Gradient potential estimates

Pointwise gradient bounds via Riesz potentials like those available for the Poisson equation actually hold for general quasilinear equations. rst, we prove a somehow surprising result, asserting the possibility of giving pointwise gradient estimates for solutions of non- linear, non-homogeneous elliptic equations, via usual linear Riesz potentials, exactly as it happens for the Poisson equation via representation formulas. This fact is in turn achieved via a \fractional variation" of De Giorgi's iteration technique, whose presentation is the other aim of the paper. This method could be useful in other contexts, as we shall explain below, since it proposes to iterate level sets of solutions via fractional derivatives also when dealing with integer order equations, which are, on the other hand, \non-dierentiabl e" in the classical sense of so called strong solutions. Finally, we demonstrate applications to the proof of optimal Lipschitz continuity criteria, and to the inference of optimal local estimates. 1.1. Linear representation. We shall rst consider the simpler case of quasilinear equations of the type