The $L^p$ trace operator on a bounded $C^1$ domain

The boundary estimate on continuous classes. Let $u\in C(\bar{\mathrm{\Omega }})\cap W^{1,p}(\mathrm{\Omega })$ and let $\{({\mathrm{\Phi }}_j,{\chi }_j)\}$ be the finite atlas and partition of [F3] and [F4], so that $u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}={\sum }_j({\chi }_{j}u){\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$ and the inequality for a sum of $N$ terms gives $\mathrm{\parallel }u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\parallel }}_{L^p(\mathrm{\partial }\mathrm{\Omega })}^p\le N^{p-1}{\sum }_j{\int }_{{\mathbb{R}}^{n-1}}\mid ({\chi }_{j}u)\circ {\mathrm{\Psi }}_j(y){\mid }^p{J}_j(y)dy$, where ${\mathrm{\Psi }}_j(y):={\mathrm{\Phi }}_j^{-1}(y,0)$ and $J_j=\sqrt{1+\mathrm{\mid }D{h}_j{\mathrm{\mid }}^2}$. For each $j$ put $u_j:=({\chi }_{j}u)\circ {\mathrm{\Phi }}_j^{-1}$ on the flattened patch ${\mathrm{\Phi }}_j(W_j)\cap \{t<0\}$ and extend it by zero; because $\mathrm{supp}{\chi }_j$ is compactly contained in $W_j$, the extension is compactly supported and continuous on the closed half-space, and $u_j\in W^{1,p}$ of the half-space with $\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{W^{1,p}}\le C_j\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}$ by [F2] and [F7]. Reflecting $t\mapsto -t$ and applying the half-space estimate [F1] to the reflected function, whose boundary value is $y\mapsto ({\chi }_{j}u)\circ {\mathrm{\Psi }}_j(y)$, gives $\int \mathrm{\mid }({\chi }_{j}u)\circ {\mathrm{\Psi }}_j{\mathrm{\mid }}^{p}dy\le C_j^{\mathrm{\prime }}\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}^p$. Since $J_j$ is bounded on the compact patch, $\int \mathrm{\mid }({\chi }_{j}u)\circ {\mathrm{\Psi }}_j{\mathrm{\mid }}^p{J}_j(y)dy\le \mathrm{\parallel }{J}_j{\mathrm{\parallel }}_{\mathrm{\infty }}\int \mathrm{\mid }({\chi }_{j}u)\circ {\mathrm{\Psi }}_j{\mathrm{\mid }}^{p}dy$, and summing the finitely many bounds yields $\mathrm{\parallel }u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\parallel }}_{L^p(\mathrm{\partial }\mathrm{\Omega })}\le C(\mathrm{\Omega },p)\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}$.

Construction of $T$ by density. Let $D$ be the class of restrictions to $\mathrm{\Omega }$ of functions in $C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$, a linear subspace of $C(\bar{\mathrm{\Omega }})\cap W^{1,p}(\mathrm{\Omega })$ that is dense in $W^{1,p}(\mathrm{\Omega })$ by [F6]. The restriction map $S:D\to L^p(\mathrm{\partial }\mathrm{\Omega })$, $S\phi :=\phi {\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$, is linear and satisfies $\mathrm{\parallel }S\phi {\mathrm{\parallel }}_{L^p(\mathrm{\partial }\mathrm{\Omega })}\le C(\mathrm{\Omega },p)\mathrm{\parallel }\phi {\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}$ by step 1.1. Since $D$ is dense in the complete space $W^{1,p}(\mathrm{\Omega })$ [F9] and carries the subspace norm, the pair $(W^{1,p}(\mathrm{\Omega }),\text{inclusion})$ is a completion of $D$; the completion universal property [F8], applied with the Banach space $L^p(\mathrm{\partial }\mathrm{\Omega })$ [F9], produces a unique bounded linear $T:W^{1,p}(\mathrm{\Omega })\to L^p(\mathrm{\partial }\mathrm{\Omega })$ with $T{\mathrm{\mid }}_D=S$ and $\mathrm{\parallel }T\mathrm{\parallel }\le C(\mathrm{\Omega },p)$.

Agreement, uniqueness and class-dependence. If $u\in C(\bar{\mathrm{\Omega }})\cap W^{1,p}(\mathrm{\Omega })$ and ${\phi }_m\in D$ with ${\phi }_m\to u$ in $W^{1,p}(\mathrm{\Omega })$ (step 1.1 and [F6]), then $T{\phi }_m={\phi }_m{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}\to Tu$ by continuity of $T$, while $\mathrm{\parallel }{\phi }_m{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}-u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\parallel }}_{L^p(\mathrm{\partial }\mathrm{\Omega })}\le C(\mathrm{\Omega },p)\mathrm{\parallel }{\phi }_m-u{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}\to 0$ by step 1.1 applied to ${\phi }_m-u\in C(\bar{\mathrm{\Omega }})\cap W^{1,p}(\mathrm{\Omega })$; hence $Tu=u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$ in $L^p(\mathrm{\partial }\mathrm{\Omega })$. If $T^{\mathrm{\prime }}$ is another bounded linear operator with the same property on $C(\bar{\mathrm{\Omega }})\cap W^{1,p}(\mathrm{\Omega })$, then $T-T^{\mathrm{\prime }}$ vanishes on the dense subspace $D$ and is bounded, hence zero by the same limiting argument; this is the asserted uniqueness. Finally, if $u=v$ are the same $W^{1,p}$ class in the sense of the quotient representation [F10], then $u-v$ is the zero class, $T(u-v)=0$ because $0\in D$ is a limit of the constant sequence and $S0=0$, so $T$ depends only on the class. The construction is chartwise exactly the transported flat trace, and the chartwise definitions agree because both equal $T$ on the dense class.