Poincaré inequalities, logarithmic growth and doubling
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 Llog L as the sharp threshold: under a natural local growth condition on the measure, doubling is forced precisely when Llog L(0,1)not↪ X. We also obtain a doubling criterion under a weaker geometric assumption when the oscillation norm is sufficiently stronger than Llog L. We show that this threshold is sharp in every fixed Euclidean dimension, using a dimension-free local Llog 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 Llog 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.
Related
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BibTeX
@misc{martin2026poincaré,
title = {{Poincaré inequalities, logarithmic growth and doubling}},
author = {Joaquim Martin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16486},
eprint = {2610.16486},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16486},
}
RIS
TY - UNPB
TI - Poincaré inequalities, logarithmic growth and doubling
AU - Joaquim Martin
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16486
UR - https://arxiv.org/abs/2610.16486
AB - 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 Llog L as the sharp threshold: under a natural local growth condition on the measure, doubling is forced precisely when Llog L(0,1)not↪ X. We also obtain a doubling criterion under a weaker geometric assumption when the oscillation norm is sufficiently stronger than Llog L. We show that this threshold is sharp in every fixed Euclidean dimension, using a dimension-free local Llog 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 Llog 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.
ER -