Korevaar-Schoen $p$-energy forms and associated $p$-energy measures on fractals

TL;DR

This paper develops a method to construct $p$-energy forms on fractals as limits of Besov-type functionals, enabling the analysis of associated measures and extending previous frameworks to self-similar sets.

Contribution

It introduces a new approach to constructing Korevaar-Schoen $p$-energy forms via subsequential limits, applicable to various fractal settings and compatible with existing theories.

Findings

Constructed $p$-energy forms as limits of Besov-type functionals.

Obtained and studied associated $p$-energy measures.

Extended the construction to self-similar fractals, preserving key properties.

Abstract

We construct good $p$-energy forms on metric measure spaces as pointwise subsequential limits of Besov-type $p$-energy functionals under certain geometric/analytic conditions. Such forms are often called Korevaar-Schoen $p$-energy forms in the literature. As an advantage of our approach, the associated $p$-energy measures are obtained and investigated. We also prove that our construction is applicable to the settings of Kigami [Mem. Eur. Math. Soc. 5 (2023)] and Cao-Gu-Qiu [Adv. Math. 405 (2022), no. 108517], yields Korevaar-Schoen $p$-energy forms comparable to the $p$-energy forms constructed in these papers, and can be further modified in the case of self-similar sets to obtain self-similar $p$-energy forms keeping most of the good properties of Korevaar-Schoen ones.