Carnot-Carath\'eodory Balls on Manifolds with Boundary

TL;DR

This paper extends the quantitative analysis of Carnot--Carathéodory balls and scaling maps from boundaryless manifolds to those with boundary, facilitating the study of subelliptic PDEs with boundary conditions.

Contribution

It introduces a new framework for Carnot--Carathéodory geometry on manifolds with boundary, including scaling maps on interior and non-characteristic boundary regions.

Findings

First to generalize Carnot--Carathéodory theory to manifolds with boundary.

Provides tools for analyzing subelliptic PDEs with boundary conditions.

Lays groundwork for future boundary value problem studies.

Abstract

Nagel, Stein, and Wainger introduced a detailed quantitative study of Carnot--Carath\'eodory balls on a smooth manifold without boundary. Most importantly, they introduced scaling maps adapted to Carnot--Carath\'eodory balls and H\"ormander vector fields. Their work was extended by many authors and has since become a key tool in the study of the interior theory of subelliptic PDEs; in particular, the study of maximally subelliptic PDEs. We introduce a generalization of this quantitative theory to manifolds with boundary, where we have scaling maps both on the interior and on the part of the boundary which is non-characteristic with respect to the vector fields. This is the first paper in a forthcoming series devoted to studying maximally subelliptic boundary value problems.