Scale Riesz map and free HUM operator

PDF p. 12: Theorem 3.6 proof; (3.15)

Assume the Axiom of Choice. Let $\mathrm{\Omega }$ be the smooth connected exterior domain, let $A=I-{\mathrm{\Delta }}_D$ and let $H_D^r=H_D^r(\mathrm{\Omega })$ be its Dirichlet spectral scale. The $L^2(\mathrm{\Omega })$ inner product is linear in its first argument. Let $T>0$, let $\phi$ be the real control cutoff, and write $W(t)=e^{it{\mathrm{\Delta }}_D}$.

For $s\in [-2,2]$ put $\sigma =-s$ and define the scale Riesz map ${\mathrm{\Lambda }}_s=A^s:H_D^s\to H_D^{-s}$, with inverse ${\mathrm{\Lambda }}_s^{-1}=A^{-s}:H_D^{\sigma }\to H_D^s$. The notation denotes the consistent extensions of the spectral powers. Write $⟨f,g{⟩}_{s,-s}$ for the pivot pairing of $H_D^s$ with $H_D^{-s}$, linear in $f$ and conjugate-linear in $g$. Thus $⟨{\mathrm{\Lambda }}_s^{-1}g,g{⟩}_{s,-s}=\mathrm{\parallel }g{\mathrm{\parallel }}_{H_D^{\sigma }}^2$ for $g\in H_D^{\sigma }$.

For $v_0\in H_D^{\sigma }$, set $v(t)=W(t)v_0$ and $h(t)=-i{\mathrm{\Lambda }}_s^{-1}(\phi v(t))$. Let $z\in C([0,T];H_D^s)$ be the unique backward mild solution of $i{\mathrm{\partial }}_{t}z+{\mathrm{\Delta }}_{D}z=\phi h$ with $z(T)=0$. Define the free Hilbert uniqueness method (HUM) operator $S:H_D^{\sigma }\to H_D^s$ by $S{v}_0=z(0)$.

Dirichlet spectral Sobolev scale and duality

Positive spectral Hilbert scale: duality and interpolation

Scale isometries and free Schrödinger evolution

Cutoff multipliers through order two and by duality

Linear and nonlinear internally controlled exterior systems

Linear forced well-posedness by Duhamel formula

Continuous and mixed time-space function spaces

The Axiom of Choice

The geometry, shifted Dirichlet Laplacian, scale and pivot convention are those of Dirichlet spectral Sobolev scale and duality, under the Choice assumption recorded in The Axiom of Choice. By Positive spectral Hilbert scale: duality and interpolation, $A^{-r\mathrm{/}2}$ is an onto isometry from $H_D^a$ to $H_D^{a+r}$ for all real $a,r$. Taking $(a,r)=(s,-2s)$ and $(-s,2s)$ gives the two mutually inverse maps above. On the common spectral core, the pivot pairing satisfies $⟨f,g{⟩}_{s,-s}=⟨A^{s\mathrm{/}2}f,A^{-s\mathrm{/}2}g{⟩}_2$; the scale isometries extend this identity continuously to all $f\in H_D^s$ and $g\in H_D^{-s}$. Substituting $f=A^{-s}g$ gives the displayed norm identity. This convention also covers negative $s$; it does not treat arbitrary negative-order elements as pointwise functions.

By Scale isometries and free Schrödinger evolution, $v\in C([0,T];H_D^{\sigma })$. The cutoff in Linear and nonlinear internally controlled exterior systems acts boundedly on this space by Cutoff multipliers through order two and by duality, and the inverse scale map is bounded; consequently $h\in C([0,T];H_D^s)$ and in particular $h\in L^1(0,T;H_D^s)$. The backward existence and uniqueness result of Linear forced well-posedness by Duhamel formula therefore defines a unique $z$ and a unique value $z(0)$. Each operation in the construction is complex-linear and bounded, so $S$ is a bounded complex-linear map. This definition alone asserts neither its coercivity nor its invertibility.

The original article is cited for the notation and construction in the proof of Theorem 3.6 and (3.15), on PDF page 12. The justification uses the independent suppliers above; the article is not used as proof of its own construction.