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

Kôhei Sasaya

Abstract

We construct canonical $p$-energy measures associated with strongly local $p$-energy forms without any assumptions of self-similarity, where $p$-energy forms are $L^p$-versions of Dirichlet forms, which have recently been studied mainly on fractals. Furthermore, we prove that the measures satisfy the chain and Leibniz rules, and that such a "good" energy measures are unique. As an application, we also prove the $p$-energy version of Mosco's domination principle. Moreover, we show a Korevaar--Schoen-type $p$-energies defined by Alonso-Ruiz and Baudoin coincide with our energy measures.

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

Abstract

We construct canonical -energy measures associated with strongly local -energy forms without any assumptions of self-similarity, where -energy forms are -versions of Dirichlet forms, which have recently been studied mainly on fractals. Furthermore, we prove that the measures satisfy the chain and Leibniz rules, and that such a "good" energy measures are unique. As an application, we also prove the -energy version of Mosco's domination principle. Moreover, we show a Korevaar--Schoen-type -energies defined by Alonso-Ruiz and Baudoin coincide with our energy measures.

Paper Structure

This paper contains 8 sections, 40 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Basic properties of $(\mathcal{F},\mathcal{E}_1)$
  5. Construction of p-energy measures
  6. Triangle and $p$-Clarkson inequalities for energy measures
  7. Chain and Leibniz rules, and uniqueness of energy measures
  8. On Korevaar--Schoen-type $p$-energy forms

Key Result

Theorem 1.4

There exists a unique set $\{\mu_{\langle f\rangle}\}_{f\in\mathcal{F}}$ of Radon measures on $(X,d)$ with the following properties. Moreover, $\mathrm{d}\mu_{\langle \varphi\circ f\rangle}=|\varphi'\circ f|^p \mathrm{d}\mu_{\langle f\rangle}$ holds for any $f\in\mathcal{F}\cap C_c(X)$ and piecewise $C^1$ function $\varphi:\mathbb{R}\to\mathbb{R}$.

Figures (1)

  • Figure 1: Sierpiński Gasket

Theorems & Definitions (95)

  • Definition 1.1: $p$-Energy form: KSdiff
  • Definition 1.2: $p$-Clarkson inequality
  • Theorem 1.4
  • Proposition 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • ...and 85 more