$p$-energies on p.c.f. self-similar sets

TL;DR

This paper extends the theory of Dirichlet forms to p-energies on p.c.f. self-similar sets for 1<p<∞, establishing existence and non-existence criteria using energy averaging methods.

Contribution

It introduces a framework for p-energies on p.c.f. fractals beyond the classical p=2 case, generalizing Sabot's criterion to all p in (1,∞).

Findings

Proves existence of symmetric p-energies on affine nested fractals.

Extends Sabot's criterion for Dirichlet forms to p≠2.

Employs energy averaging method for construction.

Abstract

We study $p$-energies on post critically finite (p.c.f.) self-similar sets for $1<p<\infty$, as limits of discrete $p$-energies on approximation graphs, extending the construction of Dirichlet forms, the $p=2$ setting. By suitably enlarging the choices of discrete $p$-energies, and employing the energy averaging method developed by Kusuoka-Zhou, we prove the existence of symmetric $p$-energies on affine nested fractals, and extend Sabot's celebrated criterion for existence and non-existence of Dirichlet forms on p.c.f. self-similar sets to the $1<p<\infty$ setting.