Analytic regularity up to the boundary for analytic-hypoelliptic sub-Laplacians

Ascertain whether solutions to Dirichlet problems for sub-Laplacians with analytic vector fields that are analytic-hypoelliptic in a domain Ω are analytic up to the boundary ∂Ω. Establish a precise boundary analyticity result under appropriate geometric (e.g., noncharacteristic) or analytic assumptions.

Background

Analytic hypoellipticity for sub-Laplacians on nilpotent groups is known only in specific settings (e.g., Hn), and fails in certain products (e.g., Hn × Rk). Boundary regularity in sub-Riemannian settings often requires noncharacteristic conditions and fine microlocal analysis.

The authors extend Courant’s theorem without noncharacteristic boundary assumptions, but note that analyticity up to the boundary, even when the operator is analytic-hypoelliptic in the interior, remains an unresolved issue.