PreprintCC0arXiv
Monotonicity of Beckner Inequalities
Abstract
In this paper, we prove a monotonicity principle for Beckner inequalities of the form (μ(f2)−μ(fp)2/p)/(2−p)≤ CE(f,f), f∈D(E), 1≤ p≤2, where E is a conservative symmetric Dirichlet form. If the inequality holds for every exponent p, it holds with the same constant C for all q∈[1,p]. This resolves an open question posed in Chapter~6 of Wang's monograph~\cite{Wangbook}. We also study weak Beckner inequalities of the form Ψp(f)≤ β(r)E(f,f)+r Osc(f)2, f∈D(E), r>0, where β is a rate function. We show that the same monotonicity principle preserves the entire rate function. We further derive consequences for polynomial rates, semigroup contractivity, and concentration.
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BibTeX
@misc{lei2026monotonicity,
title = {{Monotonicity of Beckner Inequalities}},
author = {Qingbo Lei and Cheng Li and Bo Wu and Jiyang Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21568},
eprint = {2609.21568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21568},
note = {CC0},
}
RIS
TY - UNPB
TI - Monotonicity of Beckner Inequalities
AU - Qingbo Lei
AU - Cheng Li
AU - Bo Wu
AU - Jiyang Wu
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21568
UR - https://arxiv.org/abs/2609.21568
AB - In this paper, we prove a monotonicity principle for Beckner inequalities of the form (μ(f^2)−μ(f^p)^2/p)/(2−p)≤ CE(f,f), f∈D(E), 1≤ p≤2, where E is a conservative symmetric Dirichlet form. If the inequality holds for every exponent p, it holds with the same constant C for all q∈[1,p]. This resolves an open question posed in Chapter~6 of Wang's monograph~\cite{Wangbook}. We also study weak Beckner inequalities of the form Ψ_p(f)≤ β(r)E(f,f)+r Osc(f)^2, f∈D(E), r>0, where β is a rate function. We show that the same monotonicity principle preserves the entire rate function. We further derive consequences for polynomial rates, semigroup contractivity, and concentration.
ER -