A sharp Gaussian harmonic-mean inequality for Neumann eigenvalues of the Ornstein-Uhlenbeck operator

Abstract

Let $N\ge 2$ and let $\Omega \subset {\mathbb{R}}^N$ be a connected Lipschitz domain, possibly unbounded, symmetric with respect to the origin, and such that $0<\gammaN(Ω)<1$. We assume that the Gaussian Sobolev embedding $H^1(Ω,\gammaN)\hookrightarrow L^2(Ω,\gammaN)$ is compact; a sufficient condition is the existence of a bounded Gaussian Sobolev extension operator from $\Omega$ to ${\mathbb{R}}^N$. Denote by \[ 0=μ_0(Ω)<μ_1(Ω)\leqμ_2(Ω)\leq\cdots \] the Neumann eigenvalues of the positive Ornstein--Uhlenbeck operator $-\Delta +x\cdot \nabla$ in $\Omega$. We prove the sharp reciprocal-sum inequality \[ \sum_{k=1}^{N}\frac{1}{μ_k(Ω)} \geq \frac{N}{μ_1(B_R)}, \] where $B_R$ is the Euclidean ball centred at the origin and satisfying $\gammaN(B_R)=\gammaN(Ω)$. Equality holds if and only if $\Omega =B_R$. The proof combines a coupled $N$-dimensional Ritz argument with a Gaussian raywise rearrangement. The angular imbalance is encoded by a symmetric trace-free matrix, whose contribution is controlled by a finite-dimensional convexity inequality.