  This paper aims to provide some tools coming from functional inequalities to
deal with quasi-stationarity for absorbed Markov processes. First, it is shown
how a Poincar\'e inequality related to a suitable Doob transform entails
exponential convergence of conditioned distributions to a quasi-stationary
distribution in total variation and in $1$-Wasserstein distance. A special
attention is paid to multi-dimensional diffusion processes, for which the
aforementioned Poincar\'e inequality is implied by an easier-to-check
Bakry-\'{E}mery condition depending on the right eigenvector for the
sub-Markovian generator, which is not always known. Under additional
assumptions on the potential, it is possible to bypass this lack of knowledge
showing that exponential quasi-ergodicity is entailed by the classical
Bakry-\'{E}mery condition.
