On the dichotomy of $p$-walk dimensions on metric measure spaces
Meng Yang
TL;DR
The paper establishes a dichotomy for the $p$-walk dimension $\beta_p$ on volume-doubling metric measure spaces equipped with a family of $p$-energies satisfying PI$_p(\beta_p)$ and CS$_p(\beta_p)$. By linking $\alpha_p(X)=\beta_p/p$ to Besov-space critical exponents and proving monotonicity and regularity properties, it reduces the dichotomy to an open-closed argument: either $\beta_p=p$ for all $p$ in the interval or $\beta_p>p$ for all $p$ in the interval. The results imply that if $\beta_2>2$ (as is known on classical fractals like the Sierpiński gasket and carpet), then the strict inequality $\beta_p>p$ holds for all $p$ in $I$, simplifying the analysis of the $p$-energy framework on these spaces. Collectively, the work links Besov regularity, intrinsic metrics, and energy-dominance to derive a robust global dichotomy for the walk dimension.
Abstract
On a volume doubling metric measure space endowed with a family of $p$-energies such that the Poincaré inequality and the cutoff Sobolev inequality with $p$-walk dimension $β_p$ hold, for $p$ in an open interval $I\subseteq (1,+\infty)$, we prove the following dichotomy: either $β_p=p$ for all $p\in I$, or $β_p>p$ for all $p\in I$.