Spectral estimates, contractions and hypercontractivity (Q1650235)

The paper is devoted to studying the weighted Laplacian on a weighted-manifold. The latter is a triplet \((M^n,g,\mu)\), where \((M^n,g)\) is a complete smooth \(n\)-dimensional Riemannian manifold endowed with a measure \(\mu=\exp(-V(x))d \mathrm{vol}_g(x)\) and having smooth positive density with respect to the Riemannian volume measure \(\mathrm{vol}_g\). The weighted Laplacian is defined as \[ \Delta_{g,\mu}f:=\exp(V)\nabla_g\cdot (\exp(-V)\nabla_g f), \] where \(\nabla_g\) denotes the Levi-Civita connection and \(g=\langle \cdot,\cdot\rangle\). The weighted-manifold \((M^n,g,\mu)\) satisfies the curvature-dimension condition \(CD(\rho,N)\), \(\rho\in\mathbb{R}\) and \(N=\infty\), if \[ \mathrm{Ric}_g + \nabla_g^2 V \geqslant \rho g \] as symmetric \(2\)-tensors on \(M\), where \(\mathrm{Ric}_g\) denotes the Ricci curvature tensor. Let \(\lambda_k(M^n,g,\mu)\) be the discrete eigenvalues of \(\Delta_{g,\mu}\) arranged in the ascending order counting the multiplicities, \(\gamma_\rho^n\) be the \(n\)-dimensional Gaussian probability measure with covariance \(\frac{1}{\rho}Id\). The main question studied in the paper is: Given an \(n\)-dimensional connected weighted-manifold \((M^n,g,\mu)\) satisfying \(CD(\rho,\infty)\) with \(\rho>0\), does it hold that \[ \lambda_k(M^n,g,\mu)\geqslant \lambda_k(\mathbb{R}^n,|\cdot|,\gamma_\rho^n)\quad\text{ for all }\quad k\geqslant 1? \] The first main results states that this question has a positive answer for any Euclidean space \((\mathbb{R}^n,|\cdot|,\mu)\) satisfying \(CD(\rho,\infty)\) with \(\rho>0\). There are also some extra results in the paper, below we mention those of them concerning multi-dimensional manifolds. Let \((\mathbb{R}^n,|\cdot|,\mu)\) denote a Euclidean weighted manifold where \(\mu=\exp(-V(x))dx\) is a probability measure satisfying \(\nabla^2 V\leqslant \rho \mathrm{Id}\). Then \[ \lambda_k(\mathbb{R}^n,|\cdot|,\mu)\leqslant \lambda_k(\mathbb{R}^n,|\cdot|,g_\rho^n)\quad\text{ for all }\quad k\geqslant 1. \] Assume \(p\in[1,2]\) and \(\nu_p^n:=c_p^n\exp(-\sum_{i=1}^{n}\frac{1}{p}|x_i|^p)\) is a probability measure. Let \(\mu:=\exp(-U)\nu_p^n\), where \(U:\mathbb R^n\to\mathbb{R}\) is convex and \(U(\pm x_1,\ldots,\pm x_n)=U(x)\). Then \[ \lambda_k(\mathbb{R}^n,|\cdot|,\mu)\geqslant \lambda_k(\mathbb{R}^n,|\cdot|,\nu_p^n) \quad\text{ for all }\quad k\geqslant 1. \] Assume \(p\in[2,\infty)\) and \(B_p^n\) is the unit ball on \(\ell_p^n\) rescaled to have volume \(1\); the uniform measure on \(B_p^n\) is denoted by \(\nu_{B_p^n}\). Then \[ \lambda_k(\mathbb{R}^n,|\cdot|,\nu_{B_p^n}) \geqslant \frac{1}{392} \lambda_k(\mathbb{R}^n,|\cdot|,g^n) \quad\text{ for all }\quad k\geqslant 1, \] where \(\lambda_k(\mathbb{R}^n,|\cdot|,g^n)\) are the eigenvalues of \(-\Delta\) on \(B_p^n\) subject to Neumann condition. Let \((M_i,g_i,\mu_i)\) denote two weighted-manifolds and let the map \(T:(M_1,g_1)\to (M_2,g_2)\) denote a Riemannian submersion pushing forward \(\mu_1\) onto \(\mu_2\) up to a finite constant. Then \[ \lambda_k(M_2,g_2,\mu_2)\geqslant \lambda_k(M_1,g_1,\mu_1) \quad\text{ for all }\quad k\geqslant 1. \]