L-p estimates for nonvariational hypoelliptic operators with VMO coefficients

Let X-1,X-2,...,X-q be a system of real smooth vector fields, satisfying Ho rmander's condition in some bounded domain Omega subset of R-n (n > q). We consider the differential operator [GRAPHICS] where the coefficients a(ij)(x) are real valued, bounded measurable functio ns, satisfying the uniform ellipticity condition: [GRAPHICS] for a.e. x is an element of Omega, every xi is an element of R-q, some cons tant mu. Moreover, we assume that the coefficients a(ij) belong to the spac e VMO ("Vanishing Mean Oscillation"), defined with respect to the subellipt ic metric induced by the vector fields X-1,X-2,...,X-q. We prove the follow ing local L-p-estimate: parallel to X(i)X(j)f parallel to(Lp(Omega')) less than or equal to c{paral lel to Lf parallel to(Lp(Omega)) + parallel to f parallel to(Lp(Omega))} for every Omega' subset of subset of Omega, 1 < p < infinity. We also prove the local Holder continuity for solutions to Lf = g for any g is an elemen t of L-p with p large enough. Finally, we prove L-p-estimates for higher or der derivatives of f, whenever g and the coefficients a(ij) are more regula r.