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Local observation has a lower initial regularity cost

PDF p. 9: (3.10) and definition of θ; PDF p. 10: positive-index interpolation argument

Statement

Assume the Axiom of Choice and hypothesis (H): $n \geq 2$ , $Θ \subset R^{n}$ is nonempty and compact with smooth boundary, its exterior $Ω = R^{n} ∖ Θ$ is connected, and $Θ$ is non-trapping. Let $Δ_{D}$ be the Dirichlet Laplacian, $A = I - Δ_{D}$ , and $H_{D}^{s} (Ω)$ the completed spectral scale with norm $∥ A^{s / 2} u ∥_{L^{2} (Ω)}$ . Write $W (t) = e^{i t Δ_{D}}$ and let $χ$ be the smooth compactly supported cutoff in the control-cutoff construction. Non-trapping means that each admissible generalized reflected ray, including glancing behavior, leaves each compact set in finite forward time.

For $0 < τ < 2$ , put $θ = 0$ when $0 < τ < 1 / 2$ , $θ = τ - 1 / 2$ when $1 / 2 \leq τ \leq 3 / 2$ , and $θ = 2 τ - 2$ when $3 / 2 < τ < 2$ . Then $0 \leq θ < τ$ . For every $T > 0$ there is $C > 0$ , depending on the fixed geometry, cutoff, $T$ and $τ$ but not on $v_{0}$ , such that every $v_{0} \in H_{D}^{θ} (Ω)$ satisfies

$\int_{0}^{T} ∥ χ W (t) v_{0} ∥_{H_{D}^{τ} (Ω)}^{2} d t \leq C ∥ v_{0} ∥_{H_{D}^{θ} (Ω)}^{2} .$

For $3 / 2 < τ < 2$ , the interpolation parameter is $a = 2 τ - 3 \in (0, 1)$ , and the relevant complex interpolation identities are $[H_{D}^{1}, H_{D}^{2}]_{a} = H_{D}^{2 τ - 2}$ and $[H_{D}^{3 / 2}, H_{D}^{2}]_{a} = H_{D}^{τ}$ .

Statement references

Exterior and truncated obstacle domains

Non-trapping obstacle

Control region and nested real cutoffs

Dirichlet Laplacian and positive shift

Dirichlet spectral Sobolev scale and duality

Continuous and mixed time-space function spaces

Scale isometries and free Schrödinger evolution

Positive spectral Hilbert scale: duality and interpolation

The Axiom of Choice

Facts & Assumptions

[F1]

The geometry, generalized-ray non-trapping convention and compact smooth cutoff are as stated in Exterior and truncated obstacle domains, Non-trapping obstacle and Control region and nested real cutoffs.

[F2]

Under Choice, $A$ is self-adjoint and $A \geq I$ ; the spectral scale is the completion of $D (A^{\infty})$ in the displayed norm, with the usual domain identification at nonnegative indices. The spectral calculus gives the norm formula and spectral support in $[1, \infty)$ . def-se-dirichlet-laplacian Dirichlet spectral Sobolev scale and duality Unbounded Borel functional calculus: domains, products, spectral mapping The Axiom of Choice

[F3]

Under these hypotheses, homogeneous local smoothing gives $∥ χ W (\cdot) v_{0} ∥_{L^{2} (0, T; H_{D}^{s + 1 / 2})} \leq C_{s} ∥ v_{0} ∥_{H_{D}^{s}}$ for $0 \leq s \leq 1$ . The time-space norm is the Bochner $L^{2}$ norm, namely the square root of the integral of the squared spatial norm. Homogeneous half-derivative local smoothing Continuous and mixed time-space function spaces

[F4]

Multiplication by $χ$ is bounded on $H_{D}^{2}$ , and $W (t)$ is a strongly continuous unitary group on $H_{D}^{2}$ . Cutoff multipliers through order two and by duality Scale isometries and free Schrödinger evolution

[F5]

The complex interpolation convention uses bounded continuous sum-valued functions on the closed strip, holomorphic inside, with endpoint-continuous boundary values vanishing at imaginary infinity, and the infimum of the maximum endpoint supremum norm. Under Choice, interpolation of the positive spectral scale is exact, and a consistent linear operator bounded at two scale endpoints is bounded at interpolated endpoints. The same operator bound applies to $L^{2}$ -in-time spectral-scale targets. Positive spectral Hilbert scale: duality and interpolation The Axiom of Choice

Proof

[F1] The geometry, generalized-ray non-trapping convention and compact smooth cutoff are as stated in Exterior and truncated obstacle domains, Non-trapping obstacle and Control region and nested real cutoffs.

