A Harnack-type inequality for non-symmetric Markov semigroups

For a strong Feller and irreducible Markov semigroup on a locally compact Polish space, the Harnack-type inequality (1.1) holds if and only if the semigroup has a unique invariant probability measure and is ultracontractive. Moreover, new sufficient conditions for this inequality to hold, as well as upper bound estimates of the underlying constant, are presented for diffusion semigroups on Riemannian manifolds.