## 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
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