A compactly supported smooth <function> belongs to $H^s(\mathbb R^n)$ for every real $s$. Part (a) therefore puts $u$ in $H^{s+4}$ for every $s$. Taking $s$ arbitrarily large and applying the Sobolev embedding theorem to every <derivative> shows that $u\in C^\infty(\mathbb R^n)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 23G / b / Solution

 Solution (source code)