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

Meng Yang

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arXiv · 2507.08577

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

Meng Yang

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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 Poincar\'e inequality. Conversely, we show that the Poincar\'e inequality together with the cutoff Sobolev inequality recovers the analytic inputs used in the first implication. Moreover, in a lower-dimensional regime, the Poincar\'e 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 Sierpi\'nski carpet for any $p>1$, thereby resolving a problem posed by Murugan and Shimizu [Comm. Pure Appl. Math. 78 (2025), no. 9, 1523--1608].

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Meng Yang. 2025-07-11. Energy inequalities for cutoff functions of $p$-energies on metric measure spaces. https://arxiv.org/abs/2507.08577

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