Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

<p style='text-indent:20px;'>We study the regularity properties of second order linear operator in <inline-formula><tex-math id="M1">\begin{document}$ {{\mathbb {R}}}^{N+1} $\end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \mathscr{L} u : = \sum\limits_{j,k 1}^{m} a_{jk}\partial_{x_j x_k}^2 + 1}^{N} b_{jk}x_k \partial_{x_j} - \partial_t u, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where id="M2">\begin{document}$ A \left( a_{jk} \right)_{j,k 1, \dots, m}, B b_{jk} N} $\end{document}</tex-math></inline-formula> are real valued matrices with constant coefficients, id="M3">\begin{document}$ symmetric and strictly positive. We prove that, if id="M4">\begin{document}$ {\mathscr{L}} satisfies Hörmander's hypoellipticity condition, id="M5">\begin{document}$ f is a Dini continuous function, then derivatives solution id="M6">\begin{document}$ to equation id="M7">\begin{document}$ functions as well. also consider case coefficients id="M8">\begin{document}$ $\end{document}</tex-math></inline-formula>'s. key step our proof Taylor formula for classical solutions id="M9">\begin{document}$ that we establish under minimal assumptions on id="M10">\begin{document}$ $\end{document}</tex-math></inline-formula>.</p>