A reference simplex is an integer-order Sobolev extension domain

Let $z\in \overline{{\widehat{S}}_d}$. Define the barycentric affine coordinates by ${\lambda }_0(x)=1-{\sum }_{i=1}^d{x}_i$ and ${\lambda }_i(x)=x_i$ for $1\le i\le d$; they sum to one and are nonnegative exactly on the closed simplex. The active set $I(z)=\{i:{\lambda }_i(z)=0\}$ has at most $d$ elements, and the gradients of the corresponding coordinates are linearly independent. Complete these affine coordinates to an invertible affine coordinate system $y=({\lambda }_i{)}_{i\in I(z)},y_{m+1},\dots ,y_d$, where $m=\mathrm{\mid }I(z)\mathrm{\mid }$. Because every inactive barycentric coordinate is strictly positive at $z$, a sufficiently small neighbourhood $U_z$ meets the simplex exactly in the orthant chart $y_1>0,\dots ,y_m>0$; for $m=0$ it is an interior chart.

The charts $U_z$ cover the compact closure, so [F4] gives a finite subcover. Shrink them to open sets $V_j$ still covering the closure, with ${\bar{V}}_j\subset U_j$. Use [F3] to choose smooth cutoffs ${\theta }_j$ supported compactly in $U_j$ and equal to one on $V_j$. Their sum $S$ is positive on a neighbourhood $N$ of the closed simplex. Choose a smooth cutoff $\chi$ supported compactly in $N$ and equal to one near the closed simplex, and set ${\eta }_j=\chi {\theta }_j\mathrm{/}S$ on $N$, zero outside $N$. Then each ${\eta }_j$ is smooth and supported in its chart, and their sum is one near the simplex. Each localized input ${\eta }_{j}u$ has norm bounded by a fixed multiple of $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,p}({\widehat{S}}_d)}$ by the product rule in [F3].

In an interior chart, the localized function is zero near the chart edge. Its zero extension to ${\mathbb{R}}^d$ has the same weak derivatives: for any test function, multiply that test by a cutoff equal to one on the localized support, apply the weak identity on the chart, and note that all derivatives of the cutoff product agree on that support. This is the zero-extension part of [F3]. In a boundary chart with $m$ active faces, the localized function is a $W^{k,p}$ function on the orthant $O_m=(0,\mathrm{\infty }{)}^m\times {\mathbb{R}}^{d-m}$ and vanishes near the artificial edge of the chart; extend by zero across only that artificial edge using the same test-function argument.

Extend from $O_m$ to ${\mathbb{R}}^d$ one coordinate at a time. At each stage, apply the reflected half-space formula in [F2] to the coordinate being extended, with the other $m-1$ positive coordinates as tangential variables. For clarity, the coordinate formula on $t<0$ is ${\sum }_{j=1}^k{a}_{j}w(x^{\mathrm{\prime }},-jt)$, with ${\sum }_j{a}_j(-j{)}^r=1$ for $0\le r<k$; for $k=0$ use even reflection. For a test compactly supported in $O_{m-1}\times \mathbb{R}$, apply the half-space proof's normal ACL trace and integration-by-parts identity on each compact tangential subdomain and then exhaust the test support by finitely many such subdomains. Tangential weak derivatives pass through the finite reflection sum by the weak test identity, and the displayed moment equations cancel every normal interface term. The change of variable $s=-jt$ bounds each reflected derivative in $L^p$ by the same finite coefficient sum as in [F2] (and by the corresponding essential supremum sum at $p=\mathrm{\infty }$). Thus this explicit operator is bounded on $W^{k,p}(O_m)$ and extends one orthant coordinate. Repeating at most $m\le d$ times gives the required extension. Pulling back through the affine chart changes the bound by a finite factor from [F1].

Apply these local bounded linear extensions to the finitely many localized pieces and add them. Their sum is bounded and linear, and on ${\widehat{S}}_d$ its restriction is ${\sum }_j{\eta }_{j}u=u$. There are finitely many charts, all have bounded coordinate images and the reflection operators add support only by fixed finite factors, so the sum is supported in a fixed bounded set. The definition of extension domain in [F6] gives the conclusion.