Suppose $x_0$ has a neighborhood on which $\nabla_\theta\Phi(x,\theta)\ne0$ for every $\theta\ne0$. There one may integrate by parts repeatedly with
$$
L=\frac{\nabla_\theta\Phi}
{i|\nabla_\theta\Phi|^2}\mathbin\cdot\nabla_\theta,
\qquad
Le^{i\Phi}=e^{i\Phi}.
$$
Each 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
$$
\boxed{
\operatorname{sing\,supp}I_\Phi(a)
\subset
\{x:\nabla_\theta\Phi(x,\theta)=0
\text{ for some }\theta\ne0\}}.
$$


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-327/3/b/i/solution