Energy inequalities for cutoff functions of $p$-energies on metric measures spaces

TL;DR

This paper establishes geometric and functional conditions for cutoff Sobolev inequalities in p-energy spaces on metric measure spaces, using advanced potential estimates and proving singularity of p-energy measures on fractals.

Contribution

It introduces new techniques to derive potential estimates and proves the singularity of p-energy measures on the Sierpiński carpet, advancing understanding of p-energies in fractal spaces.

Findings

Derived Wolff potential estimates for superharmonic functions.

Proved elliptic Harnack inequality for harmonic functions.

Showed p-energy measure is singular on the Sierpiński carpet.

Abstract

For $p>1$, and for a $p$-energy on a metric measure space, we provide various geometric and functional conditions for the validity of the cutoff Sobolev inequality. In particular, we employ a technique of Trudinger and Wang [Amer. J. Math. 124 (2002), no. 2, 369--410] to derive a Wolff potential estimate for superharmonic functions, and a method of Holopainen [Contemp. Math. 338 (2003), 219--239] to prove the elliptic Harnack inequality for harmonic functions. As applications, we make progress toward the capacity conjecture of Grigor'yan, Hu, and Lau [Springer Proc. Math. Stat. 88 (2014), 147--207], and we prove that the $p$-energy measure is singular with respect to the Hausdorff measure on the Sierpi\'nski carpet for all $p>1$, resolving a problem posed by Murugan and Shimizu [Comm. Pure Appl. Math. 78 (2025), no. 9, 1523--1608].