Global Schauder estimates for a class of degenerate Kolmogorov equations

Abstract

We consider a class of possibly degenerate second order elliptic operators $\mathcal{A}$ on ${\mathbb{R}}^n$. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Hölder spaces both for elliptic equations and for parabolic Cauchy problems involving $\mathcal{A}$. The Hölder function spaces are defined with respect to a non-euclidean metric related to the operator $\mathcal{A}$.