Let $w\in L^1\_{ loc}(\R^n)$ 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 $\R^n$ which are removable for the distributional divergence in $L^{ \infty}\_{ 1/w}$ are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for $L^p\_{ 1/w}$, $1\textless{ }p\textless{ }+\infty$, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.
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