W 2,p a priori estimates for nonvariational operators: the sharp maximal function technique

We consider a nonvariational degenerate elliptic operator, structured on a system of left invariant, 1-homogeneous, Hormander vector fields on a Carnot group, where the coefficient matrix is symmetric, uniformly positive on a bounded domain and the coefficients are locally VMO. We discuss a new proof (given in [BT] and also based on results in [BF]) of the interior estimates in Sobolev spaces, first proved in [BB-To]. The present proof extends to this context Krylov' technique, introduced in [K1], consisting in estimating the sharp maximal function of second order derivatives.