On divergence operators: Free space and vanishing charges

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