Removable singularities for div v = f in weighted Lebesgue spaces

Let $w\in L^1\mathrm{\_}loc({\mathbb{R}}^n)$ be apositive weight. Assuming that a doubling condition and an $L^1$ Poincar'e inequality on balls for the measure $w(x)dx$, as well as a growth condition on $w$, we prove that the compact subsets of ${\mathbb{R}}^n$ which are removable for the distributional divergence in $L^{\mathrm{\infty }}\mathrm{\_}1\mathrm{/}w$ are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for $L^p\mathrm{\_}1\mathrm{/}w$, $1\text{textless}p\text{textless}+\mathrm{\infty }$, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.