Local negative Sobolev space in the compactness argument

PDF p. 9: Lemma 3.2 proof (a)

Assume Countable Choice. Let ${\mathrm{\Omega }}^{\mathrm{\prime }}=\mathrm{\Omega }\cap B_{R+3}(0)$ be the bounded open truncated exterior used for the localization. The complex Sobolev space $H^2({\mathrm{\Omega }}^{\mathrm{\prime }})$ consists of the $L^2$ classes with all weak derivatives through order two in $L^2$, with norm $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^2}=({\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le 2}\mathrm{\parallel }{D}^{\alpha }v{\mathrm{\parallel }}_{L^2}^2{)}^{1\mathrm{/}2}$. Let $H_0^2({\mathrm{\Omega }}^{\mathrm{\prime }})$ be the closure of $C_c^{\mathrm{\infty }}({\mathrm{\Omega }}^{\mathrm{\prime }};\mathbb{C})$ in that norm.

In the localized compactness argument, write $H^{-2}({\mathrm{\Omega }}^{\mathrm{\prime }})=(H_0^2({\mathrm{\Omega }}^{\mathrm{\prime }}){)}^{\mathrm{\prime }}$ for the space of bounded conjugate-linear functionals $F:H_0^2({\mathrm{\Omega }}^{\mathrm{\prime }})\to \mathbb{C}$. Equip it with the norm $\mathrm{\parallel }F{\mathrm{\parallel }}_{H^{-2}}={\sup }_{\mathrm{\parallel }v{\mathrm{\parallel }}_{H^2}\le 1}\mathrm{\mid }F(v)\mathrm{\mid }$. Its evaluation pairing $⟨F,v⟩=F(v)$ is linear in $F$ and conjugate-linear in $v$. Here the prime denotes this antidual; the continuous linear dual is identified with it by conjugating functional values.

Control region and nested real cutoffs

Complex integer-order Sobolev notation

The dual space X^* of a normed space and its dual norm

The open set and localization notation are supplied by Control region and nested real cutoffs. The Sobolev classes, norm, test functions and closure are defined in Complex integer-order Sobolev notation under Countable Choice; in particular, the closure definition does not assert a boundary trace characterization.

For each bounded conjugate-linear $F$, the functional $v\mapsto \overline{F(v)}$ is bounded and complex-linear, with the same supremum norm. Conversely, conjugating the values of a bounded complex-linear functional gives a bounded conjugate-linear functional. Thus this operation is an isometric conjugate-linear bijection with the continuous linear dual of The dual space X^* of a normed space and its dual norm, and the displayed norm and evaluation pairing are well-defined.

The interior distribution convention of Distributional equation convention for complex Schrödinger solutions uses a complex-linear test-function pairing without conjugation. It is a separate convention from the antidual evaluation above; the definition alone asserts no boundary trace or distributional equation.

For this comparison, additionally assume the Axiom of Choice as in The Axiom of Choice. Under the exterior geometry fixed above, this local space is distinguished from the exterior Dirichlet spectral space $H_D^{-2}(\mathrm{\Omega })$ of Dirichlet spectral Sobolev scale and duality. The notation here does not assert that the two spaces are equal or identify their elements. This additional assumption is used only to invoke that spectral-scale interface; the local antidual definition above retains its stated Countable Choice hypothesis.

The article introduces $H^{-2}({\mathrm{\Omega }}^{\mathrm{\prime }})=(H_0^2({\mathrm{\Omega }}^{\mathrm{\prime }}){)}^{\mathrm{\prime }}$ in the proof of Lemma 3.2(a). Its citation records attribution; the definition and pairing convention are justified by the independent suppliers above.