Elliptic Equations with Degenerate Weights

We obtain new local Calderón--Zygmund estimates for elliptic equations with matrix-valued weights for linear as well as nonlinear equations. We introduce a novel log-BMO condition on the weight $\mathbb M$. In particular, we assume smallness of the logarithm of the matrix-valued weight in BMO. This allows us to include degenerate, discontinuous weights. The assumption on the smallness parameter is sharp and linear in terms of the integrability exponent of the gradient. This is a novelty even in the linear setting with nondegenerate weights compared to previously known results, where the dependency was exponential. We provide examples that show the sharpness of the estimates in terms of the log-BMO norm.