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Poincaré inequality and energy of separating sets

Emanuele Caputo, Nicola Cavallucci

TL;DR

The paper develops new geometric characterizations of the $1$-Poincaré inequality in doubling metric measure spaces by replacing curve-based criteria with boundary-energy conditions of separating sets. It introduces the $L$-truncated Riesz potential $R_{x,y}^L$ and the weighted measure ${\frak m}_{x,y}^L$, and proves that the $1$-PI property is quantitatively equivalent to uniform lower bounds on multiple energies of separating set boundaries, including weighted perimeter, codimension-1 Hausdorff measure, Minkowski content, variational capacity, and approximate moduli. The main contribution is a unified framework connecting PI to a family of boundary-energies defined on separating sets, with precise equivalence results and explicit constants depending only on the structural data of the space. This approach provides practical, testable geometric criteria for PI that can be applied to fractal and irregular spaces, and it clarifies the role of essential versus topological boundaries in these energy comparisons.

Abstract

We study geometric characterizations of the Poincaré inequality in doubling metric measure spaces in terms of properties of separating sets. Given a couple of points and a set separating them, such properties are formulated in terms of several possible notions of energy of the boundary, involving for instance the perimeter, codimension type Hausdorff measures, capacity, Minkowski content and approximate modulus of suitable families of curves. We prove the equivalence within each of these conditions and the $1$-Poincaré inequality.

Poincaré inequality and energy of separating sets

TL;DR

The paper develops new geometric characterizations of the -Poincaré inequality in doubling metric measure spaces by replacing curve-based criteria with boundary-energy conditions of separating sets. It introduces the -truncated Riesz potential and the weighted measure , and proves that the -PI property is quantitatively equivalent to uniform lower bounds on multiple energies of separating set boundaries, including weighted perimeter, codimension-1 Hausdorff measure, Minkowski content, variational capacity, and approximate moduli. The main contribution is a unified framework connecting PI to a family of boundary-energies defined on separating sets, with precise equivalence results and explicit constants depending only on the structural data of the space. This approach provides practical, testable geometric criteria for PI that can be applied to fractal and irregular spaces, and it clarifies the role of essential versus topological boundaries in these energy comparisons.

Abstract

We study geometric characterizations of the Poincaré inequality in doubling metric measure spaces in terms of properties of separating sets. Given a couple of points and a set separating them, such properties are formulated in terms of several possible notions of energy of the boundary, involving for instance the perimeter, codimension type Hausdorff measures, capacity, Minkowski content and approximate modulus of suitable families of curves. We prove the equivalence within each of these conditions and the -Poincaré inequality.
Paper Structure (11 sections, 24 theorems, 135 equations, 1 figure)

This paper contains 11 sections, 24 theorems, 135 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Acknowledgments
  4. Preliminaries
  5. Curves in metric spaces
  6. Modulus and approximate modulus of a family of curves
  7. The Poincaré inequality and the Riesz potential
  8. Several energies to measure boundaries
  9. Relations between the energies under Poincaré inequality
  10. Relations between the different energies weighted with the Riesz potential
  11. Proof of the main theorem

Key Result

Theorem 1.2

Let $({\rm X},{\sf d},\mathfrak m)$ be a doubling metric measure space. Then the following conditions are quantitatively equivalent:

Figures (1)

  • Figure 1: Logical scheme of the proof of Theorem \ref{['theo:main-intro-p=1-riproposed']}.

Theorems & Definitions (60)

  • Definition 1.1
  • Theorem 1.2
  • Remark 2.1
  • Proposition 2.2: Hei01, CaputoCavallucci2024
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 50 more