Removable singularities for div v=f in weighted Lebesgue spaces

Abstract

Let w ∈ L 1 loc(R n) be a positive weight. Assuming 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∞ 1/w are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for L p 1/w, 1 < p < +∞, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.