Global Schauder estimates for a class of degenerate Kolmogorov equations

TL;DR

This paper establishes global Schauder estimates in Hölder spaces for a class of possibly degenerate second order elliptic and parabolic equations, including hypoelliptic Ornstein-Uhlenbeck operators with unbounded coefficients, using a non-euclidean metric.

Contribution

It provides the first global Schauder estimates for degenerate Kolmogorov equations with unbounded coefficients in non-euclidean Hölder spaces.

Findings

Proved global Schauder estimates for elliptic and parabolic equations.

Extended estimates to hypoelliptic Ornstein-Uhlenbeck operators.

Used a non-euclidean metric to define Hölder spaces.

Abstract

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