Estimates of the derivatives for parabolic operators with unbounded coefficients

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