= Solution
For $1\leq p<\infty$, choose nonzero $\phi\in C_c(\mathbb R^{n-1})$ and $\psi\in C_c(\mathbb R)$ with $\psi(0)=1$. Define
$$
u_k(x',x_n)=\phi(x')\psi(kx_n).
$$
These functions are continuous and belong to the <Lebesgue space> $L^p(\mathbb R^n)$, while their restrictions to the hyperplane are all the same:
$$
u_k(x',0)=\phi(x').
$$
Changing variables in the normal coordinate gives
$$
\lVert u_k\rVert_{L^p(\mathbb R^n)}^p
=\frac1k\lVert\phi\rVert_{L^p(\mathbb R^{n-1})}^p
\lVert\psi\rVert_{L^p(\mathbb R)}^p
\longrightarrow0.
$$
If the claimed bounded $T$ existed, then
$$
0<\lVert\phi\rVert_p
=\lVert Tu_k\rVert_p
\leq\lVert T\rVert\,\lVert u_k\rVert_p
\longrightarrow0,
$$
a contradiction. This scaling argument is the <failure of an Lp hyperplane trace>.
Back to article page
