General Hardy-type Inequalities and f-Ergodicity of Markov Processes
This talk is divided into two parts, which mainly introduce some recent research progresses and unresolved problems. The first part is about the optimal constant of general Hardy-type inequalities with non-negative kernels. It covers the classic Hardy-type inequalities and the inequalities with Riemann-Liouville integral operator. In this talk, we give the variational formula and basic estimation for the optimal constants. At the same time, some unresolved problems are introduced. In the second part, we are concerned with the f-ergodicity of Markov processes. Since the motivation of this research is Markov decision process (MDP), we give some basic concept and the main result of MDP. The key point lies in that the semigroup Pt has exponential f-ergodicity if and only if the operator Ptf has exponential convergence. Finally, some further research problems are presented.
