Modulus and Poincare inequalities on non-self-similar Sierpinski carpets

MODULUS AND POINCARE INEQUALITIES ON NON-SELF-SIMILAR SIERPINSKI CARPETS

It is well known that the classical self-similar Sierpinski 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=(a₁,a₂,...) 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.

Let S_a,m denote the union of the squares at stage m and let S_a be the intersection of the sets S_a,m. Then S_a is a compact planar set, topologically equivalent to the usual Sierpinski carpet. The figure above shows the set S_{(1/3,1/5,1/7,...)}. In the case when the sequence a is in l², the carpet S_a has positive area.

In [2] we prove the following results:

(*) a is in l¹ if and only if S_a (equipped with Euclidean metric and Lebesgue measure) satisfies the 1-Poincare inequality.

(**) For any a in l², 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 Sierpinski carpets", preprint, 2011.