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

Abstract

For a class of regular strongly local p-energy forms on complete doubling metric measure spaces, 1<p<∞, 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≲ 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 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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BibTeX RIS

BibTeX

@misc{baudoin2026macroscopic,
  title = {{Macroscopic Poincaré Inequalities and Weak-Type Characterizations of $p$-Energy Forms}},
  author = {Fabrice Baudoin and Andrea Pinamonti and Meng Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29789},
  eprint = {2610.29789},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29789},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Macroscopic Poincaré Inequalities and Weak-Type Characterizations of p-Energy Forms
AU  - Fabrice Baudoin
AU  - Andrea Pinamonti
AU  - Meng Yang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29789
UR  - https://arxiv.org/abs/2610.29789
AB  - For a class of regular strongly local p-energy forms on complete doubling metric measure spaces, 1<p<∞, 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≲ 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 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.
ER  -

Abstract

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