The sharp maximal function approach to Lp estimates for operators structured on Hörmander’s vector fields

We consider a nonvariational degenerate elliptic operator structured on a system of left invariant, 1-homogeneous, Hörmander's vector fields on a Carnot group in Rn, where the matrix of coefficients is symmetric, uniformly positive on a bounded domain of Rn and the coefficients are bounded, measurable and locally VMO in the domain. We give a new proof of the interior Lp estimates on the second order derivatives with respect to the vector fields, first proved by Bramanti-Brandolini in [Rend. Sem. Mat. dell'Univ. e del Politec. di Torino, Vol. 58, 4 (2000), 389-433], extending to this context Krylov' technique, introduced in [Comm. in P.D.E.s, 32 (2007), 453-475], consisting in estimating the sharp maximal function of the second order derivatives.