Failure of an Lp hyperplane trace
= Failure of an Lp hyperplane trace
For $1\leq p<\infty$, no bounded map from $L^p(\mathbb R^n)$ to $L^p(\mathbb R^{n-1})$ can agree with restriction on continuous functions. Indeed,
$$
u_k(x',x_n)=\phi(x')\psi(kx_n),\qquad \psi(0)=1,
$$
has a fixed nonzero trace while $\lVert u_k\rVert_p=k^{-1/p}\lVert\phi\rVert_p\lVert\psi\rVert_p\to0$.