Simultaneous Sobolev extension and localization

For bounded smooth $G$, take the common compactly supported endpoint extension from the first fact. It is defined already on $L^2(G)$ and is bounded at the second weak-derivative endpoint. The whole-space $Er$ lemma therefore applies with $D=G$: for each $0\le s\le 2$ and each ambient $H^s$ extension $F$ of $u$, $E_{G}u=(E_G{r}_G)F$ and $\mathrm{\parallel }{E}_{G}u{\mathrm{\parallel }}_{H^s}\le C_s\mathrm{\parallel }F{\mathrm{\parallel }}_{H^s}$. The value is independent of $F$ because it is the original $L^2$ map applied to $u$. Infimizing gives the bounded quotient extension for every order. Its support remains in the same fixed compact set because the formula on positive-order inputs is unchanged.

For any compatible complex Banach couple, denote the evaluation norms from ${\mathcal{F}}_b$ and ${\mathcal{F}}_d$ by $N_b$ and $N_d$. Fix $0<\theta <1$, $f\in {\mathcal{F}}_b$ and $\epsilon >0$, and put $f_{\epsilon }(z)=\exp (\epsilon (z-\theta {)}^2)f(z)$ and $c_{\theta }=\max \{{\theta }^2,(1-\theta {)}^2\}$. For $z=x+it\in S$, the scalar factor has modulus $\exp (\epsilon ((x-\theta {)}^2-t^2))\le \exp (\epsilon c_{\theta })\exp (-\epsilon t^2)$. Multiplication by this entire scalar function preserves sum-norm continuity and norm-holomorphy: the latter follows directly from the product difference quotient. The displayed bound preserves boundedness on $S$ and makes each continuous $X_j$ boundary map decay in its endpoint norm. Hence $f_{\epsilon }\in {\mathcal{F}}_d$, $f_{\epsilon }(\theta )=f(\theta )$, and $\mathrm{\parallel }{f}_{\epsilon }{\mathrm{\parallel }}_{{\mathcal{F}}_d}\le e^{\epsilon c_{\theta }}\mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathcal{F}}_b}$. Thus the two classes have identical sets of values at $\theta$. The inclusion ${\mathcal{F}}_d\subseteq {\mathcal{F}}_b$ gives $N_b(u)\le N_d(u)$, while the damped representatives give $N_d(u)\le e^{\epsilon c_{\theta }}{N}_b(u)$ after infimizing over all $f$ representing $u$. Letting $\epsilon \downarrow 0$ proves $N_d(u)=N_b(u)$. No convergence of $f_{\epsilon }$ in either strip norm is needed.

For $0<s<2$, apply the quotient/interpolation lemma with $\theta =s\mathrm{/}2$. Its decaying-strip interpolation space is $H^s(G)$ with equivalent norms; step 1.2 identifies that evaluation space isometrically with the bounded-strip convention of the Statement. At $s=0,2$ the brackets are the specified endpoint spaces, and the endpoint identifications are supplied directly by F2 and F3, without an endpoint strip-evaluation argument. Quotient completeness, definiteness and independence of the chosen ambient extension are supplied by the whole-space $Er$ lemma in F3. Consequently the domain interpolation identity and the common extension have precisely the asserted restriction-quotient meaning.

For the smooth exterior domain, use the final helper’s bounded collar $G=\mathrm{\Omega }\cap B_L$, its finite-reflection collar extension and its independently justified identity tail. That helper proves the common $L^2\mathrm{/}{W}^{2,2}$ endpoint bounds and then applies the same whole-space and quotient arguments; it also gives explicit maps $J,P$ with $PJ=I$ and equivalent collar/tail norms for each $0\le s\le 2$. Thus both the bounded compact-output assertion and the exterior localization/retraction in the Statement hold in exactly the restriction-quotient Bessel convention.