Carnot-Carathéodory Balls on Manifolds with Boundary

2025/07/04 by Street, Brian
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.03501

Abstract

Nagel, Stein, and Wainger introduced a detailed quantitative study of Carnot--Carathéodory balls on a smooth manifold without boundary. Most importantly, they introduced scaling maps adapted to Carnot--Carathéodory balls and Hörmander 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.

Citations

• Function spaces and trace theorems for maximally subelliptic boundary value problems
• Sub-Hermitian Geometry and the Quantitative Newlander-Nirenberg Theorem
• Coordinates Adapted to Vector Fields III: Real Analyticity
• Coordinates Adapted to Vector Fields II: Sharp Results
• Coordinates Adapted to Vector Fields: Canonical Coordinates

Related

• Regularity for Subelliptic PDE Through Uniform Estimates in Multi-Scale\n Geometries
• Metric geometry of nonregular weighted Carnot-Carathéodory spaces
• Function spaces and trace theorems for maximally subelliptic boundary value problems
• Submanifolds and differential forms on Carnot manifolds, after M. Gromov and M. Rumin
• Compactness methods for Γ^1,α boundary Schauder estimates in Carnot groups