The trace commutes with smooth cutoffs and is chart local

Multiplicativity (i). Let $u\in W^{1,p}(\mathrm{\Omega })$ and let $u_m\in D$ with $u_m\to u$ in $W^{1,p}(\mathrm{\Omega })$ by [F4]. Each $u_m$ extends to a smooth compactly supported function, so $\eta u_m\in C(\bar{\mathrm{\Omega }})\cap W^{1,p}(\mathrm{\Omega })$ has classical restriction $(\eta {\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }})(u_m{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }})=(\eta {\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }})T{u}_m$ by [F2]; hence $T(\eta u_m)=(\eta {\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }})T{u}_m$. By [F3] $\eta u_m\to \eta u$ in $W^{1,p}(\mathrm{\Omega })$, so $T(\eta u_m)\to T(\eta u)$ in $L^p(\mathrm{\partial }\mathrm{\Omega })$ by [F1]; and $(\eta {\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }})T{u}_m\to (\eta {\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }})Tu$ because $\eta {\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$ is bounded and multiplication by a bounded continuous function is continuous on $L^p(\mathrm{\partial }\mathrm{\Omega })$ (Holder). The identity follows, and the bound $\mathrm{\parallel }T(\eta u)\mathrm{\parallel }\le C(\mathrm{\Omega },p)\mathrm{\parallel }\eta u{\mathrm{\parallel }}_{W^{1,p}}\le C(\mathrm{\Omega },p)C_{\eta }\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}}$ follows from [F1] and [F3].

Locality (ii). Let ${\mathrm{\Omega }}^{\mathrm{\prime }}\subseteq \mathrm{\Omega }$ be a bounded $C^1$ domain with $\mathrm{\partial }\mathrm{\Omega }\cap \mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$ relatively open in $\mathrm{\partial }\mathrm{\Omega }$, and let $T_{{\mathrm{\Omega }}^{\mathrm{\prime }}}$ be the trace operator relative to ${\mathrm{\Omega }}^{\mathrm{\prime }}$, which is defined by [F1] precisely when ${\mathrm{\Omega }}^{\mathrm{\prime }}$ is a bounded $C^1$ domain of the type covered there. Define $R:W^{1,p}(\mathrm{\Omega })\to L^p(\mathrm{\partial }\mathrm{\Omega }\cap \mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }})$, $Ru:=(T_{{\mathrm{\Omega }}^{\mathrm{\prime }}}(u{\mathrm{\mid }}_{{\mathrm{\Omega }}^{\mathrm{\prime }}})){\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }\cap \mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}}$, a bounded linear map by [F1] and [F3]. On the dense class $D$ of smooth restrictions, $Ru$ is the classical restriction of $u$ to $\mathrm{\partial }\mathrm{\Omega }\cap \mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}$, which equals $(Tu){\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }\cap \mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}}$ by [F2]. Two bounded linear maps that agree on the dense subspace $D$ are equal, so the identity holds on all of $W^{1,p}(\mathrm{\Omega })$; the two surface measures agree on the common piece by [F6].

Chart transport (iii). Fix a compact ambient patch $K\subset W$ and a smooth ambient cutoff $\zeta \in C_c^{\mathrm{\infty }}(W)$ equal to one near $K$. For $u$ supported in $K\cap \bar{\mathrm{\Omega }}$, choose $u_m\in D$ tending to $u$ by [F4]. Then $\zeta u_m\to u$ in $W^{1,p}(\mathrm{\Omega })$ by [F3]. Flatten, reflect, and extend each chart-supported function by zero within the half-space. These operations are bounded by [F4] (apply the local composition formula on interior patches and exhaust the chart with the uniform compact ambient derivative bounds); zero extension across the artificial chart edge is licensed by the cutoff support margin. The flattened functions are continuous, compactly supported and Sobolev, so their flat traces equal their classical restrictions by [F5], although they need only be $C^1$, since the chart is $C^1$. Those restrictions equal the transported $T(\zeta u_m)$ by [F2]. Pass to the limit using both trace bounds. The surface formula [F6] gives the claimed norm comparison since its density is bounded above and below on the compact patch.

Conclusion. Step 1.1 proves (i) with the stated bound; step 1.2 proves (ii) by uniqueness of the bounded extension from the dense smooth class; step 1.3 proves (iii), including the equivalence of the transported boundary norms up to the chart Jacobian.