On the Resistance Conjecture

TL;DR

This paper proves the resistance conjecture linking parabolic Harnack inequalities with volume doubling, capacity bounds, and Poincaré inequalities, using the cutoff Sobolev inequality in $p$-Dirichlet spaces.

Contribution

It establishes that key inequalities imply the cutoff Sobolev inequality in a general setting, unifying analysis on metric spaces, fractals, graphs, and manifolds for all $p \, \in\ (1,\infty)$.

Findings

Proves the resistance conjecture in a broad setting.

Shows finite martingale dimension for certain Dirichlet spaces.

Extends Sobolev function characterizations and methods of Jones and Koskela.

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 Poincar\'e 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, Poincar\'e 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 Poincar\'e-inequalities, and extend the methods of Jones and Koskela to the general setting of $p$-Dirichlet spaces.