> [!theorem] > > Let $U \subset \real^n$ be bounded and [[Open Set|open]] such that $\partial U$ is [[Differentiable Boundary|of class]] $C^1$. Then there exists a neighbourhood $V \in \cn^o(\ol{U})$ and a linear operator on the [[Space of Uniformly Continuously Differentiable Functions|UC]] functions > $ > E: UC^1(U) \to C_c^1(\real^n) \quad f \mapsto F > $ > where $\supp{F} \subset V$. Moreover, there exists $C \ge 0$, independent from $f$, such that > $ > \norm{F}_{W^{1, p}(V)} \le C\norm{f}_{W^{1, p}(U)} > $ > > *Proof*. Let $x \in \partial U$, then over a [[Higher Order Reflection|reflection]], there exists $V_x \in \cn^o(x)$, $C_x \ge 0$, and a local $C^1$ extension $F_x \in C^1(V_x)$ such that $\norm{F_x}_{W^{1, p}(V_x)} \le C_x \norm{f}_{W^{1, p}(U)}$. Note that $V_x$ and $C_x$ do not depend on $f$. > > Since $\partial U$ is [[Compactness|compact]] and $\bracs{V_p}_{p \in \partial U}$ is an [[Open Cover|open cover]] of $\partial U$, there exists a finite subcover $\bracs{V_i}_1^k$. Let $V_0 = U$, then $V = \bigcup_{i = 0}^k V_i$ is a desired neighbourhood. Let $F_i$ be the associated extensions on $V_i$, and $F_0 = f$. Let $\bracs{\zeta_i}_0^k$ be a smooth [[Partition of Unity|partition of unity]] subordinate to $\bracs{V_i}_0^k$. By [[Gluing Lemma|gluing]] with $0$ outside of $\supp{\zeta_i}$, each $\zeta_iF_i \in C^1_c(\real^n)$ with $\supp{\zeta_iF_i} \subset V$. Define $F = \sum_{i = 0}^k \zeta_i F_i$, then $F \in C_c^1(\real^n)$ is an extension of $f$ with $\supp{F} \subset V$ and > $ > \begin{align*} > \norm{F}_{W^{1, p}(V)} &\le \sum_{i = 0}^k \norm{\zeta_i F_i}_{W^{1, p}(V_i)} \\ > &\le \sum_{i = 0}^kC_{i, k, p}\norm{F_i}_{W^{1, p}(V_i)} \\ > &\le \sum_{i = 0}^k C_{i, k, p}C_{x}\norm{f}_{W^{1, p}(U)} \\ > &\le C\norm{f}_{W^{1, p}(U)} > \end{align*} > $ > Note that the selected neighbourhoods for the finite subcover, and the obtained partition of unity are all chosen independently of $f$. Thus the same choices can be made for each $f$, resulting in a uniform bound in the norm of the extension.