A bounded right inverse of the trace, supported in a prescribed collar

Construction inside a prescribed collar. Let $U$ be an open neighbourhood of the compact boundary. Choose a finite boundary atlas $\{({\mathrm{\Phi }}_j,W_j)\}$ with $W_j\subseteq U$ (shrinking the chart neighbourhoods of the boundary, which is possible because $U$ is open and contains $\mathrm{\partial }\mathrm{\Omega }$) and a subordinate finite ambient partition $\{{\rho }_j\}$ with $\mathrm{supp}{\rho }_j\subseteq W_j$ and ${\sum }_j{\rho }_j=1$ on a neighbourhood of $\mathrm{\partial }\mathrm{\Omega }$, by [F4]. For $g\in W^{\theta ,p}(\mathrm{\partial }\mathrm{\Omega })$ and each $j$, transport the localised datum: $g_j:=({\rho }_{j}g)\circ {\mathrm{\Psi }}_j^{-1}\in W^{\theta ,p}({\mathbb{R}}^{n-1})$; lift it flat, $w_j:=R_{+}{g}_j$, so that $T_{+}{w}_j=g_j$ and $\mathrm{\parallel }{w}_j\mathrm{\parallel }\le C\mathrm{\parallel }{g}_j\mathrm{\parallel }$ by [F1]; then transport back through the chart and multiply by a fixed cutoff equal to $1$ on a neighbourhood of $\mathrm{supp}{\rho }_j$ and supported in $W_j\cap U$. The result $u_j$ is a class in $W^{1,p}(\mathrm{\Omega })$ with support in $U$ and $\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}\le C_j\mathrm{\parallel }{g}_j{\mathrm{\parallel }}_{W^{\theta ,p}({\mathbb{R}}^{n-1})}$ by [F3].

The traces of the pieces. By the chart-transport and multiplicativity parts of [F2], applied to the flattened piece and its cutoff, $T{u}_j=({\rho }_j{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }})g={\rho }_{j}g$ on $\mathrm{\partial }\mathrm{\Omega }$: the transported flat lift has flat trace $g_j$, and multiplication by the cutoff, which equals one near the support of ${\rho }_j$ on the boundary, leaves the localised datum unchanged.

The operator $R_U$. Define $R_{U}g:={\sum }_j{u}_j$. This is linear in $g$ (every construction is linear), its image is contained in the classes supported in ${\bigcup }_j{W}_j\subseteq U$, and $\mathrm{\parallel }{R}_{U}g{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}\le {\sum }_j\mathrm{\parallel }{u}_j\mathrm{\parallel }\le C(\mathrm{\Omega },p,U)\mathrm{\parallel }g{\mathrm{\parallel }}_{W^{\theta ,p}(\mathrm{\partial }\mathrm{\Omega })}$ by [F5], the atlas-independence of [F2] and step 1.1. Its trace is $T(R_{U}g)={\sum }_{j}T{u}_j={\sum }_j{\rho }_{j}g=g$ by step 2.1 and linearity of $T$.

Conclusion and non-uniqueness. Taking $U={\mathbb{R}}^n$ gives $R$, and the construction for general $U$ gives $R_U$ with the claimed dependence of its bound. To exhibit distinct right inverses, fix a nonzero $v\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and the bounded nonzero linear functional $\mathrm{\ell }(g)={\int }_{\mathrm{\partial }\mathrm{\Omega }}gdS$ on the boundary space (Hölder and its $L^p$ term give boundedness). Since $Tv=0$ by classical restriction, $\tilde{R}g=Rg+\mathrm{\ell }(g)v$ is another bounded linear right inverse, distinct from $R$. For the collar version choose $v\in C_c^{\mathrm{\infty }}(\mathrm{\Omega }\cap U)$; this open set is nonempty since $U$ contains the boundary. No canonical choice is claimed.