Almost uniform domains and Poincar\'e inequalities

TL;DR

This paper constructs numerous measure-dense subsets of Euclidean and metric spaces, including Heisenberg groups, that support Poincaré inequalities despite having empty interior, expanding understanding of Sobolev extension domains.

Contribution

It introduces a new method to identify subsets supporting Poincaré inequalities without relying on rectilinear or self-similar structures, applicable to a broad class of metric spaces.

Findings

Existence of measure-dense subsets supporting Poincaré inequalities in Euclidean and metric spaces.

First examples of such subsets in step-2 Carnot groups like the Heisenberg group.

New approach to isoperimetric inequalities on spaces without Semmes families of curves.

Abstract

Here we show existence of numerous subsets of Euclidean and metric spaces that, despite having empty interior, still support Poincar\'e inequalities. Most importantly, our methods do not depend on any rectilinear or self-similar structure of the underlying space. We instead employ the notion of uniform domain of Martio and Sarvas. Our condition relies on the measure density of such subsets, as well as the regularity and relative separation of their boundary components. In doing so, our results hold true for metric spaces equipped with doubling measures and Poincar\'e inequalities in general, and for the Heisenberg groups in particular. To our knowledge, these are the first examples of such subsets on any step-2 Carnot group. Such subsets also give, in general, new examples of Sobolev extension domains on doubling metric measure spaces. When specialized to the plane, we give general sufficient conditions for planar subsets, possibly with empty interior, to be Ahlfors 2-regular and to satisfy a (1,2)-Poincar\'e inequality. In the Euclidean case, our construction also covers the non-self-similar Sierpi\'nski carpets of Mackay, Tyson, and Wildrick, as well as higher dimensional analogues not treated in the literature. The analysis of the Poincar\'e inequality with exponent p=1, for these carpets and their higher dimensional analogues, includes a new way of proving an isoperimetric inequality on a space without constructing Semmes families of curves.