We study the regularity properties of a general second order H\"ormander operator with Dini continous coefficients $a_{ ij}$. Precisely if $X_0, X_1,\cdots X_m$ are smooth self adjoint vector fields satisfying the H\"ormander condition, we consider the linear operator in $\mathbb{ R}^{ N}$, with $N>m+1$: \begin{ equation*} \mathcal{ L} u := \sum_{ i, j= 1}^{ m} a_{ ij} X_{ i}X_{ j} u - X_0 u. \end{ equation*} The vector field $X_0$ plays a role similar to the time derivative in a parabolic problem so that it is a vector of degree two. We prove that, if $f$ is a Dini continuous function, then the second order derivatives of the solution $u$ to the equation $\mathcal{ L} u = f$ are Dini continuous functions as well. A key step in our proof is a Taylor formula in this anisotropic setting, that we establish under minimal regularity assumptions.
Categories: math.AP