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

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 Sierpiński carpet for all $p>1$, resolving a problem posed by Murugan and Shimizu [Comm. Pure Appl. Math. 78 (2025), no. 9, 1523–1608].