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arXiv · 2610.29789

Macroscopic Poincaré Inequalities and Weak-Type Characterizations of $p$-Energy Forms

Fabrice Baudoin·Andrea Pinamonti·Meng Yang
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Abstract

For a class of regular strongly local $p$-energy forms on complete doubling metric measure spaces, $1<p<\infty$, we derive a macroscopic Poincaré inequality from a Poincaré inequality and a capacity upper bound at a doubling scale $Ψ$. The macroscopic inequality compares $L^p$-oscillations at any two scales and yields a uniform two-scale comparison $J_s\lesssim J_r$, $0<r<s$, for the Korevaar-Schoen functionals, strengthening their usual weak monotonicity property. Conversely, under a density assumption on compactly supported functions, the macroscopic inequality yields a regular strongly local Korevaar-Schoen $p$-energy satisfying a $Ψ$-Poincaré inequality. We also show that the macroscopic inequality forces lower $p$-scaling of $Ψ$ on scales not exceeding $\mathrm{diam} X$; together with doubling, this gives two-sided power bounds. Finally, we establish a weak-type characterization of the energy domain of Brezis-Van Schaftingen-Yung type.

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Fabrice Baudoin, Andrea Pinamonti, Meng Yang. 2026-10-21. Macroscopic Poincaré Inequalities and Weak-Type Characterizations of $p$-Energy Forms. https://arxiv.org/abs/2610.29789

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