Estimates of the derivatives for parabolic operators with unbounded coefficients

We consider a class of second-order uniformly elliptic operators A with unbounded coefficients in RN. Using a Bernstein approach we provide several uniform estimates for the semigroup T(t) generated by the realization of the operator A in the space of all bounded and continuous or Hölder continuous functions in RN. As a consequence, we obtain optimal Schauder estimates for the solution to both the elliptic equation λu − Au = f (λ > 0) and the nonhomogeneous Dirichlet Cauchy problem D_tu = Au + g. Then, we prove two different kinds of pointwise estimates of T(t) that can be used to prove a Liouville-type theorem. Finally, we provide sharp estimates of the semigroup T(t) in weighted Lp-spaces related to the invariant measure associated with the semigroup.