On C¹, C², and weak type-(1,1) estimates for linear elliptic operators

We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the L¹-mean sense satisfies the Dini condition. This in particular answers a question recently raised by Yanyan Li and allows us to improve a result of Haïm Brezis. We also prove a weak type-(1,1) estimate under a stronger assumption on the modulus of continuity. The corresponding results for nondivergence form equations are also established.