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Cutoff multipliers through order two and by duality

PDF p. 8: paragraph following (3.4)

Statement

Assume the Axiom of Choice. Let $n \geq 2$ , let $Θ \subset R^{n}$ be nonempty and compact with smooth boundary, and let its exterior $Ω = R^{n} ∖ Θ$ be connected. Let $φ, χ, ψ$ be the real smooth cutoffs of Control region and nested real cutoffs. Put $A = I - Δ_{D}$ , where $Δ_{D}$ is the Dirichlet Laplacian, and let $H_{D}^{σ} (Ω)$ be the spectral scale of Dirichlet spectral Sobolev scale and duality. Thus for $σ \geq 0$ it is $D (A^{σ / 2})$ with norm $∥ A^{σ / 2} u ∥_{2}$ ; for $s > 0$ , $H_{D}^{- s}$ is the pivot antidual of $H_{D}^{s}$ , the space of bounded conjugate-linear scalar functionals under the extended $L^{2}$ pairing.

For every $σ \in [- 2, 2]$ , multiplication by each of $φ, χ, ψ$ extends to a bounded complex-linear operator on $H_{D}^{σ} (Ω)$ . For negative orders the extension is determined by $⟨ a f, u ⟩_{- s, s} = ⟨ f, a u ⟩_{- s, s}$ for $a \in {φ, χ, ψ}$ , $f \in H_{D}^{- s}$ and $u \in H_{D}^{s}$ .

Facts & Assumptions

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

[F1]

The cutoffs are smooth and real-valued; $φ$ is constant outside the annulus $R < ∣ x ∣ < R + 1$ , while $χ$ and $ψ$ are compactly supported. They are constant near the obstacle boundary. Control region and nested real cutoffs

[F2]

$W^{2, \infty} (Ω)$ consists of functions with essentially bounded weak derivatives through order two, including the function itself. Complex integer-order Sobolev notation

[F3]

Under Choice, the spectral cutoff lemma gives bounded multiplication on every $H_{D}^{σ}$ , $- 2 \leq σ \leq 2$ , for each $a \in W^{2, \infty} (Ω)$ ; its negative-order action is defined using multiplication by $\overline{a}$ on the positive space. Endpoint multiplier bounds extended to negative spectral orders The Axiom of Choice

[F4]

The scale and its pivot pairing are those defined in Dirichlet spectral Sobolev scale and duality.

[F5]

Assuming Countable Choice, the classical derivatives through order two of a $C^{2}$ function on an open subset of $R^{n}$ are its weak derivatives. Classical derivatives agree with weak derivatives The stated Axiom of Choice implies Countable Choice. The Axiom of Choice implies countable choice

Proof

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

[F1] The cutoffs are smooth and real-valued; $φ$ is constant outside the annulus $R < ∣ x ∣ < R + 1$ , while $χ$ and $ψ$ are compactly supported. They are constant near the obstacle boundary. Control region and nested real cutoffs

[F2] $W^{2, \infty} (Ω)$ consists of functions with essentially bounded weak derivatives through order two, including the function itself. Complex integer-order Sobolev notation

[F3] Under Choice, the spectral cutoff lemma gives bounded multiplication on every $H_{D}^{σ}$ , $- 2 \leq σ \leq 2$ , for each $a \in W^{2, \infty} (Ω)$ ; its negative-order action is defined using multiplication by $\overline{a}$ on the positive space. Endpoint multiplier bounds extended to negative spectral orders The Axiom of Choice

[F4] The scale and its pivot pairing are those defined in Dirichlet spectral Sobolev scale and duality.

[F5] Assuming Countable Choice, the classical derivatives through order two of a $C^{2}$ function on an open subset of $R^{n}$ are its weak derivatives. Classical derivatives agree with weak derivatives The stated Axiom of Choice implies Countable Choice. The Axiom of Choice implies countable choice

Given: The hypotheses and cutoffs in the Statement.

Goal: Establish the bounded multiplier operators and the stated negative-order action.

1.1F1F2F5given

Every derivative of $φ$ of positive order is supported in the closed annulus $R \leq ∣ x ∣ \leq R + 1$ ; smoothness makes each such derivative bounded there. The functions $χ$ and $ψ$ and all their derivatives are bounded because they are smooth and compactly supported. By [F5], these classical derivatives through order two are the weak derivatives on the open set $Ω$ ; hence the restrictions of all three cutoffs to $Ω$ lie in $W^{2, \infty} (Ω)$ .

2.1F3F4step 1.1∎

Apply the spectral cutoff lemma to each restriction and to the scale in the Statement. It gives the bounded operator for every $σ \in [- 2, 2]$ . For $s > 0$ , its antidual formula contains $\overline{a}$ ; since each cutoff is real, this is $a$ , giving the displayed formula. The lemma also identifies this extension with ordinary multiplication on $L^{2}$ , so the operators agree with the original multiplier action.

Source notes

The original article is cited for attribution of the cutoff statement on PDF page 8. The deduction above uses the independently supplied cutoff multiplier lemma; constancy near the boundary is sufficient here but is not required by that lemma through order two. No multiplier assertion at higher Dirichlet orders is made.

Depends on

Used by · 17

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2