Almost uniform domains and Poincaré inequalities

Abstract

Abstract Here we show existence of many subsets of Euclidean spaces that, despite having empty interior, still support Poincaré inequalities with respect to the restricted Lebesgue measure. Most importantly, despite the explicit constructions in our proofs, our methods do not depend on any rectilinear or self‐similar structure of the underlying space. We instead employ the uniform domain condition of Martio and Sarvas. This 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 Poincaré inequalities in general, and for the Heisenberg groups in particular. To our knowledge, these are the first examples of such subsets on any (nonabelian) Carnot group. Such subsets also give new examples of Sobolev extension domains, also in the general setting of doubling metric measure spaces. In the Euclidean case, our construction also includes the non‐self‐similar Sierpiński carpets of Mackay, Tyson and Wildrick, as well as higher dimensional analogues not treated in the literature. When specialized to the plane, our results lead to new, general sufficient conditions for a planar subset to be 2‐Ahlfors regular and to satisfy the Loewner condition. Two of these conditions, uniform separation and regularity of boundary components, are also necessary. The sufficiency is obtained with an additional measure density condition.

Keywords

• 26A45
• 30L99 (primary)
• 28A75
• 28A80
• 31E05 (secondary)