# Global estimates in Sobolev spaces for homogeneous H\"ormander sums of squares

## TL;DR

This paper establishes global regularity estimates for a class of homogeneous Hörmander sums of squares operators in Sobolev spaces, using a combination of local analysis, homogeneity, and a global lifting technique.

## Contribution

It provides the first comprehensive global Sobolev space estimates for homogeneous Hörmander sums of squares operators, leveraging homogeneity and a novel lifting approach.

## Key findings

- Proved global regularity estimates in Sobolev spaces for homogeneous Hörmander sums of squares.
- Established a connection between local properties and global estimates through a lifting technique.
- Demonstrated the effectiveness of homogeneity in deriving regularity results.

## Abstract

Let $\mathcal{L}=\sum_{j=1}^m X_j^2$ be a H\"ormander sum of squares of vector fields in space $\mathbb{R}^n$, where any $X_j$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in space. In this paper we prove global estimates and regularity properties for $\mathcal{L}$ in the $X$-Sobolev spaces $W^{k,p}_X(\mathbb{R}^n)$, where $X = \{X_1,\ldots,X_m\}$. In our approach, we combine local results for general H\"ormander sums of squares, the homogeneity property of the $X_j$'s, plus a global lifting technique for homogeneous vector fields.