Bounded C^k domains admit integer-order Sobolev extension

Since $\mathrm{\partial }\mathrm{\Omega }$ is compact and $V$ is an open neighbourhood of $\bar{\mathrm{\Omega }}$, choose $\rho >0$ with $\{x:\mathrm{dist}(x,\bar{\mathrm{\Omega }})<2\rho \}\subseteq V$. Choose finitely many boundary charts ${\mathrm{\Phi }}_i$, with the larger patches compactly contained in their original chart neighbourhoods, on nested patches $W_i^{\mathrm{\prime }}⋐W_i$, so that the larger patches have $\rho$-neighbourhoods in $V$ and the smaller patches cover $\mathrm{\partial }\mathrm{\Omega }$. The inner patches may be taken thin enough in flattened normal coordinates that reflection by any factor $j\le k$ keeps the support of a function localized there inside the larger flattened patch. Compactness also gives an open set ${\mathrm{\Omega }}_0\subset \subset \mathrm{\Omega }$ with $\bar{\mathrm{\Omega }}\subseteq {\mathrm{\Omega }}_0\cup {\bigcup }_{i=1}^m{W}_i^{\mathrm{\prime }}$.

Let $N:={\mathrm{\Omega }}_0\cup {\bigcup }_{i=1}^m{W}_i^{\mathrm{\prime }}$, an open neighbourhood of $\bar{\mathrm{\Omega }}$ inside $V$. Apply [F4] on $N$ to this finite cover and choose a compact neighbourhood $K⋐N$ of $\bar{\mathrm{\Omega }}$. Only finitely many partition supports meet $K$; these pieces still sum to one on a neighbourhood of $\bar{\mathrm{\Omega }}$. Group them by their assigned cover member and extend them by zero outside $N$. This gives ${\zeta }_0,{\zeta }_1,\dots ,{\zeta }_m\in C_c^{\mathrm{\infty }}(V;[0,1])$ with ${\sum }_i{\zeta }_i=1$ near $\bar{\mathrm{\Omega }}$, $\mathrm{supp}{\zeta }_0\subset {\mathrm{\Omega }}_0⋐\mathrm{\Omega }$, and $\mathrm{supp}{\zeta }_i\subset W_i^{\mathrm{\prime }}$ for $i\ge 1$. Each support is compact in $V$.

To apply [F3] on patches reaching the boundary, let $T:A\to B$ be either direction of a chart restricted to corresponding open half-patches. The derivatives of $T$ and $T^{-1}$ through order $k$ have uniform bounds inherited from the compact ambient chart. Exhaust $A$ by nested open sets $A_j⋐A$, $j\ge 1$, and put $B_j=T(A_j)⋐B$. For $v\in W^{k,p}(B)$, [F3] applied on each matched pair gives the weak composition formulas and $\mathrm{\parallel }v\circ T{\mathrm{\parallel }}_{W^{k,p}(A_j)}\le C\mathrm{\parallel }v{\mathrm{\parallel }}_{W^{k,p}(B_j)}\le C\mathrm{\parallel }v{\mathrm{\parallel }}_{W^{k,p}(B)}$, with $C$ independent of $j$. Every test support in $A$ lies in some $A_j$, so these same formula fields are weak derivatives on $A$. Increasing the integrals by [F9] for finite $p$, or taking essential bounds on the countable union for $p=\mathrm{\infty }$, proves the identical norm bound on $A$. Thus no compact-containment hypothesis is being assumed of the entire half-patch.

Let $u\in W^{k,p}(\mathrm{\Omega };\mathbb{K})$. By [F5], since the ambient cutoff has bounded derivatives through order $k$, each product ${\zeta }_{i}u{\mathrm{\mid }}_{\mathrm{\Omega }}$ belongs to $W^{k,p}(\mathrm{\Omega };\mathbb{K})$, is supported in $\mathrm{supp}{\zeta }_i\cap \mathrm{\Omega }$, and satisfies $\mathrm{\parallel }{\zeta }_{i}u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}\le C_i\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}$ with $C_i$ determined by the cutoff; moreover $u={\sum }_{i=0}^m{\zeta }_{i}u$ almost everywhere on $\mathrm{\Omega }$.

