Regularized lower-order observation remainder

PDF p. 8: indices I, k, τ, ρ and (3.7)

Assume the Axiom of Choice. Let $\mathrm{\Omega }$ be the smooth exterior domain, $H_D^s(\mathrm{\Omega })$ its Dirichlet spectral scale, $R=(I-{\mathrm{\Delta }}_D{)}^{-1}$, $W(t)=e^{it{\mathrm{\Delta }}_D}$, and $\psi$ the real smooth compactly supported cutoff specified in Control region and nested real cutoffs. Let $T>0$ and $\mathcal{I}=[-2,2)$.

For $\sigma \in \mathcal{I}$, set $k(\sigma )=0$ if $\sigma \ge 0$ and $k(\sigma )=1$ if $\sigma <0$, and put $\tau =\sigma +2k(\sigma )$. Set $\rho (\sigma )=\sigma +1$ when $-1<\sigma <0$, and $\rho (\sigma )=0$ otherwise. For $v_0\in H_D^{\sigma }(\mathrm{\Omega })$, write $v(t)=W(t)v_0$ and define the lower-order observation remainder

$Q_{\sigma }(v_0)={\int }_0^T\mathrm{\parallel }\psi R^{k(\sigma )}v(t){\mathrm{\parallel }}_{H_D^{\rho (\sigma )}(\mathrm{\Omega })}^{2}dt\mathrm{.}$

Here $R^0$ is the identity and $R^1=R$. The multiplication by $\psi$ is the bounded spectral-scale multiplier supplied by lem-se-spectral-cutoff-duality. The integral is with respect to Lebesgue time measure. In particular, outside $(-1,0)$ the spatial norm in the integrand is the $L^2(\mathrm{\Omega })$ norm.

Control region and nested real cutoffs

Dirichlet spectral Sobolev scale and duality

Scale isometries and free Schrödinger evolution

Endpoint multiplier bounds extended to negative spectral orders

Positive spectral Hilbert scale: duality and interpolation

Continuous and mixed time-space function spaces

The Axiom of Choice

The domain and cutoff are fixed by Control region and nested real cutoffs, and the operator and scale notation by Dirichlet spectral Sobolev scale and duality. The Axiom of Choice is retained as an assumption of the scale and flow suppliers The Axiom of Choice. The interval and piecewise formulas determine unique values of $k$, $\tau$ and $\rho$. Directly from the cases, $\tau \in [0,2)$ and $0\le \rho \le \tau$; if $-1<\sigma <0$, then $\rho =\tau -1\in (0,1)$.

By prop-se-scale-flow, $R^{k(\sigma )}v(t)=W(t)R^{k(\sigma )}{v}_0$, $R^{k(\sigma )}{v}_0\in H_D^{\tau }(\mathrm{\Omega })$, and $\mathrm{\parallel }{R}^{k(\sigma )}{v}_0{\mathrm{\parallel }}_{H_D^{\tau }}=\mathrm{\parallel }{v}_0{\mathrm{\parallel }}_{H_D^{\sigma }}$. The same supplier makes $t\mapsto R^{k(\sigma )}v(t)$ continuous in $H_D^{\tau }$. The scale embeddings from Positive spectral Hilbert scale: duality and interpolation give a continuous inclusion $H_D^{\tau }\hookrightarrow H_D^{\rho }$ because $\rho \le \tau$. The smooth compactly supported $\psi$ has bounded derivatives through order two, so Endpoint multiplier bounds extended to negative spectral orders makes multiplication by $\psi$ bounded on $H_D^{\rho }$.

Consequently the integrand is a continuous nonnegative scalar function on $[0,T]$, and its Lebesgue integral is finite and independent of representatives; the time and norm conventions are those of Continuous and mixed time-space function spaces. More explicitly, for a constant depending on the fixed scale level and cutoff, $0\le Q_{\sigma }(v_0)\le T{C}_{\psi }^2\mathrm{\parallel }{v}_0{\mathrm{\parallel }}_{H_D^{\sigma }}^2$. At $\rho =0$, Dirichlet spectral Sobolev scale and duality identifies $H_D^0$ with $L^2$. This definition alone does not assert compactness of the resulting quadratic quantity.

The article supplies the indices and the notation (3.7) on PDF page 8. Its citation is attribution; the well-definedness argument uses the independent suppliers stated above.