John and uniform domains in generalized Siegel boundaries

TL;DR

This paper investigates geometric properties of domains in a generalized Siegel boundary setting, establishing conditions for $( ext{epsilon}, ext{delta})$-domain classification and analyzing boundary measure regularity in a sub-Riemannian context.

Contribution

It provides new criteria for domain regularity and boundary measure analysis in the setting of vector fields defining a generalized Siegel boundary.

Findings

Established conditions for domains to be $( ext{epsilon}, ext{delta})$-domains.

Analyzed Ahlfors regularity of boundary measures induced by the vector fields.

Connected geometric domain properties with sub-Riemannian metric structures.

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-Carath\'eodory metric. We also analyze the Ahlfors regularity of the natural surface measure induced at the boundary by the vector fields.