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Integer resolvent commutator and localized first-order bounds

PDF p. 10: commutator identity and bounds in Lemma 3.3

Statement

Assume the Axiom of Choice. Let $Ω = R^{n} ∖ Θ$ , $n \geq 2$ , be the smooth connected exterior of a nonempty compact obstacle, and let $φ, ψ$ be the real cutoffs of Control region and nested real cutoffs. Let $Δ_{D}$ be the Dirichlet Laplacian, $A = I - Δ_{D}$ , $R = A^{- 1}$ , and $H_{D}^{s} (Ω)$ the spectral scale of Dirichlet spectral Sobolev scale and duality. Write $C = [Δ, φ]$ for the distributional first-order operator $C u = (Δ φ) u + 2 \nabla φ \cdot \nabla u = 2 div (u \nabla φ) - (Δ φ) u .$ It extends boundedly from $L^{2} (Ω)$ to $H_{D}^{- 1} (Ω)$ and from $H_{D}^{1} (Ω)$ to $L^{2} (Ω)$ , and, for $0 \leq α \leq 1$ , from $H_{D}^{α} (Ω)$ to $H_{D}^{α - 1} (Ω)$ . These extensions are consistent, and $C u = C (ψ u)$ for $u \in L^{2} (Ω)$ and hence for $u \in H_{D}^{α} (Ω)$ , $0 \leq α \leq 1$ .

For $- 2 \leq σ < 0$ , $τ = σ + 2$ , and $v \in H_{D}^{σ} (Ω)$ , put $y = R v$ and set $ρ (σ) = σ + 1$ when $- 1 < σ < 0$ , and $ρ (σ) = 0$ otherwise. Then, as an identity in $H_{D}^{τ} (Ω)$ , $φ R v - R (φ v) = - R C R v,$ where $C y$ is regarded as an element of $H_{D}^{σ}$ . Consequently there is a constant $C_{φ}$ independent of $v$ such that $∥ φ R v ∥_{H_{D}^{τ}} \leq ∥ φ v ∥_{H_{D}^{σ}} + C_{φ} ∥ ψ R v ∥_{H_{D}^{ρ (σ)}} .$

Facts & Assumptions

[F1]

The cutoffs are real and smooth, $φ$ is constant outside a compact annulus, and $ψ = 1$ on a neighborhood of $supp \nabla φ$ . Thus the derivatives of $φ$ occurring below are bounded and compactly supported, and $φ$ vanishes near the obstacle boundary. Control region and nested real cutoffs

[F2]

The Dirichlet realization has domain $H^{2} (Ω) \cap H_{0}^{1} (Ω)$ . The scale has $H_{D}^{0} = L^{2}$ , $H_{D}^{1} = H_{0}^{1}$ , and $H_{D}^{2} = D (A)$ ; $H_{D}^{- 1}$ is the pivot antidual of $H_{D}^{1}$ . The $H_{D}^{1}$ norm controls the $L^{2}$ norms of a function and its first weak derivatives. Dirichlet Laplacian and positive shift Dirichlet spectral Sobolev scale and duality Complex integer-order Sobolev notation

[F3]

Under Choice, $R : H_{D}^{s} \to H_{D}^{s + 2}$ is an onto isometry. The spectral scale embeds contractively from a higher index to a lower one, has the stated dense common core, and admits the endpoint operator interpolation bound. Scale isometries and free Schrödinger evolution Positive spectral Hilbert scale: duality and interpolation Spectral cutoff cores and unitary identifications of completed scales The Axiom of Choice

[F4]

Multiplication by $φ$ and $ψ$ is bounded on each $H_{D}^{s}$ for $- 2 \leq s \leq 2$ , consistently with ordinary multiplication on $L^{2}$ . Cutoff multipliers through order two and by duality

[F5]

Distributional derivatives are signed transposes of test-function derivatives on complex-linear distributions, with bilinear test pairing and no conjugation. In the pivot pairing below the distribution is evaluated at the conjugated test function. Distributional derivative

[F6]

The weak Leibniz rule applies on an arbitrary open set to a complex $H^{k} = W^{k, 2}$ function and a smooth multiplier whose classical derivatives through order $k$ are bounded, for $k = 1, 2$ . Its Countable Choice assumption follows here by applying the assumed Axiom of Choice to the family of values of a sequence of nonempty sets and composing the resulting choice function with that sequence. Weak Leibniz rule with a smooth factor The Axiom of Choice

