Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

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