Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Heat kernel-based p-energy norms on metric measure spaces

Jin Gao, Zhenyu Yu, Junda Zhang

TL;DR

The paper develops a unified framework for comparing heat kernel–based $p$-energies and Besov–Korevaar–Schoen energies on metric measure spaces, including fractals and fractal glue-ups. It introduces weak-monotonicity properties $(KE)_p$, $(NE)_p$, and $(VE)_p$ and proves their equivalences under two-sided heat kernel estimates, while also delivering BBM-type characterizations for $p eq 2$ and automatic monotonicity for $p=2$ in the Dirichlet-form setting. Under (UHE) and (LHE), it shows that heat-kernel energies and Besov–KS energies are equivalent, yielding BBM-type convergence and Gagliardo–Nirenberg inequalities on nested fractals and their blow-ups. The framework extends classical BBM results from smooth manifolds to fractal spaces, providing a robust bridge between probabilistic heat-kernel methods and deterministic Besov/Korevaar–Schoen approaches in fractal analysis, with explicit applications to nested fractals and fractal glue-ups.

Abstract

We investigate heat kernel-based and other $p$-energy norms (1<p<\infty) on bounded and unbounded metric measure spaces, in particular, on nested fractals and their blowups. With the weak-monotonicity properties for these norms, we generalise the celebrated Bourgain-Brezis-Mironescu (BBM) type characterization for p\neq2. When there admits a heat kernel satisfying the two-sided estimates, we establish the equivalence of various $p$-energy norms and weak-monotonicity properties, and show that these weak-monotonicity properties hold when p=2 (in the case of Dirichlet form). Our paper's key results concern the equivalence and verification of various weak-monotonicity properties on fractals. Consequently, many classical results on p-energy norms hold on nested fractals and their blowups, including the BBM type characterization and Gagliardo-Nirenberg inequality.

Heat kernel-based p-energy norms on metric measure spaces

TL;DR

The paper develops a unified framework for comparing heat kernel–based -energies and Besov–Korevaar–Schoen energies on metric measure spaces, including fractals and fractal glue-ups. It introduces weak-monotonicity properties , , and and proves their equivalences under two-sided heat kernel estimates, while also delivering BBM-type characterizations for and automatic monotonicity for in the Dirichlet-form setting. Under (UHE) and (LHE), it shows that heat-kernel energies and Besov–KS energies are equivalent, yielding BBM-type convergence and Gagliardo–Nirenberg inequalities on nested fractals and their blow-ups. The framework extends classical BBM results from smooth manifolds to fractal spaces, providing a robust bridge between probabilistic heat-kernel methods and deterministic Besov/Korevaar–Schoen approaches in fractal analysis, with explicit applications to nested fractals and fractal glue-ups.

Abstract

We investigate heat kernel-based and other -energy norms (1<p<\infty) on bounded and unbounded metric measure spaces, in particular, on nested fractals and their blowups. With the weak-monotonicity properties for these norms, we generalise the celebrated Bourgain-Brezis-Mironescu (BBM) type characterization for p\neq2. When there admits a heat kernel satisfying the two-sided estimates, we establish the equivalence of various -energy norms and weak-monotonicity properties, and show that these weak-monotonicity properties hold when p=2 (in the case of Dirichlet form). Our paper's key results concern the equivalence and verification of various weak-monotonicity properties on fractals. Consequently, many classical results on p-energy norms hold on nested fractals and their blowups, including the BBM type characterization and Gagliardo-Nirenberg inequality.
Paper Structure (19 sections, 33 theorems, 222 equations, 1 figure)

This paper contains 19 sections, 33 theorems, 222 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Historical background and motivation
  3. Preliminaries
  4. Basic definitions
  5. Besov and Korevaar-Schoen norms
  6. Heat kernel-based $p$-energy norm
  7. Weak-monotonicity properties and main results
  8. Structure of the paper
  9. Consequence of $(KE)_{{ \if@compatibility \mathchar"011B {} \mathchar"011B }}$ and $(NE)_{{ \if@compatibility \mathchar"011B {} \mathchar"011B }}$: BBM type characterization
  10. BBM type characterization of heat kernel-based $p$-energy semi-norms
  11. BBM type characterization of Besov-Korevaar-Schoen $p$-energy semi-norms
  12. Equivalences on metric measure spaces
  13. Equivalence of integral-type semi-norms
  14. Equivalence between $(KE)_{{ \if@compatibility \mathchar"011B {} \mathchar"011B }}$ and $(NE)_{{ \if@compatibility \mathchar"011B {} \mathchar"011B }}$
  15. Connected homogeneous p.c.f self-similar sets and their glue-ups
  16. ...and 4 more sections

Key Result

Theorem 1.1

If a metric measure space $(M,d,{ \if@compatibility \mathchar"0116 {} \mathchar"0116 } )$ admits a heat kernel $\{p_{t}\}_{t>0}$ satisfying (UHE) and (LHE), then we have the following equivalences for ${ \if@compatibility \mathchar"011B {} \mathchar"011B }>0$:

Figures (1)

  • Figure 1: A u.f.r. fractal.

Theorems & Definitions (75)

  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Theorem 1.1
  • Definition 3
  • Theorem 1.2
  • Remark 3
  • Proposition 2.1
  • proof
  • ...and 65 more