Let $\Omega$ be an open subset of $\mathbb{R}^n$ and consider a distribution $\Phi\in\mathcal{D}'(\Omega)$. Assuming the derivatives (in all directions) of $\Phi$ are locally integrable functions on $\Omega$, is $\Phi$ necessarily a locally integrable function, too?
When $n=1$, one can easily show that this is true since one can integrate the derivative of $\Phi$ to obtain an (locally) absolutely continuous function, which must be $\Phi$ up to constant. What happens when $n>1$? Can we characterize such $\Phi$s in a nice way, perhaps as a dual space of some other elementary spaces?