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arXiv · 2610.16486

Poincaré inequalities, logarithmic growth and doubling

Joaquim Martin
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Abstract

We investigate when a Poincaré inequality forces the underlying measure to be doubling, with particular attention to the space \(X\) controlling the oscillation. We identify \(L\log L\) as the sharp threshold: under a natural local growth condition on the measure, doubling is forced precisely when \(L\log L(0,1)\not\hookrightarrow X\). We also obtain a doubling criterion under a weaker geometric assumption when the oscillation norm is sufficiently stronger than \(L\log L\). We show that this threshold is sharp in every fixed Euclidean dimension, using a dimension-free local \(L\log L\)-Poincaré inequality for log-concave measures and non-doubling Gaussian examples. We then turn to infinite-dimensional spaces, where doubling is impossible for the measures under consideration. Nevertheless, the \(L\log L\)-Poincaré inequality continues to hold, and we establish sharp characterizations for infinite product models, Gaussian measures on Hilbert spaces under power-type small-ball estimates, and Wiener measure with the uniform norm.

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Joaquim Martin. 2026-10-12. Poincaré inequalities, logarithmic growth and doubling. https://arxiv.org/abs/2610.16486

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