Estimates for Ratios of Low-Order Eigenvalues of the Laplacian with Muckenhoupt Weights on Euclidean Domains
Abstract
We study low-order eigenvalue ratio estimates for the Laplacian with Muckenhoupt A2 weights. For weights bounded above and below, we prove a universal upper bound for λ2/λ1 with constant 1 + 4/n C/c, linear in the ratio of the weight bounds, and recover the classical Payne-Pólya-Weinberger bound as a special case. For radial decreasing weights, we prove that the ball maximizes λ2/λ1 among domains of equal volume, assuming that an admissible radial test function exists (Definition 5.8) and that on the ball the second eigenvalue is attained by the l=1 mode. The argument adapts the method of Ashbaugh--Benguria to the weighted setting via a weighted Chiti comparison theorem and a monotonicity property of the admissible radial test function (Definition 5.8). Higher-order eigenvalue ratios are estimated in terms of the (n−1)-Hausdorff measure of the boundary, under an explicit balance condition on the weight at the covering scale. The generalized Payne-Pólya-Weinberger conjecture for eigenvalue sums remains open in general; conditional results are presented for restricted families of domains.
Related
- Muckenhoupt weights for the Hardy-Littlewood maximal operator on hyperbolic spaces
- A Note on Capacity Muckenhoupt Weights with respect to the Hausdorff Content
- Properties and applications of Muckenhoupt A1 growth weights
- Local Matrix Muckenhoupt Weights and Quantitative Weighted Inequalities Achieving Global Best Known Exponents
- Weighted norm inequalities for operators with new Muckenhoupt type weights in the Bessel setting and applications
BibTeX
@misc{mao2026estimates,
title = {{Estimates for Ratios of Low-Order Eigenvalues of the Laplacian with Muckenhoupt Weights on Euclidean Domains}},
author = {Jing Mao and He-Jun Sun and Lingzhong Zeng},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02701},
eprint = {2611.02701},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02701},
}
RIS
TY - UNPB
TI - Estimates for Ratios of Low-Order Eigenvalues of the Laplacian with Muckenhoupt Weights on Euclidean Domains
AU - Jing Mao
AU - He-Jun Sun
AU - Lingzhong Zeng
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02701
UR - https://arxiv.org/abs/2611.02701
AB - We study low-order eigenvalue ratio estimates for the Laplacian with Muckenhoupt A_2 weights. For weights bounded above and below, we prove a universal upper bound for λ_2/λ_1 with constant 1 + 4/n C/c, linear in the ratio of the weight bounds, and recover the classical Payne-Pólya-Weinberger bound as a special case. For radial decreasing weights, we prove that the ball maximizes λ_2/λ_1 among domains of equal volume, assuming that an admissible radial test function exists (Definition 5.8) and that on the ball the second eigenvalue is attained by the l=1 mode. The argument adapts the method of Ashbaugh--Benguria to the weighted setting via a weighted Chiti comparison theorem and a monotonicity property of the admissible radial test function (Definition 5.8). Higher-order eigenvalue ratios are estimated in terms of the (n−1)-Hausdorff measure of the boundary, under an explicit balance condition on the weight at the covering scale. The generalized Payne-Pólya-Weinberger conjecture for eigenvalue sums remains open in general; conditional results are presented for restricted families of domains.
ER -