First‐order Sobolev spaces, self‐similar energies and energy measures on the Sierpiński carpet

Abstract

For any , we construct ‐energies and the corresponding ‐energy measures on the Sierpiński carpet. A salient feature of our Sobolev space is the self‐similarity of energy. An important motivation for the construction of self‐similar energy and energy measures is to determine whether or not the Ahlfors regular conformal dimension is attained on the Sierpiński carpet. If the Ahlfors regular conformal dimension is attained, we show that any optimal Ahlfors regular measure attaining the Ahlfors regular conformal dimension must necessarily be a bounded perturbation of the ‐energy measure of some function in our Sobolev space, where is the Ahlfors regular conformal dimension. Under the attainment of the Ahlfors regular conformal dimension, the ‐Newtonian Sobolev space corresponding to any optimal Ahlfors regular metric and measure is shown to coincide with our Sobolev space with comparable norms, where is the Ahlfors regular conformal dimension.