Estimates of the derivatives for parabolic operators with unbounded coefficients

Abstract

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