Characterizations of \(L^{p}\) -Vector Fields Divergence Distributions

We characterize the set $$\mathcal {D}iv_{p}(\mathbb {R}^d)$$ of all tempered distributions $$\mu $$ on $$\mathbb {R}^d$$ for which the equation $$div \,u =\mu $$ has a solution in $$L^{p}(\mathbb {R}^d,\mathbb {R}^d)$$ when $$d\geq 2$$ . Our first characterization of $$\mathcal {D}iv_{p}(\mathbb {R}^d)$$ is linked to the dual of the Riesz potential space $$\mathcal {R}^{1,p'}(\mathbb {R}^d):=\left \{I_{1}f \;:\; f \in L^{p'}(\mathbb {R}^d) \right \}$$ , where $$\frac {d}{d-1}