What is it about?
This article studies a family of Poincaré-type inequalities defined using rearrangement-invariant function spaces on metric measure spaces. The authors analyze under which conditions these inequalities force the underlying measure to satisfy the doubling property, a key structural hypothesis in analysis on metric spaces. The paper introduces a general framework that includes classical Lebesgue and Orlicz–Sobolev settings as particular cases and provides examples that illustrate when the doubling condition can and cannot be recovered.
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Why is it important?
Poincaré inequalities and the doubling condition are central tools in modern analysis on metric spaces, as they allow the extension of many Euclidean techniques to much more general settings. Understanding exactly which “strength” of Poincaré inequality is needed to guarantee doubling clarifies the borderline between spaces where powerful analytical methods work and those where they fail. By working with general rearrangement-invariant spaces, the article unifies and extends several previous results (for instance in Lebesgue and Orlicz contexts), offering a flexible framework that can be adapted to different function spaces arising in analysis and PDEs.
Perspectives
From my personal point of view, the project was an opportunity to connect abstract rearrangement-invariant space theory with concrete geometric questions on metric measure spaces, and to systematize earlier partial results into a single theorem that captures the “gain” on the left-hand side needed for doubling
Walter Andrés Ortíz Vargas
Universidad Internacional de La Rioja
Read the Original
This page is a summary of: A note on rearrangement Poincaré inequalities and the doubling condition, Proceedings of the American Mathematical Society, June 2024, American Mathematical Society (AMS),
DOI: 10.1090/proc/16795.
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