Internal Schauder estimates for Hörmander type equations with Dini continuous source

Abstract

We study the regularity properties of a general second order Hörmander operator with Dini continous coefficients $a_{ij}$. Precisely if $X_0,X_1,\cdots X_m$ are smooth self adjoint vector fields satisfying the Hörmander 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.

arXiv admin note: substantial text overlap with arXiv:2102.10381