Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

We study the regularity properties of the second order linear degenerate parabolic operators. We prove that, if the operator Lsatisfies Hörmander’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 of the secondo order derivatives. 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.