Extremal functions for modules of systems of measures

TL;DR

This paper investigates extremal functions for Fuglede's p-module of systems of measures in various geometric settings, extending Rodin's method to new contexts including Carnot groups and the Heisenberg group.

Contribution

It generalizes Rodin's explicit method for finding extremal functions to Euclidean spaces, Carnot groups, and the Heisenberg group, providing new estimates and exact calculations.

Findings

Explicit extremal functions for Euclidean and Carnot group settings.

Estimates for conformal modules of geometric domains.

Calculation of modules and extremal measures for spherical ring domains.

Abstract

We study Fuglede's $p$-module of systems of measures in condensers in Euclidean spaces and on polarizable Carnot groups. We apply and generalize a result by Rodin, which provides an explicit method for finding the extremal function and the 2-module of a foliated family of curves in $\mathbb R^2$, to a variety of settings. In the planar case, we apply Rodin's method to obtain estimates for the conformal module of a parallelogram and of a ring domain using directional dilatations. In $\mathbb R^n,$ we identify the extremal function and compute the $p$-module of images of families of connecting curves and of separating sets with respect to the plates of a condenser under homeomorphisms of certain regularity. Then we calculate the module and find the extremal measures for the spherical ring domain on polarizable Carnot groups and extend Rodin's theorem to the spherical ring domain on the Heisenberg group.