*Preprint*

**Inserted:** 6 jun 2023

**Last Updated:** 7 jun 2023

**Year:** 2023

**Abstract:**

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 $N>m+1$:

$\mathcal{L} u := \sum_{i, j= 1}^{m} a_{ij} X_{i} X_{j} u - X_0 u.$

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.