The kernel of the trace is the closure of the test functions

The forward inclusion. Every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ is continuous on $\bar{\mathrm{\Omega }}$ and vanishes on a neighbourhood of $\mathrm{\partial }\mathrm{\Omega }$, so $T\phi =0$ by [F1]. Since $T$ is bounded and $W_0^{1,p}(\mathrm{\Omega })$ is the closure of $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ (Zero-boundary Sobolev space as a norm closure), every class in $W_0^{1,p}(\mathrm{\Omega })$ is a limit of test functions and hence has zero trace.

The half-space zero-extension computation. Let $w\in W^{1,p}(H)$ be compactly supported with $T_{+}w=0$, and let $\bar{w}$ be its extension by zero to ${\mathbb{R}}^n$. Choose $w_m\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n){\mathrm{\mid }}_H$ with $w_m\to w$ in $W^{1,p}(H)$ by [F2] and let ${\tilde{w}}_m\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ extend $w_m$. Fix $\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ and $i$. For each $m$, integrating the identity ${\mathrm{\partial }}_i({\tilde{w}}_m\phi )={\mathrm{\partial }}_i{\tilde{w}}_m\phi +{\tilde{w}}_m{\mathrm{\partial }}_i\phi$ over $H$ and using [F6]: for $i\mathrm{\ne }n$ the integral of the tangential derivative vanishes (its inner $x_i$-integral has compact support), and for $i=n$ it equals $-{\int }_{{\mathbb{R}}^{n-1}}{\tilde{w}}_m(x^{\mathrm{\prime }},0)\phi (x^{\mathrm{\prime }},0)d{x}^{\mathrm{\prime }}$; hence ${\int }_H{w}_m{\mathrm{\partial }}_i\phi =-{\int }_H({\mathrm{\partial }}_i{w}_m)\phi -{\delta }_{in}{\int }_{{\mathbb{R}}^{n-1}}(T_{+}{w}_m)\phi (\cdot ,0)$. Passing to the limit by [F7] (the volume pairings converge because $w_m\to w$ and ${\mathrm{\partial }}_i{w}_m\to {\mathrm{\partial }}_{i}w$ in $L^p$ and $\phi ,{\mathrm{\partial }}_i\phi$ are bounded with compact support) and using $T_{+}{w}_m\to T_{+}w=0$ in $L^p({\mathbb{R}}^{n-1})$ gives ${\int }_{H}w{\mathrm{\partial }}_i\phi =-{\int }_H({\mathrm{\partial }}_{i}w)\phi$ for every $i$ and every test function. By the definition of the weak derivative on ${\mathbb{R}}^n$, the zero extension satisfies ${\int }_{{\mathbb{R}}^n}\bar{w}{\mathrm{\partial }}_i\phi =-{\int }_{{\mathbb{R}}^n}\overline{{\mathrm{\partial }}_{i}w}\phi$, so $\bar{w}\in W^{1,p}({\mathbb{R}}^n)$ with $D_i\bar{w}$ the zero extension of $D_{i}w$.

Approximation in the half-space by test functions of $H$. Let $w$ be as in step 1.2. In the notation of [F5], the translates ${\bar{w}}_{\epsilon }:=\bar{w}(\cdot -\epsilon e_n)$ converge to $\bar{w}$ in $W^{1,p}({\mathbb{R}}^n)$ as $\epsilon \downarrow 0$, hence their restrictions to $H$ converge to $w$ in $W^{1,p}(H)$ by [F3]. Each ${\bar{w}}_{\epsilon }$ is supported in $\{x_n\ge \epsilon \}$, so for $0<\delta <\epsilon \mathrm{/}2$ the mollifications ${\rho }_{\delta }\ast {\bar{w}}_{\epsilon }$ lie in $C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ with support in $\{x_n>0\}$ and converge to ${\bar{w}}_{\epsilon }$ in $W^{1,p}({\mathbb{R}}^n)$ by [F8]; their restrictions lie in $C_c^{\mathrm{\infty }}(H)$ and converge to $w$ in $W^{1,p}(H)$. Hence every compactly supported class in $W^{1,p}(H)$ with zero flat trace is a limit of test functions of $H$.

The chart pieces. Let $u\in W^{1,p}(\mathrm{\Omega })$ with $Tu=0$. Choose the finite atlas and partition of [F1] and write $u={\sum }_j{\chi }_{j}u+u_0$, where $u_0:=(1-{\sum }_j{\chi }_j)u$ is supported away from $\mathrm{\partial }\mathrm{\Omega }$. For each $j$ the class ${\chi }_{j}u$ is supported in the chart, $T({\chi }_{j}u)=({\chi }_j{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }})Tu=0$ by [F1], and by the chart-transport part of [F1] its flattening $w_j:=({\chi }_{j}u)\circ {\mathrm{\Phi }}_j^{-1}$, reflected into $H$, is a compactly supported class in $W^{1,p}(H)$ with $T_{+}{w}_j=0$; by [F7] the flattening is bounded, and by step 2.1, $w_j$ is a $W^{1,p}(H)$-limit of test functions of the half-space. Multiply the half-space approximants by a fixed smooth cutoff in the flattened ambient patch equal to one near the support of $w_j$, before pulling them back. The resulting pullbacks have compact support inside $\mathrm{\Omega }$ and converge to ${\chi }_{j}u$ by [F3] and [F7]; because the charts are only $C^1$, these functions need only be $C^1$, not smooth. For each such compactly supported Sobolev approximant, [F4] and [F8] give a smooth approximation with mollifier radius smaller than its distance to $\mathrm{\partial }\mathrm{\Omega }$. These lie in $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and can be chosen with errors tending to zero. Thus ${\chi }_{j}u\in W_0^{1,p}(\mathrm{\Omega })$ for every $j$.

The interior piece and conclusion. The function $1-{\sum }_j{\chi }_j$ is bounded with bounded first derivatives and vanishes on a neighbourhood of $\mathrm{\partial }\mathrm{\Omega }$, so $u_0$ vanishes a.e. outside a compact subset of the open set $\mathrm{\Omega }$; by [F4] its zero extension lies in $W^{1,p}({\mathbb{R}}^n)$. By [F2] choose ${\psi }_m\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ with ${\psi }_m\to {\bar{u}}_0$ in $W^{1,p}({\mathbb{R}}^n)$, and fix $\xi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $\xi =1$ on a neighbourhood of $\mathrm{supp}{u}_0$; then $\xi {\psi }_m\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and $\xi {\psi }_m\to \xi {\bar{u}}_0=u_0$ in $W^{1,p}(\mathrm{\Omega })$ by [F3]. Hence $u_0\in W_0^{1,p}(\mathrm{\Omega })$. Since $W_0^{1,p}(\mathrm{\Omega })$ is a linear subspace and $u={\sum }_j{\chi }_{j}u+u_0$ with every summand in it by step 3.1, $u\in W_0^{1,p}(\mathrm{\Omega })$. Together with the forward inclusion of step 1.1, this proves $\{Tu=0\}=W_0^{1,p}(\mathrm{\Omega })$.