Construction of $p$-energy measures associated with strongly local $p$-energy forms

TL;DR

This paper constructs canonical $p$-energy measures linked to strongly local $p$-energy forms, proving their key properties and uniqueness, and relating them to existing measures in the literature.

Contribution

It introduces a new method to construct and analyze canonical $p$-energy measures without relying on self-similarity, extending the theory of $L^p$-Dirichlet forms.

Findings

Established the chain and Leibniz rules for the measures.

Proved the uniqueness of these energy measures.

Showed the equivalence with existing Korevaar–Schoen-type measures.

Abstract

We construct canonical $p$-energy measures associated with strongly local $p$-energy forms without assuming self-similarity. Here, $p$-energy forms are $L^p$-analogues of Dirichlet forms, which have recently been studied mainly on fractals. Furthermore, we prove that these measures satisfy the chain and Leibniz rules, and that such "good" energy measures are unique. A key ingredient is a $p$-energy analogue of Le Jan's domination principle. Moreover, we show that the Korevaar$-$Schoen-type $p$-energy measures defined by Alonso-Ruiz and Baudoin (2025, Nonlinear Anal.) coincide with our canonical $p$-energy measures.