Let $u \in \mathscr E'(\mathbb R^n)$ we a distribution with compact support. Then $u \in C^\infty(\mathbb R^n)$ if and only if for any $N \in \mathbb N$ there exists $C_N > 0$ such that $$ |\hat u(\xi)| \leq C_N (1+|\xi|)^{-N}, \quad \xi \in \mathbb R^n, $$ where $\hat u$ denotes the Fourier transform. Thus, we can check global smoothness of $u$ looking at its Fourier transform.
Now let $x_0 \in \mathbb R^n$. We say that $x_0 \not\in \mathop{\mathrm{sing}} \mathop{\mathrm{supp}} u$ if there exists $\varphi \in C^\infty_c(\mathbb R^n)$ such that $\varphi u \in C^\infty(\mathbb R^n)$. In terms of Fourier transform this means that for any $N \in \mathbb N$ there exists $C_N > 0$ such that $$ \left|\int_{\mathbb R^n} \hat u(\eta) \hat \varphi(\xi-\eta) d\eta \right| \leq C_N (1+|\xi|)^{-N}, \quad \xi \in \mathbb R^n. $$ This condition is less restrictive than the first one but it is not explicit. Given $\hat u$ I can't directly check it to say whether $u$ is smooth at point $x_0$ or not. My question is whether there are some explicit sufficient conditions in terms of $\hat u$ which guarantee that $u$ is smooth at $x_0$ and are more general than the condition of global smoothness?