** 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 [1] for any finite p. We consider
non-self-similar carpets. For a sequence
**a**=(a_{1},a_{2},...) of reciprocals of odd
integers, carry out the following recursive procedure:

- At stage 0, begin with the unit square [0,1]
^{2}. - For each positive integer m, at stage m, divide each current
square into a
_{m}^{-2}essentially disjoint congruent subsquares and remove the central subsquare.

In [2] we prove the following results:

(*) **a** is in l^{1} if and only if S_{a}
(equipped with Euclidean metric and Lebesgue measure) satisfies the
1-Poincare inequality.

(**) For any **a** in l^{2}, the carpet
S_{a} 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.

[1] J. Heinonen and P. Koskela, "Quasiconformal maps in metric spaces
with controlled geometry", *Acta Math.* **181** (1998), no. 1,
1–61.

[2] 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.