[F7]

Under Countable Choice, a locally integrable complex function determines its regular distribution by $T_{u} (η) = \int_{Ω} u η$ , depending only on its almost-everywhere class. The Countable Choice needed here follows from the existing Choice assumption as explained in [F6]. Locally integrable functions as regular distributions The Axiom of Choice

[F8]

For a smooth complex multiplier $a$ and any complex distribution $T$ , multiplication is defined by $(a T) (η) = T (a η)$ , with no conjugation. The distributional Leibniz rule holds for every multi-index on any open set, without a choice assumption. Multiplication of a distribution by a smooth function Leibniz rule for distributions

Proof

[F1] The cutoffs are real and smooth, $φ$ is constant outside a compact annulus, and $ψ = 1$ on a neighborhood of $supp \nabla φ$ . Thus the derivatives of $φ$ occurring below are bounded and compactly supported, and $φ$ vanishes near the obstacle boundary. Control region and nested real cutoffs

[F2] The Dirichlet realization has domain $H^{2} (Ω) \cap H_{0}^{1} (Ω)$ . The scale has $H_{D}^{0} = L^{2}$ , $H_{D}^{1} = H_{0}^{1}$ , and $H_{D}^{2} = D (A)$ ; $H_{D}^{- 1}$ is the pivot antidual of $H_{D}^{1}$ . The $H_{D}^{1}$ norm controls the $L^{2}$ norms of a function and its first weak derivatives. Dirichlet Laplacian and positive shift Dirichlet spectral Sobolev scale and duality Complex integer-order Sobolev notation

[F3] Under Choice, $R : H_{D}^{s} \to H_{D}^{s + 2}$ is an onto isometry. The spectral scale embeds contractively from a higher index to a lower one, has the stated dense common core, and admits the endpoint operator interpolation bound. Scale isometries and free Schrödinger evolution Positive spectral Hilbert scale: duality and interpolation Spectral cutoff cores and unitary identifications of completed scales The Axiom of Choice

[F4] Multiplication by $φ$ and $ψ$ is bounded on each $H_{D}^{s}$ for $- 2 \leq s \leq 2$ , consistently with ordinary multiplication on $L^{2}$ . Cutoff multipliers through order two and by duality

[F5] Distributional derivatives are signed transposes of test-function derivatives on complex-linear distributions, with bilinear test pairing and no conjugation. In the pivot pairing below the distribution is evaluated at the conjugated test function. Distributional derivative

[F6] The weak Leibniz rule applies on an arbitrary open set to a complex $H^{k} = W^{k, 2}$ function and a smooth multiplier whose classical derivatives through order $k$ are bounded, for $k = 1, 2$ . Its Countable Choice assumption follows here by applying the assumed Axiom of Choice to the family of values of a sequence of nonempty sets and composing the resulting choice function with that sequence. Weak Leibniz rule with a smooth factor The Axiom of Choice

[F7] Under Countable Choice, a locally integrable complex function determines its regular distribution by $T_{u} (η) = \int_{Ω} u η$ , depending only on its almost-everywhere class. The Countable Choice needed here follows from the existing Choice assumption as explained in [F6]. Locally integrable functions as regular distributions The Axiom of Choice

[F8] For a smooth complex multiplier $a$ and any complex distribution $T$ , multiplication is defined by $(a T) (η) = T (a η)$ , with no conjugation. The distributional Leibniz rule holds for every multi-index on any open set, without a choice assumption. Multiplication of a distribution by a smooth function Leibniz rule for distributions

Given: The geometry, cutoffs, Choice assumption, scale, and indices in the Statement.

Goal: Establish the commutator extensions, localization, resolvent identity, and estimate.

