An end point Orlicz type estimate for nonlinear elliptic equations.

Abstract We study nonlinear elliptic equations of p -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 Φ (t) = t q p : Φ (| F | p) ∈ L 1 (Q 2 R) ⟹ Φ (| D u | p) ∈ L 1 (Q R). In the previous results, the estimates obtained were strictly above the natural exponents, and the function such as Φ (t) = t log (1 + t) was ruled out for the candidate of Φ. But with our approach, Φ can be selected up to the end point case of the estimates, and the functions Φ (t) = t and Φ (t) = t log (1 + t) can be additionally selected for Φ. [ABSTRACT FROM AUTHOR]