On divergence operators: Free space and vanishing charges

We use localized topologies to prove existence and optimal regularity results for the divergence equation $\mathrm{div} (v) = F$ in critical cases $v \in L_1(\Omega;\mathbb{R}^m)$ or $v \in C_0(\Omega;\mathbb{R}^m)$, i.e. we characterize those $F$ for which a solution $v$ exists whose norm is bounded by an appropriate norm of $F$. We assume $\Omega$ satisfies a Poincar\'e inequality or an extension property. We apply the general theory to give examples of admissible $F$ in each case.