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

Abstract

Let $N\geq2$ and let $\Omega\subset\R^N$ be a connected Lipschitz domain, possibly unbounded, symmetric with respect to the origin, and such that $0<\gammaN(\Omega)<1$. We assume that the Gaussian Sobolev embedding $H^1(\Omega,\gammaN)\hookrightarrow L^2(\Omega,\gammaN)$ is compact; a sufficient condition is the existence of a bounded Gaussian Sobolev extension operator from $\Omega$ to $\R^N$. Denote by \[ 0=\mu_0(\Omega)<\mu_1(\Omega)\leq\mu_2(\Omega)\leq\cdots \] the Neumann eigenvalues of the positive Ornstein--Uhlenbeck operator $-\Delta+x\cdot\nabla$ in $\Omega$. We prove the sharp reciprocal-sum inequality \[ \sum_{k=1}^{N}\frac{1}{\mu_k(\Omega)} \geq \frac{N}{\mu_1(B_R)}, \] where $B_R$ is the Euclidean ball centred at the origin and satisfying $\gammaN(B_R)=\gammaN(\Omega)$. 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.