Abstract

This thesis investigates the question of whether a doubling metric measure space supports a Poincar\'e inequality and explains the relationship between the existence of such an inequality and the non-triviality of the respective modulus. It discusses in detail a general class of modified Sierpi\'nski carpets presented by Mackay, Tyson, and Wildrick~\cite{M & T & K}, which are the first examples of spaces that support Poincar\'e inequalities for a renormalized Lebesgue measure that are also compact subsets of Euclidean space with empty interior. It describes the intricate relationship between the sequence used in the construction of a modified Sierpi\'nski carpet and the validity of Poincar\'e inequalities on that space.