A generalization of Poincaré and log-Sobolev inequalities
Conditions for the validity of the relation \[ \sup_{p\in[1,2)} \{[\mu(f^2)-\mu(| f| ^p)^{2/p}]/\varphi(p)\}\leq\varepsilon(f,f), \quad f\in D(L), \] are provided, where \(\varepsilon(f,f)=-\mu(fLf)\), \(L\) generates in \(L^2(\mu)\) a Markov semigroup with invariant probability measure \(\mu\), and \(\varphi\in C[1,2]\) is a decreasing function, \(\varphi(p)>0\) for \(p<2\). The inequality studied by \textit{R. Latala} and \textit{K. Oleszkiewicz} in [Lect. Notes Math. 1745, 147--168 (2000; Zbl 0986.60017)] is recovered when \(\varphi(p)=C(p-2)^{\alpha}\), \(C>0\), and \(\alpha\in [1,2]\). Connections with other relevant inequalities having an ``additivity'' property are discussed. Thereby, some results on the exponential convergence of Markov semigroups are generalized.
scientific article; zbMATH DE number 2168217
scientific article