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

The singular support of $u\in {\mathcal{D}}^{\prime }(X)$ is the complement of the largest open subset of $X$ on which $u$ is represented by a smooth function.

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

The claim is false because vanishing of the amplitude at one spatial point need not control its derivatives nearby. Take $X=\mathbb{R}$, $k=1$,Then $0\in Z(a)$, but distributionallyup to the Fourier-transform sign convention. Thus $0$ lies in the singular support of $I_{\Phi }(a)$ despite belonging to $Z(a)$.