Abstract. The aim of this paper is to prove logarithmic Sobolev inequalities for
measures on discrete product spaces, by proving inequalities for an appropriate
Wasserstein-like distance. A logarithmic Sobolev inequality is, roughly speaking,
a contractivity property of relative entropy with respect to some Markov semigroup.
It is much easier to prove contractivity for a distance between measures, than for rela-
tive entropy, since for distances well known linear tools, like estimates through matrix