Semiclassical Moser--Trudinger type inequalities for orthonormal families in ℝ2
Abstract
For H1(ℝ2)-orthonormal families {un}n=1N with one-body density ρ=∑n=1N |un|2, we prove the whole-space endpoint Moser--Trudinger inequality ∫ℝ2(e4πρ(x)−1−4πρ(x))/ρ(x) dx ≤ C(N−1+∫ℝ2ρ(x) dx), where the integrand on the left-hand side is defined to be zero on {x: ρ(x)=0}. We develop a heat-semigroup refinement of Rumin's density method to treat both the endpoint coefficient 4π and the whole-space setting. Our inequality is sharp in two respects: the optimality of the coefficient 4π and the linear growth in the rank N. Moreover, through a semiclassical analysis of the density generated by normalized Dirichlet eigenfunctions, we establish the collective asymptotic behaviors of the quantities appearing in the inequality in various regimes. These semiclassical behaviors show that the rank-dependent term N−1 on the right-hand side is indispensable.
Related
- The compactness of Moser-Trudinger type inequalities in the unit ball
- Extremals for Trudinger-Moser inequalities via polygonal profiles
- Best Constants for Anisotropic Moser-Trudinger Inequalities with the Exact Growth Condition and Their Applications
- On sharp singular Moser-Trudinger type inequalities and applications to zero mass (p,Q)-Laplace subelliptic equations in the Heisenberg group
- Unique Positive Extremals and Sharp Stability Constants for Singular Moser-Trudinger Inequalities
BibTeX
@misc{nguyen2026semiclassical,
title = {{Semiclassical Moser--Trudinger type inequalities for orthonormal families in $\mathbbR^2$}},
author = {Phuoc-Tai Nguyen and Tuan-Dat Tran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16922},
eprint = {2610.16922},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16922},
}
RIS
TY - UNPB
TI - Semiclassical Moser--Trudinger type inequalities for orthonormal families in ℝ^2
AU - Phuoc-Tai Nguyen
AU - Tuan-Dat Tran
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16922
UR - https://arxiv.org/abs/2610.16922
AB - For H^1(ℝ^2)-orthonormal families {u_n}_n=1^N with one-body density ρ=∑_n=1^N |u_n|^2, we prove the whole-space endpoint Moser--Trudinger inequality ∫_ℝ^2(e^4πρ(x)−1−4πρ(x))/ρ(x) dx ≤ C(N−1+∫_ℝ^2ρ(x) dx), where the integrand on the left-hand side is defined to be zero on {x: ρ(x)=0}. We develop a heat-semigroup refinement of Rumin's density method to treat both the endpoint coefficient 4π and the whole-space setting. Our inequality is sharp in two respects: the optimality of the coefficient 4π and the linear growth in the rank N. Moreover, through a semiclassical analysis of the density generated by normalized Dirichlet eigenfunctions, we establish the collective asymptotic behaviors of the quantities appearing in the inequality in various regimes. These semiclassical behaviors show that the rank-dependent term N−1 on the right-hand side is indispensable.
ER -