An end point Orlicz type estimate for nonlinear elliptic equations

We study nonlinear elliptic equations of [Formula presented]-Laplacian type in divergence form to establish a natural Calderón–Zygmund type theory of an Orlicz space type, where the Lebesgue space is the special case with [Formula presented]: [Formula presented]In the previous results, the estimates obtained were strictly above the natural exponents, and the function such as [Formula presented] was ruled out for the candidate of [Formula presented]. But with our approach, [Formula presented] can be selected up to the end point case of the estimates, and the functions [Formula presented] and [Formula presented] can be additionally selected for [Formula presented].