Nonlinear Calderón-Zygmund Theory in the Limiting Case

We prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems. Specifically, we deal with the limiting case of the classical theory of Calderón and Zygmund in the setting of nonlinear, possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions. A prototype of the results obtained here tells for instance that (Formula presented.) being a Borel measure with locally finite mass on the open subset Ω ⊂ Rnand p> 2 - 1 / n, then (Formula presented.) for every σ ∈ (0,1).The case σ= 1 is obviously forbidden already in the classical linear case of the Poisson equation - ▵u= μ.