Dirichlet spectral Sobolev scale and duality

PDF p. 8: definition of HσD and pairing

Assume the Axiom of Choice. Let $n\ge 2$, let $\mathrm{\Theta }\subset {\mathbb{R}}^n$ be nonempty and compact with smooth boundary, and let its exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$ be connected. Work in the complex Hilbert space $L^2(\mathrm{\Omega })$, with inner product $⟨u,v{⟩}_2={\int }_{\mathrm{\Omega }}u\bar{v}dx$, linear in the first argument. Let ${\mathrm{\Delta }}_D$ be the Dirichlet Laplacian with domain $H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$, and put $A=I-{\mathrm{\Delta }}_D$.

For $\sigma \in \mathbb{R}$, let $D(A^{\mathrm{\infty }})={\bigcap }_{k\ge 1}D(A^k)$ and define $H_D^{\sigma }(\mathrm{\Omega })$ as its completion in the norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{H_D^{\sigma }}=\mathrm{\parallel }{A}^{\sigma \mathrm{/}2}u{\mathrm{\parallel }}_2$, where the powers are defined by the spectral calculus. For $\sigma \ge 0$, this completion is identified with $D(A^{\sigma \mathrm{/}2})$, equipped with that norm. In particular $H_D^0(\mathrm{\Omega })=L^2(\mathrm{\Omega })$, $H_D^1(\mathrm{\Omega })=H_0^1(\mathrm{\Omega })$ and $H_D^2(\mathrm{\Omega })=H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$, as spaces with their respective indicated norms.

For $s\ge 0$, identify $H_D^{-s}(\mathrm{\Omega })$ isometrically with the antidual of $H_D^s(\mathrm{\Omega })$: the antidual is the space of bounded conjugate-linear scalar functionals, with its operator norm. The pairing $⟨f,g{⟩}_{-s,s}$ is linear in $f$ and conjugate-linear in $g$ and extends ${\int }_{\mathrm{\Omega }}f\bar{g}dx$ from pairs in $D(A^{\mathrm{\infty }})$. The integral is therefore a pivot notation for this extension; arbitrary negative-order elements need not be functions for which the integral is defined.

Write $R=A^{-1}$. It extends to an isometric isomorphism $R:H_D^{\sigma }(\mathrm{\Omega })\to H_D^{\sigma +2}(\mathrm{\Omega })$ for every real $\sigma$. The group $e^{it{\mathrm{\Delta }}_D}$ extends consistently to a unitary group on every $H_D^{\sigma }(\mathrm{\Omega })$ and commutes there with $R$.

Exterior and truncated obstacle domains

Complex integer-order Sobolev notation

Dirichlet Laplacian and positive shift

Exterior Dirichlet realization, H² domain and form domain

Positive spectral Hilbert scale: duality and interpolation

Unbounded Borel functional calculus: domains, products, spectral mapping

The Axiom of Choice

The geometry is fixed by Exterior and truncated obstacle domains, and the weak Sobolev spaces and the closure defining $H_0^1(\mathrm{\Omega })$ are fixed by Complex integer-order Sobolev notation. Under Choice, Dirichlet Laplacian and positive shift supplies the displayed Dirichlet realization and its self-adjoint shift $A\ge I$ on the nonzero complex Hilbert space $L^2(\mathrm{\Omega })$. The powers and their domains are defined by Unbounded Borel functional calculus: domains, products, spectral mapping, with the Choice assumption of that supplier recorded by The Axiom of Choice.

The positive spectral-scale result Positive spectral Hilbert scale: duality and interpolation applies to this $A$: it supplies the normed completion, the positive-domain identification, and the isometric pivot-antidual identification. In particular, it gives a definite norm on the common core and a complete space at each real index. Its pivot pairing extends the $L^2$ inner product by continuity, so its notation does not assert that a general negative-order element is an integrable function.

The realization theorem Exterior Dirichlet realization, H² domain and form domain gives $D(A^{1\mathrm{/}2})=H_0^1(\mathrm{\Omega })$ with $\mathrm{\parallel }{A}^{1\mathrm{/}2}u{\mathrm{\parallel }}_2^2=\mathrm{\parallel }u{\mathrm{\parallel }}_2^2+\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2^2$ and $D(A)=H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$. The latter space carries the scale norm $\mathrm{\parallel }Au{\mathrm{\parallel }}_2$, which that theorem states is equivalent to its $H^2$ norm. The same spectral-scale result supplies the stated extension of $A^{-1}$, the unitary evolution and their commutation.

The article is cited for the scale, pivot pairing and notation on PDF page 8. The mathematical justifications are the independent suppliers listed above; the article is not used as proof of its own definition or its properties.