[F2] Under Choice, $A$ is self-adjoint and $A \geq I$ ; the spectral scale is the completion of $D (A^{\infty})$ in the displayed norm, with the usual domain identification at nonnegative indices. The spectral calculus gives the norm formula and spectral support in $[1, \infty)$ . def-se-dirichlet-laplacian Dirichlet spectral Sobolev scale and duality Unbounded Borel functional calculus: domains, products, spectral mapping The Axiom of Choice

[F3] Under these hypotheses, homogeneous local smoothing gives $∥ χ W (\cdot) v_{0} ∥_{L^{2} (0, T; H_{D}^{s + 1 / 2})} \leq C_{s} ∥ v_{0} ∥_{H_{D}^{s}}$ for $0 \leq s \leq 1$ . The time-space norm is the Bochner $L^{2}$ norm, namely the square root of the integral of the squared spatial norm. Homogeneous half-derivative local smoothing Continuous and mixed time-space function spaces

[F4] Multiplication by $χ$ is bounded on $H_{D}^{2}$ , and $W (t)$ is a strongly continuous unitary group on $H_{D}^{2}$ . Cutoff multipliers through order two and by duality Scale isometries and free Schrödinger evolution

[F5] The complex interpolation convention uses bounded continuous sum-valued functions on the closed strip, holomorphic inside, with endpoint-continuous boundary values vanishing at imaginary infinity, and the infimum of the maximum endpoint supremum norm. Under Choice, interpolation of the positive spectral scale is exact, and a consistent linear operator bounded at two scale endpoints is bounded at interpolated endpoints. The same operator bound applies to $L^{2}$ -in-time spectral-scale targets. Positive spectral Hilbert scale: duality and interpolation The Axiom of Choice

Given: The geometry, Choice assumption, $T > 0$ , $0 < τ < 2$ , the cutoff, scale and initial datum in the statement.

Goal: Prove the lower-order estimate and the displayed interpolation identities.

1.1F1F2F3given

Since the spectral measure of $A$ is supported in $[1, \infty)$ , the spectral norm formula gives $∥ u ∥_{H_{D}^{r}} \leq ∥ u ∥_{H_{D}^{q}}$ for $0 \leq r \leq q$ and $u \in H_{D}^{q}$ . If $0 < τ < 1 / 2$ , apply this with $r = τ$ , $q = 1 / 2$ to the output of smoothing with $s = 0$ . If $1 / 2 \leq τ \leq 3 / 2$ , apply smoothing with $s = τ - 1 / 2 \in [0, 1]$ . These two cases give the required estimate after squaring, with $θ$ as defined.

1.2F2F3F4

For the remaining range, define the consistent linear observation operator $B v_{0} : t \mapsto χ W (t) v_{0}$ . Smoothing at $s = 1$ bounds $B : H_{D}^{1} \to L^{2} (0, T; H_{D}^{3 / 2})$ . The multiplier bound and unitarity give $∥ B v_{0} ∥_{L^{2} (0, T; H_{D}^{2})} \leq T^{1 / 2} M ∥ v_{0} ∥_{H_{D}^{2}}$ , where $M$ is the $H_{D}^{2}$ multiplier norm. Strong continuity ensures that this endpoint path is Bochner measurable. Both endpoint maps agree on $H_{D}^{2} \subset H_{D}^{1}$ , so they define one consistent operator on the compatible sum.

2.1F3F5step 1.2

Set $a = 2 τ - 3 \in (0, 1)$ . The scale interpolation identities give $[H_{D}^{1}, H_{D}^{2}]_{a} = H_{D}^{1 + a} = H_{D}^{2 τ - 2}$ and $[H_{D}^{3 / 2}, H_{D}^{2}]_{a} = H_{D}^{3 / 2 + a / 2} = H_{D}^{τ}$ . Interpolating the endpoint bounds of step 1.2, using the time-target form of the operator theorem, therefore bounds $B : H_{D}^{2 τ - 2} \to L^{2} (0, T; H_{D}^{τ})$ . Squaring gives the asserted integral estimate for this range.

3.1givenstep 1.1step 2.1∎

In the first range $θ = 0 < τ$ ; in the middle range $θ = τ - 1 / 2 \geq 0$ and $θ < τ$ ; in the final range $θ = 2 τ - 2 > 0$ and $θ < τ$ because $τ < 2$ . Thus the estimates in steps 1.1 and 2.1 cover every stated order and complete the proof.

Source notes

The original article is cited for attribution of (3.10), its piecewise index and the interpolation calculation on PDF pages 9–10. The proof above uses the stated mathematical suppliers; the article is not used as a proof premise.

Depends on

Used by · 1

Lemma 3.3: observability modulo a compact remainder

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2