Estimates for Ratios of Low-Order Eigenvalues of the Laplacian with Muckenhoupt Weights on Euclidean Domains

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 $1 + \frac{4}{n}\,\frac{C}{c}$, 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.