On a Class of Hypoelliptic Operators with Unbounded Coefficients in R Balint Farkas and Luca Lorenzi

We consider a class of non-trivial perturbations A of the degenerate OrnsteinUhlenbeck operator in R . In fact we perturb both the diffusion and the drift part of the operator (say Q and B) allowing the diffusion part to be unbounded in R . Assuming that the kernel of the matrix Q(x) is invariant with respect to x ∈ R and the Kalman rank condition is satisfied at any x ∈ R by the same m < N , and developing a revised version of Bernstein’s method we prove that we can associate a semigroup {T (t)} of bounded operators (in the space of bounded and continuous functions) with the operator A . Moreover, we provide several uniform estimates for the spatial derivatives of the semigroup {T (t)} both in isotropic and anisotropic spaces of (Hölder-) continuous functions. Finally, we prove Schauder estimates for some elliptic and parabolic problems associated with the operator A .