Gradient estimates for elliptic equations with measurable nonlinearities

We obtain Calderón–Zygmund type estimate for nonlinear elliptic equations of p-Laplacian type, under the condition that the associated nonlinearity is allowed to be merely measurable in one spatial variable, but has locally small mean oscillation in the remaining spatial variables. This is the minimal regularity requirement on the associated nonlinearity for Calderón–Zygmund type estimate, in the sense that if the associated nonlinearity is allowed to be merely measurable with respect to two independent spatial variables then Calderón–Zygmund type estimate fails in general.