Interior piece: ${\zeta }_{0}u$ is supported in the compact set $\mathrm{supp}{\zeta }_0\subset \mathrm{\Omega }$, so by [F7] its extension by zero $E_0({\zeta }_{0}u)$ lies in $W^{k,p}({\mathbb{R}}^n;\mathbb{K})$, agrees with ${\zeta }_{0}u$ on $\mathrm{\Omega }$, is supported in $\mathrm{supp}{\zeta }_0\subseteq V$, and satisfies $\mathrm{\parallel }{E}_0({\zeta }_{0}u){\mathrm{\parallel }}_{W^{k,p}({\mathbb{R}}^n)}=\mathrm{\parallel }{\zeta }_{0}u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}$.

Boundary pieces: fix $i\ge 1$. The flattened function $w_i:=({\zeta }_{i}u)\circ {\mathrm{\Phi }}_i^{-1}$ is defined on ${\mathrm{\Phi }}_i(W_i\cap \mathrm{\Omega })={\mathrm{\Phi }}_i(W_i)\cap \{t<0\}$. The cutoff support is compactly contained laterally in the chart; extending $w_i$ by zero across the artificial edges inside this half-space gives a $W^{k,p}(H_{-})$ class, since the cutoff vanishes near those edges and tests in $H_{-}$ stay away from $t=0$. Step 2.2 applied to the corresponding half-patches bounds its norm by $C_i\mathrm{\parallel }{\zeta }_{i}u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}$. Conjugating the upper-half-space operator of [F2] by the coordinate flip gives an extension from $H_{-}=\{t<0\}$, so the zero-extended $w_i$ has an extension ${\tilde{w}}_i\in W^{k,p}({\mathbb{R}}^n;\mathbb{K})$. Its support remains inside the larger flattened patch by the choice in step 1.1, and its extension formula is linear. Pulling back by ${\mathrm{\Phi }}_i$ and multiplying by a cutoff ${\psi }_i\in C_c^{\mathrm{\infty }}(W_i)$ equal to one near $\mathrm{supp}{\zeta }_i$ gives $E_i(u):={\psi }_i\cdot ({\tilde{w}}_i\circ {\mathrm{\Phi }}_i)\in W^{k,p}({\mathbb{R}}^n;\mathbb{K}),$ by [F3] on matched ambient patches and [F5], followed by [F7] to extend the compactly supported product from $W_i$ by zero. These operations give $\mathrm{\parallel }{E}_i(u){\mathrm{\parallel }}_{W^{k,p}({\mathbb{R}}^n)}\le C_i\mathrm{\parallel }{\zeta }_{i}u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}$. It agrees with ${\zeta }_{i}u$ almost everywhere on $\mathrm{\Omega }$, and its support lies in $\mathrm{supp}{\psi }_i\cap {\mathrm{\Phi }}_i^{-1}(\mathrm{supp}{\tilde{w}}_i)$, a compact subset of $V$.

Define $E(u):=E_0({\zeta }_{0}u)+{\sum }_{i=1}^m{E}_i(u)$. Each step above is linear in $u$, so $E$ is linear; on $\mathrm{\Omega }$ the sum equals ${\sum }_{i=0}^m{\zeta }_{i}u=u$ almost everywhere by step 3.1; the support of $E(u)$ is contained in the union of finitely many compact subsets of $V$, hence compact in $V$; and [F8] together with the bounds of steps 3.1, 4.1 and 4.2 gives $\mathrm{\parallel }E(u){\mathrm{\parallel }}_{W^{k,p}({\mathbb{R}}^n)}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}$ for a constant $C$ independent of $u$.

Therefore $E$ is a bounded linear extension operator in the sense of [F8], and $\mathrm{\Omega }$ is a $W^{k,p}$-extension domain for every $1\le p\le \mathrm{\infty }$, including $p=\mathrm{\infty }$ through the chart and half-space interfaces used above. For $k=0$ the $L^p$ extension property is immediate on any open $\mathrm{\Omega }$: extension by zero of an $L^p(\mathrm{\Omega };\mathbb{K})$ class lies in $L^p({\mathbb{R}}^n;\mathbb{K})$ with the same norm, is linear, and restricts back to the class, so zero extension is the required operator; its support is contained in $\bar{\mathrm{\Omega }}$, which is compact in $V$ under the bounded-domain hypotheses.