Vectorial nonlinear potential theory

We settle the longstanding problem of establishing pointwise potential estimates for vectorial solutions u : Omega -> R-N to the non-homogeneous p-Laplacean system -div(vertical bar Du vertical bar(p-2) Du) = mu in Omega subset of R-n, where mu is an R-N-valued Borel measure with finite total mass. In particular, for solutions u is an element of W-loc(1,p-1) (R-n) with a suitable decay at infinity, the global estimates via Riesz and Wolff potentials, vertical bar Du(x(0))vertical bar(p-1) less than or similar to integral R-n d vertical bar mu vertical bar(x)/vertical bar x - x(0)vertical bar(n-1) and vertical bar u(x(0))vertical bar less than or similar to W-1,p(mu) (x(0), infinity) = integral(infinity)(0) (vertical bar mu vertical bar B-e(x(0)))/e(n-p))(1/(p-1)) de/e respectively, hold at every point x(0) such that the corresponding potentials are finite. The estimates allow sharp descriptions of fine properties of solutions which are the exact analog of the ones in classical linear potential theory. For instance, sharp characterizations of Lebesgue points of u and Du and optimal regularity criteria for solutions are provided exclusively in terms of potentials.