John and uniform domains in generalized Siegel boundaries

Abstract

Given the pair of vector fields $X={\partial }_x+|z{|}^{2m}y{\partial }_t$ and $Y={\partial }_y-|z{|}^{2m}x{\partial }_t,$ where $(x,y,t)=(z,t)\in {\mathbb{R}}^3=\mathbb{C}\times \mathbb{R}$, we give a condition on a bounded domain $\Omega \subset {\mathbb{R}}^3$ which ensures that $\Omega$ is an $(\epsilon ,\delta )$-domain for the Carnot-Carathéodory metric. We also analyze the Ahlfors regularity of the natural surface measure induced at the boundary by the vector fields.

Minor corrections and updated references