Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 327 / 3 / b / i / Solution

Suppose $x_0$ has a neighborhood on which ${\nabla }_{\theta }\Phi (x,\theta )\ne 0$ for every $\theta \ne 0$. There one may integrate by parts repeatedly withEach adjoint application lowers the effective symbol order. After enough repetitions, the integral and all its $x$-derivatives converge absolutely and define a smooth function near $x_0$. Therefore