It is well known that if $f\in \mathcal{D}'(\mathbb{R}^3,\mathbb{R}^3)$ and $\textbf{curl} f= 0$ then there exists a $u\in \mathcal{D}'(\mathbb{R}^3)$ such that $\nabla u = f$.
Question. Does the same result still hold with $f\in \mathcal{S}'(\mathbb{R}^3,\mathbb{R}^3)$ and $u\in \mathcal{S}'(\mathbb{R}^3)$?
This is a question that I asked some time ago on MSE without having a proper answer. https://math.stackexchange.com/questions/2405993/poincar%c3%a9s-lemma-in-the-space-of-tempered-distributions