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

Abstract

For $p>1$, and for a $p$-energy on a volume doubling metric measure space, we provide several geometric and functional conditions for the validity of the cutoff Sobolev inequality. In particular, we prove that the elliptic Harnack inequality, two-sided capacity bounds, and some additional geometric and analytic assumptions imply the cutoff Sobolev inequality, without assuming the Poincaré inequality. Conversely, we show that the Poincaré inequality together with the cutoff Sobolev inequality recovers the analytic inputs used in the first implication. Moreover, in a lower-dimensional regime, the Poincaré inequality and the capacity upper bound imply the cutoff Sobolev inequality. As an application, we prove that the $p$-energy measure is singular with respect to the Hausdorff measure on the Sierpiński carpet for any $p>1$, thereby resolving a problem posed by Murugan and Shimizu [Comm. Pure Appl. Math. 78 (2025), no. 9, 1523--1608].