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Poincaré inequality and energy of separating sets

  • Emanuele Caputo und Nicola Cavallucci EMAIL logo
Veröffentlicht/Copyright: 28. März 2025

Abstract

We study geometric characterizations of the Poincaré inequality in doubling metric measure spaces in terms of properties of separating sets. Given a couple of points and a set separating them, such properties are formulated in terms of several possible notions of energy of the boundary, involving for instance the perimeter, codimension type Hausdorff measures, capacity, Minkowski content and approximate modulus of suitable families of curves. We prove the equivalence within each of these conditions and the 1-Poincaré inequality.

MSC 2020: 30L15; 53C23; 49J52

Communicated by Juha Kinnunen


Award Identifier / Grant number: 948021

Funding statement: The first author was supported by the Academy of Finland (Grant No. 321896). He currently acknowledges the support by the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 948021).

Acknowledgements

The first author also thanks the kind hospitality of the University of Ottawa and of Augusto Gerolin. We thank Francesco Nobili for a careful reading of the manuscript.

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Received: 2024-10-09
Revised: 2025-02-11
Published Online: 2025-03-28
Published in Print: 2025-07-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 6.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/acv-2024-0109/html
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