  Equivalence of the spectral gap, exponential integrability of hitting times
and Lyapunov conditions are well known. We give here the correspondance (with
quantitative results) for reversible diffusion processes. As a consequence, we
generalize results of Bobkov in the one dimensional case on the value of the
Poincar\'e constant for logconcave measures to superlinear potentials. Finally,
we study various functional inequalities under different hitting times
integrability conditions (polynomial, ...). In particular, in the one
dimensional case, ultracontractivity is equivalent to a bounded Lyapunov
condition.
