Sharp L2-stability and gradient stability hierarchies of the L2-Poincaré inequalities on Euclidean balls and Gaussian Poincaré inequality
Abstract
We study the sharp L2-Poincaré inequalities on Euclidean balls and with respect to the Gaussian measure, and we focus on their stability. In both settings, the sharp constant is the first nonzero eigenvalue of a self-adjoint operator and the set of optimizers is a finite dimensional linear space. Using the spectral decomposition, we establish sharp L2-stability estimates and sharp gradient stability estimates, together with the stability of the stability inequalities in both norms, with explicit optimal constants and with the complete characterization of the equality cases. We then show that this process can be continued. The Poincaré deficit is equal to an infinite sum of L2-distances to the increasing spaces of optimizers of the successive stability inequalities. It is also equal to an infinite sum of L2-gradient distances to the same spaces. The coefficients of the first sum are the spectral gaps μj−μj−1, and the coefficients of the second sum are μ1 times the gaps of the reciprocal spectrum, μ1(μj−1−1−μj−1). On the Euclidean ball, all the constants are expressed through the roots of Bessel functions, and we prove that the first two spectral levels are given by the degree one and the degree two Neumann modes in all dimensions, while the third level is given by the first radial mode when N=2,3 and by the degree three modes when N≥4. In the Gaussian case, all the constants are rational numbers and the L2-sum has all coefficients equal to $1$; we prove that this property characterizes the arithmetic spectrum.
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BibTeX
@misc{lam2026sharp,
title = {{Sharp $L^2$-stability and gradient stability hierarchies of the $L^2$-Poincaré inequalities on Euclidean balls and Gaussian Poincaré inequality}},
author = {Nguyen Lam and Guozhen Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22626},
eprint = {2609.22626},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22626},
}
RIS
TY - UNPB
TI - Sharp L^2-stability and gradient stability hierarchies of the L^2-Poincaré inequalities on Euclidean balls and Gaussian Poincaré inequality
AU - Nguyen Lam
AU - Guozhen Lu
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22626
UR - https://arxiv.org/abs/2609.22626
AB - We study the sharp L^2-Poincaré inequalities on Euclidean balls and with respect to the Gaussian measure, and we focus on their stability. In both settings, the sharp constant is the first nonzero eigenvalue of a self-adjoint operator and the set of optimizers is a finite dimensional linear space. Using the spectral decomposition, we establish sharp L^2-stability estimates and sharp gradient stability estimates, together with the stability of the stability inequalities in both norms, with explicit optimal constants and with the complete characterization of the equality cases. We then show that this process can be continued. The Poincaré deficit is equal to an infinite sum of L^2-distances to the increasing spaces of optimizers of the successive stability inequalities. It is also equal to an infinite sum of L^2-gradient distances to the same spaces. The coefficients of the first sum are the spectral gaps μ_j−μ_j−1, and the coefficients of the second sum are μ_1 times the gaps of the reciprocal spectrum, μ_1(μ_j−1^−1−μ_j^−1). On the Euclidean ball, all the constants are expressed through the roots of Bessel functions, and we prove that the first two spectral levels are given by the degree one and the degree two Neumann modes in all dimensions, while the third level is given by the first radial mode when N=2,3 and by the degree three modes when N≥4. In the Gaussian case, all the constants are rational numbers and the L^2-sum has all coefficients equal to $1$; we prove that this property characterizes the arithmetic spectrum.
ER -