Global Schauder estimates for nondivergence stationary operators modeled on homogeneous H\"ormander vector fields

TL;DR

This paper establishes global regularity and Schauder estimates for a class of non-divergence operators modeled on homogeneous Hörmander vector fields, extending classical results to more general geometric settings.

Contribution

It provides the first global Schauder estimates for non-divergence operators associated with homogeneous Hörmander vector fields that are not necessarily left-invariant.

Findings

Proves global regularity results for the operators.

Establishes Schauder estimates in the context of homogeneous Hörmander vector fields.

Extends classical PDE regularity theory to non-invariant geometric structures.

Abstract

In this paper we prove global regularity results and Schauder estimates for non-divergence stationary operators of the form L=\sum_{i,j=1}^m a_{ij}(x) X_i X_j, where X_1, ..., X_m are homogeneous (but not necessarily left-invariant) H\"ormander vector fields in R^n (n>m), and [a_{ij}(x)] is a symmetric uniformly positive-definite matrix with H\"older-continuous entries w.r.t. the control distance induced by the vector fields X_1, ..., X_m.