Mesh-uniform fractional boundary flux trace

The boundary of the fixed polyhedral domain lies in finitely many supporting hyperplanes $P_1,\dots ,P_m$. Choose any whole-space extension $Z\in H^{1+s}({\mathbb{R}}^d)$ of $\zeta$. Its distributional derivative has Fourier multiplier $2\pi i{\xi }_j$; hence by [F3], $\mathrm{\mid }2\pi {\xi }_j\mathrm{\mid }⟨\xi {⟩}^s\le 2\pi ⟨\xi {⟩}^{1+s}$ and $\mathrm{\parallel }\mathrm{\nabla }Z{\mathrm{\parallel }}_{H^s}^2:={\sum }_{j=1}^d\mathrm{\parallel }{\mathrm{\partial }}_{j}Z{\mathrm{\parallel }}_{H^s}^2\le (2\pi {)}^{2}d\mathrm{\parallel }Z{\mathrm{\parallel }}_{H^{1+s}}^2$. For each $P_j$, choose $d$ affinely independent points in that plane and a point off it to form a nondegenerate simplex having a facet in $P_j$; this only names the plane trace in [F2] and imposes no mesh condition. Apply the scalar Fourier hyperplane trace estimate in [F2] to every component ${\mathrm{\partial }}_{k}Z$ and sum the finitely many squared inequalities. After a rigid coordinate change this gives the componentwise vector trace and $\mathrm{\parallel }{T}_{P_j}\mathrm{\nabla }Z{\mathrm{\parallel }}_{L^2(P_j)}^2:={\sum }_{k=1}^d\mathrm{\parallel }{T}_{P_j}({\mathrm{\partial }}_{k}Z){\mathrm{\parallel }}_{L^2(P_j)}^2\le C_s\mathrm{\parallel }\mathrm{\nabla }Z{\mathrm{\parallel }}_{H^s}^2$.

Group the boundary facets by their supporting planes. On a facet $F\subset P_j$, the scalar normal derivative is the dot product of the same plane trace with a fixed unit normal, so $\mathrm{\parallel }{\mathrm{\partial }}_n\zeta {\mathrm{\parallel }}_F^2\le \mathrm{\parallel }{T}_{P_j}\mathrm{\nabla }Z{\mathrm{\parallel }}_{L^2(F)}^2$. The boundary facets in one supporting plane have disjoint relative interiors: their relative interiors are boundary points of the polyhedral domain, and the matching mesh has disjoint cell interiors and whole-face intersections. The face-height convention gives $h_F\le h_K\mathrm{/}d\le h\mathrm{/}d$. Consequently ${\sum }_{F\in {\mathcal{F}}_h^{\mathrm{\partial }}}{h}_F\mathrm{\parallel }{\mathrm{\partial }}_n\zeta {\mathrm{\parallel }}_F^2\le \frac{h}{d}{\sum }_{j=1}^m\mathrm{\parallel }{T}_{P_j}\mathrm{\nabla }Z{\mathrm{\parallel }}_{L^2(P_j)}^2\le Ch\mathrm{\parallel }Z{\mathrm{\parallel }}_{H^{1+s}({\mathbb{R}}^d)}^2,$ using step 1.1 for the last inequality.

For every $\epsilon >0$, the restriction norm supplies an extension $Z_{\epsilon }$ with $\mathrm{\parallel }{Z}_{\epsilon }{\mathrm{\parallel }}_{H^{1+s}({\mathbb{R}}^d)}\le \mathrm{\parallel }\zeta {\mathrm{\parallel }}_{H^{1+s}(\mathrm{\Omega })}+\epsilon$. Apply step 2.1 to $Z_{\epsilon }$ and let $\epsilon \downarrow 0$. Taking square roots proves the first estimate; since ${\mathrm{\ell }}_{\mathrm{\Omega }}>0$ is fixed, writing $C_T=C{\mathrm{\ell }}_{\mathrm{\Omega }}^{-1\mathrm{/}2}$ gives the displayed normalization.