On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary

We study the differentiability of very weak solutions v∈L1(Ω) of (v,L⋆φ)0=(f,φ)0 for all vanishing at the boundary whenever f is in L1(Ω,δ), with δ=dist(x,∂Ω), and L* is a linear second order elliptic operator with variable coefficients. We show that our results are optimal. We use symmetrization techniques to derive the regularity in Lorentz spaces or to consider the radial solution associated to the increasing radial rearrangement function of f.