Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 105 / 4 / a / Solution

Extend $u$ by zero to ${\mathbb{R}}^n$; its compact support inside the ball makes the extension smooth. The Fourier transform of a derivative givesThe assumption says that $L_0$ is a uniformly elliptic operator, soMoreover,The Plancherel theorem therefore yieldsAll integrands vanish outside the original support where appropriate, so