Beurling's criterion and extremal metrics for Fuglede modulus

Abstract

We formulate a necessary and sufficient condition for an admissible metric to be extremal for the Fuglede p-modulus of a system of measures. When p=2, this characterization generalizes Beurling's criterion, a sufficient condition for an admissible metric to be extremal for the extremal length of a planar curve family. In addition, we prove that every non-negative Borel function in R^n with positive and finite p-norm is extremal for the p-modulus of some curve family.

12 pages, 2 figures (version 4: minor improvements, updated numbering and corrected typos, final version)