×

Which log-Sobolev inequality implies the spectral gap inequality. (Quand l’inégalité log-Sobolev implique l’inégalité de trou spectral.) (French) Zbl 0913.60072

Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXII. Berlin: Springer. Lect. Notes Math. 1686, 30-35 (1998).
Consider the following set-up: \(\mu\) is a probability supported by a locally compact separable space \(E\); \((\mathcal E,\mathcal F)\) is a symmetric Dirichlet form on \((E,\mu)\) such that \(1\in\mathcal F\) and \(\mathcal E(u,1)=0\), \(\forall u\in \mathcal F\); define \(E_2(u)=\int u^2\log u^2 d\mu- \int u^2 d\mu \log\int u^2 d\mu\), for positive \(u\in L^2(\mu)\). \((\mathcal E,\mathcal F)\) is said to satisfy a log-Sobolev inequality if, for \(u\in\mathcal F\), \(u>0\), one has \[ E_2(u)\leq \frac{1}{\Lambda}\Big(\mathcal E(u,u)+m\int u^2 d\mu\Big), \] for some \(\Lambda>0\) and \(m\geq 0\). If \(m=0\), the inequality is said to be tight. \((\mathcal E,\mathcal F)\) satisfies the spectral gap inequality if the spectral gap of the infinitesimal generator associated to \((\mathcal E,\mathcal F)\) is non-zero (i.e. there is a constant \(\lambda_0\) such that the spectrum is contained in \(\{0\}\cup [\lambda_0,\infty\mathclose]\)). It is known that \((\mathcal E,\mathcal F)\) satisfies a tight log-Sobolev inequality if and only if it satisfies log-Sobolev and spectral gap inequalities. Consider the folowing property: If \(u_n\in\mathcal F\) are such that \(\int u_n d\mu=0\), \(\negthinspace{u_n}_{\infty}\leq 1\) and \(\mathcal E(u_n)\to 0\), then \(u_n\to 0\) in \(\mu\)-measure. The paper shows that under this property and a log-Sobolev inequality, the spectral gap inequality holds, and hence a tight log-Sobolev inequality. The result is applied to Riemann manifolds of finite volume with the Dirichlet form associated to the Laplace-Beltrami operator.
For the entire collection see [Zbl 0893.00035].

MSC:

60J45 Probabilistic potential theory
60J25 Continuous-time Markov processes on general state spaces
31C12 Potential theory on Riemannian manifolds and other spaces
PDF BibTeX XML Cite
Full Text: Numdam EuDML