MODULUS AND POINCARE INEQUALITIES ON NON-SELF-SIMILAR SIERPIŃSKI CARPETS
It is well known that the classical self-similar Sierpiński carpet, equipped with the Euclidean metric and Hausdorff measure in its dimension log 8/log 3 does not satisfy the p-Poincare inequality of Heinonen and Koskela  for any finite p. We consider non-self-similar carpets. For a sequence a=(a1,a2,...) of reciprocals of odd integers, carry out the following recursive procedure:
In  we prove the following results:
(*) a is in l1 if and only if Sa (equipped with Euclidean metric and Lebesgue measure) satisfies the 1-Poincare inequality.
(**) For any a in l2, the carpet Sa satisfies the p-Poincare inequality for each p>1.
These are the first known examples of compact Euclidean sets without interior which support Poincare inequalities when equipped with the Euclidean metric and Lebesgue measure.
 J. Heinonen and P. Koskela, "Quasiconformal maps in metric spaces with controlled geometry", Acta Math. 181 (1998), no. 1, 1–61.
 J. M. Mackay, J. T. Tyson and K. Wildrick, "Modulus and Poincare inequalities on non-self-similar Sierpiński carpets", GAFA 23 (2013), no. 3, 985-1034.