Lyapunov conditions for logarithmic Sobolev and Super Poincar\'e inequality

We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincar'e inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincar'e and weak Poincar'e inequalities was studied in Bakry and al. This approach allows us to recover and extend in an unified way some known criteria in the euclidean case (Bakry-Emery, Wang, Kusuoka-Stroock …).