Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 23F / b / Solution

For $1\le p<\infty$, choose nonzero $\varphi \in C_c({\mathbb{R}}^{n-1})$ and $\psi \in C_c(\mathbb{R})$ with $\psi (0)=1$. DefineThese functions are continuous and belong to the Lebesgue space $L^p({\mathbb{R}}^n)$, while their restrictions to the hyperplane are all the same:Changing variables in the normal coordinate givesIf the claimed bounded $T$ existed, thena contradiction. This scaling argument is the failure of an Lp hyperplane trace.