= Solution
For a smooth $u$ and fixed $y$, the <fundamental theorem of calculus> gives, for $0<x<1$,
$$
|u(0,y)|\leq |u(x,y)|+\int_0^1|D_xu(s,y)|\,ds.
$$
Average this inequality over $x\in(0,1)$, use <Holder inequality> on that unit interval, raise to the power $p$, and integrate in $y$. This proves the estimate behind the <W1p trace theorem on a half-space>:
$$
\|u(0,\cdot)\|_{L^p(\mathbb R)}
\leq C_p\bigl(\|u\|_{L^p(U)}+\|D_xu\|_{L^p(U)}\bigr)
\leq C_p\|u\|_{W^{1,p}(U)}.
$$
Use the <Sobolev extension operator> from part d, approximate $Eu$ in $W^{1,p}(\mathbb R^2)$ by smooth functions, and define $Tu$ as the $L^p(\mathbb R)$ limit of their restrictions to $x=0$. The trace inequality makes this limit independent of the approximation and proves that
$$
T:W^{1,p}(U)\longrightarrow L^p(\mathbb R)
$$
is linear and bounded. For a smooth function that extends continuously to the boundary, $Tu=u|_{\partial U}$, so this is the <W1p trace theorem on a half-space>[trace operator] required.
Solved by gpt-5.6-sol high.
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