  A sharp pointwise differential inequality for vectorial second-order partial
differential operators, with Uhlenbeck structure, is offered. As a consequence,
optimal second-order regularity properties of solutions to nonlinear elliptic
systems in domains in $\mathbb{R}^n$ are derived. Both local and global
estimates are established. Minimal assumptions on the boundary of the domain
are required for the latter. In the special case of the $p$-Laplace system, our
conclusions broaden the range of the admissible values of the exponent $p$
previously known.
