Modulus and Poincar\'e inequalities on non-self-similar Sierpinski carpets

TL;DR

This paper characterizes non-self-similar Sierpinski carpets that support Poincaré inequalities and curve families of nontrivial modulus, providing new examples of complex metric spaces with these properties.

Contribution

It introduces a characterization of certain non-self-similar carpets supporting Poincaré inequalities, expanding the class of known metric measure spaces with these features.

Findings

Identifies conditions under which non-self-similar carpets support Poincaré inequalities.

Provides new examples of compact doubling metric measure spaces with these properties.

Shows these spaces can embed isometrically into Euclidean space despite lacking manifold points.

Abstract

A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincar\'e inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poincar\'e inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.