Weak Poincar Inequalities, Decay of Markov Semigroups and Concentration of Measures

For a reasonable class of weak Poincar inequalities, the decay of the corresponding Markov semigroups obtained earlier by Rckner and the first named author is improved by removing an extra L 2norm. Next, a concentration estimate of the reference measure is presented for the weak Poincar inequality, which is sharp as illustrated by some examples of onedimensional diffusion processes.