Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

We study the regularity properties of the second order linear operator in $R^{N+1}$: egin{equation*} 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 $A = left( a_{jk} ight)_{j,k= 1, dots, m}, B= left( b_{jk} ight)_{j,k= 1, dots, N}$ are real valued matrices with constant coefficients, with $A$ symmetric and strictly positive. We prove that, if the operator $L$ 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 $L u = f$ 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 $L u = f$ that we establish under minimal regularity assumptions on $u$.