Beurling's criterion and extremal metrics for Fuglede modulus

TL;DR

This paper generalizes Beurling's criterion to characterize extremal metrics for Fuglede p-modulus, providing necessary and sufficient conditions, and shows that any suitable Borel function can be extremal for some curve family.

Contribution

It extends Beurling's criterion to Fuglede p-modulus and proves that any positive finite p-norm Borel function can be extremal for some curve family.

Findings

Generalization of Beurling's criterion for p-modulus

Characterization of extremal metrics for Fuglede p-modulus

Any positive finite p-norm Borel function can be extremal

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.