Poincar inequality and the Lp convergence of semi-groups

Abstract. We prove that for symmetric Markov processes of diffusion type admitting a “carre ́ du champ”, the Poincare ́ inequality is equivalent to the exponential convergence of the associated semi-group in one (resp. all) Lp(µ) spaces for 1 < p < +∞. We also give the optimal rate of convergence. Part of these results extends to the stationary, not necessarily symmetric situation