On singularity of $p$-energy measures on metric measure spaces

TL;DR

This paper demonstrates that under certain geometric and analytic conditions, the $p$-energy measure on metric measure spaces is singular with respect to the underlying measure, especially on fractals like the Sierpiński gasket.

Contribution

It establishes the singularity of $p$-energy measures under volume doubling, Poincaré, and Sobolev inequalities with high walk dimension, extending to fractals and spaces with specific regularity.

Findings

$p$-energy measures are singular under certain inequalities and conditions.

On fractals, the singularity holds for all $p$ above the conformal dimension.

Equivalence between Poincaré, Sobolev inequalities, and resistance estimates under regularity.

Abstract

For $p\in(1,+\infty)$, we prove that for a $p$-energy on a metric measure space, under the volume doubling condition, the conjunction of the Poincar\'e inequality and the cutoff Sobolev inequality both with $p$-walk dimension strictly larger than $p$ implies the singularity of the associated $p$-energy measure with respect to the underlying measure. We also prove that under the slow volume regular condition, the conjunction of the Poincar\'e inequality and the cutoff Sobolev inequality is equivalent to the resistance estimate. As a direct corollary, on a large family of fractals and metric measure spaces, including the Sierpi\'nski gasket and the Sierpi\'nski carpet, we obtain the singularity of the $p$-energy measure with respect to the underlying measure for all $p$ strictly great than the Ahlfors regular conformal dimension.