On the Resistance Conjecture

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.