  In this paper we are concerned with global maximal regularity estimates for
elliptic equations with degenerate weights. We consider both the linear case
and the non-linear case. We show that higher integrability of the gradients can
be obtained by imposing a local small oscillation condition on the weight and a
local small Lipschitz condition on the boundary of the domain. Our results are
new in the linear and non-linear case. We show by example that the relation
between the exponent of higher integrability and the smallness parameters is
sharp even in the linear or the unweighted case.
