η$ . Thus $η χ = η$ . The imported estimate applied to $χ$ at order $s = 1$ , together with the multiplier bounds on $H_{D}^{1}$ and $H_{D}^{2}$ , gives $∥ K_{w} ∥_{H_{D}^{1} \to H_{D}^{2}} \leq C$ for $w = (λ \pm i ϵ)^{2}$ in its stated parameter range, and $∥ K_{w} ∥_{H_{D}^{1} \to H_{D}^{2}} \leq C (1 + ∣ Re w ∣)^{- 1 / 2}$ for $Re w < 0$ . Indeed, on $L^{2}$ , $K_{w} = M_{η} (χ (- Δ_{D} - w)^{- 1} χ) M_{η}$ ; bounded extensions give the same identity on $H_{D}^{1}$ . The constants absorb the two multiplier norms.

1.2F1F5F7given

The spectrum of $- Δ_{D}$ lies in $[0, \infty)$ . Every parameter in the Statement and its complex conjugate is therefore in the resolvent set: the squared parameter is nonreal unless $λ = 0$ , when it is $- ϵ^{2} < 0$ . The Borel calculus gives $((- Δ_{D} - z)^{- 1})^{*} = (- Δ_{D} - \overline{z})^{- 1}$ on $L^{2}$ . Since $η$ is real, $M_{η}^{*} = M_{η}$ , so $K_{z}^{*} = K_{\overline{z}}$ on $L^{2}$ . Both parameter families are closed under conjugation and preserve the stated bound.

2.1F2step 1.1step 1.2∎

For $f \in H_{D}^{- 2}$ define a conjugate-linear functional on $H_{D}^{1}$ by $v \mapsto ⟨ f, K_{\overline{z}} v ⟩_{- 2, 2}$ . Step 1.1 bounds its norm by $∥ f ∥_{H_{D}^{- 2}} ∥ K_{\overline{z}} ∥_{H_{D}^{1} \to H_{D}^{2}}$ . Pivot antiduality therefore gives a unique $T f \in H_{D}^{- 1}$ with $⟨ T f, v ⟩_{- 1, 1} = ⟨ f, K_{\overline{z}} v ⟩_{- 2, 2}$ and the same operator bound. For $f \in L^{2}$ this pairing equals $⟨ K_{z} f, v ⟩_{2}$ by step 1.2, so $T$ extends the original $K_{z}$ . The spectral core is dense in $H_{D}^{- 2}$ and lies in $L^{2}$ , making this extension unique. Applying the bounds of step 1.1 to $\overline{z}$ proves both asserted estimates.

Source notes

The negative-order statement is a local adjoint consequence of the cited positive-order imported estimate. The imported supplier retains its own source attribution and reading limits; no negative-order estimate is attributed directly to that source here.

Depends on

Used by · 2

Dependency tree · 2 levels

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