I am wondering if the following statement holds.
If $u\in \mathscr{S}'$ and $u=\nabla p$ for some $p\in \mathscr{D}'$, then $p$ is in $\mathscr{S}'$.
Here $\mathscr{S}'$ is the space of tempered distributions and $\mathscr{D}'$ is the space of distributions.
I raised this question when I want to show the following claim:
If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \mathscr{S}'$ such that $u=\nabla p$ in $\mathscr{S}'$.
Another guess for my question is based on the generalized Lions' lemma:
If $f\in \mathscr{D}'$ and $\nabla f \in H^{-1}$, then $f\in L^2$.
Thank you for your time.