1.1F1F2F5F6F7F8

For $u \in L^{2}$ , Cauchy–Schwarz gives $\int_{K} ∣ u ∣ \leq ∣ K ∣^{1 / 2} ∥ u ∥_{2}$ for every compact $K \subset Ω$ , so $u \in L_{l o c}^{1}$ and [F7] identifies it with a regular distribution. Smooth multiplication in [F8] agrees with pointwise multiplication on such distributions, since $(a T_{u}) (η) = \int_{Ω} u a η = T_{a u} (η)$ . Apply the distributional Leibniz rule in [F8] with $a = φ$ and $α = 2 e_{j}$ in each coordinate: $\partial_{j}^{2} (φ u) - φ \partial_{j}^{2} u = (\partial_{j}^{2} φ) u + 2 (\partial_{j} φ) \partial_{j} u$ . Apply its first-order case also to $(\partial_{j} φ) u$ and sum over $j$ . These finite identities give $Δ (φ u) - φ Δ u = (Δ φ) u + 2 \nabla φ \cdot \nabla u = 2 div (u \nabla φ) - (Δ φ) u$ in distributions. Thus defining $C u$ by this divergence expression using [F5] gives exactly $[Δ, φ] u$ , even when $u$ has no $L^{2}$ derivatives. For $ζ \in C_{c}^{\infty} (Ω)$ its pivot pairing is $- 2 \int_{Ω} u \nabla φ \cdot \nabla \overline{ζ} - \int_{Ω} (Δ φ) u \overline{ζ}$ . Cauchy–Schwarz and boundedness of the coefficients bound its absolute value by a constant times $∥ u ∥_{2} ∥ ζ ∥_{H_{D}^{1}}$ . Density of test functions in $H_{0}^{1}$ therefore defines a bounded antidual element $C u \in H_{D}^{- 1}$ . For $u \in H_{D}^{1} = H_{0}^{1}$ , apply [F6] with $k = 1$ and the smooth multipliers $\partial_{j} φ$ , whose derivatives through order one are bounded by [F1]. Summing the resulting identities gives the first displayed expression and its $L^{2}$ norm is bounded by a constant times $∥ u ∥_{H_{D}^{1}}$ . The two definitions agree as distributions, hence as pivot-antidual elements.

2.1F1F3F4F5step 1.1

Apply the scale operator interpolation bound to these consistent endpoint maps. The intermediate indices are $α$ and $α - 1$ , yielding the asserted bounds for $0 < α < 1$ ; the endpoints are step 1.1. Since $ψ = 1$ on a neighborhood of the support of $\nabla φ$ , the functions $(1 - ψ) \nabla φ$ and $(1 - ψ) Δ φ$ vanish, and also $\nabla ψ$ vanishes on the support of $\nabla φ$ . The divergence formula then gives $C u = C (ψ u)$ for $u \in L^{2}$ in distributions. It follows at every stated nonnegative scale level by consistency of the embeddings and extensions.

2.2F1F2F3F6step 1.1

For $v \in L^{2}$ , let $y = R v \in D (A)$ . Apply [F6] with $k = 2$ , $u = y \in H^{2}$ and $η = φ$ ; [F1] supplies every required bounded classical derivative, including $φ$ itself. This gives $φ y \in H^{2}$ and its weak Laplacian formula; approximation in $H_{0}^{1}$ by compactly supported smooth functions shows $φ y \in H_{0}^{1}$ . Thus $φ y \in D (A)$ and $A (φ y) = φ A y - C y$ in $L^{2}$ . Applying $R$ gives $φ R v - R (φ v) = - R C R v$ . Here $C y \in L^{2}$ by step 1.1.

3.1F3F4step 1.1step 2.1step 2.2∎

Let $- 2 \leq σ < 0$ . If $τ \leq 1$ , then $y \in H_{D}^{τ} \subset L^{2}$ , and steps 1.1 and 2.1 give $∥ C y ∥_{H_{D}^{σ}} \leq ∥ C y ∥_{H_{D}^{- 1}} \leq C ∥ ψ y ∥_{2}$ , because $σ \leq - 1$ . If $1 < τ < 2$ , put $α = τ - 1 = ρ (σ)$ . Then $y \in H_{D}^{α}$ and interpolation and localization give $∥ C y ∥_{H_{D}^{σ}} \leq C ∥ ψ y ∥_{H_{D}^{α}}$ because $α - 1 = σ$ . These estimates also show that $C R : H_{D}^{σ} \to H_{D}^{σ}$ is bounded. The common spectral core lies in $L^{2}$ and is dense in $H_{D}^{σ}$ ; the multiplier and resolvent maps on both sides of step 2.2 are bounded on the required levels. Passing to the limit proves the identity in $H_{D}^{τ}$ . Finally, use the triangle inequality, the isometry $R : H_{D}^{σ} \to H_{D}^{τ}$ , and the preceding bounds for $C y$ . This proves the displayed estimate.

Source notes

The article is cited for attribution of the commutator identity and its application in the proof of Lemma 3.3, PDF page 10. The argument above uses the stated mathematical suppliers and does not use the article as proof of itself.

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Nonlinear Schrödinger equation in an exterior domain, v2