Universal potential estimates

We prove a class of endpoint pointwise estimates for solutions to quasilinear, possibly degenerate elliptic equations in terms of linear and nonlinear potentials of Wolff type of the source term. Such estimates allow to bound size and oscillations of solutions and their gradients pointwise, and entail in a unified approach virtually all kinds of regularity properties in terms of the given datum and regularity of coefficients. In particular, local estimates in Hölder, Lipschitz,Morrey and fractional spaces, as well as Calderón–Zygmund estimates, follow as a corollary in a unified way. Moreover, estimates for fractional derivatives of solutions by mean of suitable linear and nonlinear potentials are also implied. The classical Wolff potential estimate by Kilpeläinen & Malý and Trudinger & Wang as well as recent Wolff gradient bounds for solutions to quasilinear equations embed in such a class as endpoint cases.