L p Estimates for Some Ultraparabolic Operators with Discontinuous Coefficients

We consider a class of ultraparabolic operators of the kind q Lʹ ai j z.­xixjq x,BD:y­t , Ds ­x1, ­x2, . . . , ­xN. i,js1 zs x,t.gRNq1., where the principal part is uniformly elliptic on Rq, qFN, and the constant matrix B is upper triangular and such that the operator obtained by freezing the coefficients ai j at any point z0gRNq1 is hypoelliptic. We prove local L p-estimates for the derivatives ­xixju i, js1, . . . , q.of a solution to the equation Lusf, under the assumption that the coefficients ai j belong to the space VMO ‘‘vanishing mean oscillation’’.with respect to a suitable metric related to B.