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On the Resistance Conjecture

Sylvester Eriksson-Bique

TL;DR

The paper proves that volume doubling, upper capacity bounds Cap$_p(\beta)$, and a Poincaré inequality PI$_p(\beta)$ imply a parabolic Harnack inequality across general metric measure $p$-Dirichlet spaces by deriving the cutoff Sobolev inequality CS$_p$ via a novel Whitney blending/extension framework. This approach unifies analysis on metric spaces, fractals, graphs, and manifolds for all $p\in(1,\infty)$ and yields consequences such as finite martingale dimension and a Cheeger-type differential structure. The method hinges on a log-Caccioppoli bound, Maz'ya truncation, and a Whitney blending construction that localizes arguments and controls energy without assuming regularity of the energy measure. The results extend and refine prior equivalences among PHI, HKE, and CS-type inequalities, broadening applicability to new energy constructions and establishing robust tools for anomalous diffusion and Dirichlet form theory.

Abstract

We give an affirmative answer to the resistance conjecture on characterization of parabolic Harnack inequalities in terms of volume doubling, upper capacity bounds and a Poincaré inequalities. The key step is to show that these three assumptions imply the so called cutoff Sobolev inequality, an important inequality in the study of anomalous diffusions, Dirichlet forms and re-scaled energies in fractals. This implication is shown in the general setting of $p$-Dirichlet Spaces introduced by the author and Murugan, and thus a unified treatment becomes possible to proving Harnack inequalities and stability phenomena in both analysis on metric spaces and fractals and for graphs and manifolds for all exponents $p\in (1,\infty)$. As an application, we also show that a Dirichlet space satisfying volume doubling, Poincaré and upper capacity bounds has finite martingale dimension and admits a type of differential structure similar to the work of Cheeger. In the course of the proof, we establish methods of extension and characterizations of Sobolev functions by Poincaré-inequalities, and extend the methods of Jones and Koskela to the general setting of $p$-Dirichlet spaces.

On the Resistance Conjecture

TL;DR

The paper proves that volume doubling, upper capacity bounds Cap, and a Poincaré inequality PI imply a parabolic Harnack inequality across general metric measure -Dirichlet spaces by deriving the cutoff Sobolev inequality CS via a novel Whitney blending/extension framework. This approach unifies analysis on metric spaces, fractals, graphs, and manifolds for all and yields consequences such as finite martingale dimension and a Cheeger-type differential structure. The method hinges on a log-Caccioppoli bound, Maz'ya truncation, and a Whitney blending construction that localizes arguments and controls energy without assuming regularity of the energy measure. The results extend and refine prior equivalences among PHI, HKE, and CS-type inequalities, broadening applicability to new energy constructions and establishing robust tools for anomalous diffusion and Dirichlet form theory.

Abstract

We give an affirmative answer to the resistance conjecture on characterization of parabolic Harnack inequalities in terms of volume doubling, upper capacity bounds and a Poincaré inequalities. The key step is to show that these three assumptions imply the so called cutoff Sobolev inequality, an important inequality in the study of anomalous diffusions, Dirichlet forms and re-scaled energies in fractals. This implication is shown in the general setting of -Dirichlet Spaces introduced by the author and Murugan, and thus a unified treatment becomes possible to proving Harnack inequalities and stability phenomena in both analysis on metric spaces and fractals and for graphs and manifolds for all exponents . As an application, we also show that a Dirichlet space satisfying volume doubling, Poincaré and upper capacity bounds has finite martingale dimension and admits a type of differential structure similar to the work of Cheeger. In the course of the proof, we establish methods of extension and characterizations of Sobolev functions by Poincaré-inequalities, and extend the methods of Jones and Koskela to the general setting of -Dirichlet spaces.
Paper Structure (17 sections, 17 theorems, 92 equations)

This paper contains 17 sections, 17 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background on the cutoff Sobolev inequality
  4. Key proof ideas
  5. Outline
  6. $p$-Dirichlet spaces
  7. Axioms and basic properties
  8. Harmonic and superharmonic functions
  9. Poincaré inequality and capacity bounds
  10. Proof of Main Theorems
  11. Proof of Theorem \ref{['thm:mainthm']}
  12. Proof of Theorem \ref{['thm:mainthmhke']}
  13. Whitney Blending
  14. Sobolev partitions of unity
  15. Discrete convolutions
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(X,d,\mu,\mathop{\mathrm{\mathcal{E}}}\nolimits, \mathop{\mathrm{\mathcal{F}}}\nolimits)$ be a geodesic metric measure Dirichlet space, then the following are equivalent:

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.4
  • Lemma 1.6
  • Definition 2.1
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 24 more