Restriction Bessel-potential spaces on polyhedral domains

Assume Countable Choice. Let $s\ge 0$ and let $D\subset {\mathbb{R}}^d$ be a nonempty open set. Use $H^s({\mathbb{R}}^d)$ with the Bessel-potential completion convention of Real-order Bessel-potential completion H^s. Realize this completion in $L^2({\mathbb{R}}^d)$ by the canonical injection ${\iota }_{s}U:={\mathcal{F}}_2^{-1}(⟨\xi {⟩}^{-s}{J}_{s}U),$ where $J_s:H^s({\mathbb{R}}^d)\to L^2({\mathbb{R}}^d)$ is the weighted Fourier isometry of The Bessel completion embeds canonically in tempered distributions and ${\mathcal{F}}_2$ is the unitary transform of Plancherel theorem. Since $s\ge 0$, multiplication by $⟨\xi {⟩}^{-s}$ is bounded on $L^2$ with norm at most one and is injective. Hence ${\iota }_s$ is linear, injective, and satisfies $\mathrm{\parallel }{\iota }_{s}U{\mathrm{\parallel }}_2\le \mathrm{\parallel }U{\mathrm{\parallel }}_{H^s}$; it sends each canonical Schwartz class to that function's almost-everywhere class. Equivalently, for a Cauchy representative $[u_j]$, ${\iota }_s[u_j]={\lim }_j{u}_j$ in $L^2$, because ${\widehat{u}}_j=⟨\xi {⟩}^{-s}{J}_s{u}_j$ converges in $L^2$ and ${\mathcal{F}}_2^{-1}$ is continuous. Identify $U$ with this $L^2$ class, and write $U{\mathrm{\mid }}_D:=({\iota }_{s}U){\mathrm{\mid }}_D$. Define the restriction space $H^s(D):=\{U{\mathrm{\mid }}_D:U\in H^s({\mathbb{R}}^d)\},\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s(D)}:=\inf \{\mathrm{\parallel }U{\mathrm{\parallel }}_{H^s({\mathbb{R}}^d)}:U{\mathrm{\mid }}_D=u\text{ a.e. on }D\}\mathrm{.}$ Two extensions represent the same element exactly when their restrictions agree almost everywhere on $D$. This is the convention for fractional-order spaces on the polyhedral domain and its affine simplex cells used in the Nitsche items on this page. No $C^1$ boundary or trace theorem is part of this definition.