  We study the regularity properties of the second order linear operator in
$\mathbb{R}^{N+1}$: \begin{equation*} \mathscr{L} u := \sum_{j,k= 1}^{m}
a_{jk}\partial_{x_j x_k}^2 u + \sum_{j,k= 1}^{N} b_{jk}x_k \partial_{x_j} u -
\partial_t u, \end{equation*} where $A = \left( a_{jk} \right)_{j,k= 1, \dots,
m}, B= \left( b_{jk} \right)_{j,k= 1, \dots, N}$ are real valued matrices with
constant coefficients, with $A$ symmetric and strictly positive. We prove that,
if the operator $\mathscr{L}$ satisfies H\"ormander's hypoellipticity
condition, and $f$ is a Dini continuous function, then the second order
derivatives of the solution $u$ to the equation $\mathscr{L} u = f$ are Dini
continuous functions as well. We also consider the case of Dini continuous
coefficients $a_{jk}$'s. A key step in our proof is a Taylor formula for
classical solutions to $\mathscr{L} u = f$ that we establish under minimal
regularity assumptions on $u$.
