Hitting times, functional inequalities, Lyapunov conditions and uniform ergodicity

The use of Lyapunov conditions for proving functional inequalities was initiated in [5]. It was shown in [4,30] that there is an equivalence between a Poincaré inequality, the existence of some Lyapunov function and the exponential integrability of hitting times. In the present paper, we close the scheme of the interplay between Lyapunov conditions and functional inequalities by