Suppose we have a non- tempered distribution $u\in \mathcal D'(\mathbb R^d)\backslash \mathcal S'(\mathbb R^d)$. Is it possible to have $\partial\_{x\_1}...\partial\_{x\_d}u \in \mathcal S'(\mathbb R^d)$ where the derivative is taken in the sense of Schwartz distributions? I cannot find an example nor prove the converse. Any reference to suggest?



The derivative of a non-tempered distribution can be tempered?
$e^x+e^y$ defines an obviously non tempered distribution on $\mathbb{R}^2$, whose *mixed* derivative $\partial\_x\partial\_y(e^x+e^y)=0$ ...


