Let $U\subset\mathbb{R}^{d}$ be an open set. The singular support of a distribution $u\in\mathcal{D}^{\prime}(U)$ is defined to be the compliment of the set of points, which have a neighbourhood in which $u$ can be identified with a smooth function. In a paper [Fourier integral operators. I](https://doi.org/10.1007/BF02392052) of Hörmander, it is claimed that


$$\DeclareMathOperator\singsupp{sing supp}\singsupp(u)=\bigcap\_{\varphi\in C^{\infty}\_{c}(U),\varphi u\in C^{\infty}(U)}\{x\in U\mid \varphi(x)=0\}.$$


However, I don't see why it is the case. Does anyone know how to see that this is the case?


My attempt:


The direction $\Leftarrow$ is quite clear I think: Let $x\notin\singsupp(u)$. Then there is an open neighbourhood $N$ of $x$ on which $u\vert\_{N}\in C^{\infty}$. Let $\chi\in C\_{c}^{\infty}(U)$ be supported in $N$ such that $\chi(x)\neq 0$. Then $\chi u\vert\_{N}\in C^{\infty}(U)$ and hence $x\notin\bigcap\_{\varphi\in C^{\infty}\_{c}(U),\varphi u\in C^{\infty}(U)}\{x\in U\mid \varphi(x)=0\}$.


What is left is the other direction. I essentially have to show that if there exists a test function $\varphi\in C^{\infty}\_{c}(U)$ such that $\varphi u\in C^{\infty}(U)$ and $\varphi(x)\neq 0$, then $x\notin\singsupp(u)$.



Singular support: equivalent definition
The basic idea is the one put forward by Pierre PC's comment above. More precisely, let $u,\varphi,x$ be as in the last paragraph of the OP. There is no loss of generality in assuming that $\varphi(x)=\lambda>0$. Then by continuity of $\varphi$ at $x$ there is an open neighborhood $V\ni x$, $V\subset U$ such that $\varphi(x')>\frac{\lambda}{2}$ for all $x'\in V$, so that by the chain rule $\psi=\frac{1}{\varphi|\_V}\in C^\infty(V)$. One then clearly has that $u|\_V=\psi(\varphi u)|\_V\in C^\infty(V)$ (*exercise:* check this!), thus proving the direction $\Rightarrow$ of the claim.


By the way, this (together with the reasoning used in the OP to get the $\Leftarrow$ direction of the claim) is essentially the same argument used to prove a similar characterization of the support of a distribution: $$\mathrm{supp}\,u = \bigcap\_{\varphi\in C^\infty\_c(U),\phi u\equiv 0}\{x\in U\ |\ \phi(x)=0\}\ .$$


