  The Harnack inequality established in [13] for generalized Mehler semigroup
is improved and generalized. As applications, the log-Harnack inequality, the
strong Feller property, the hyper-bounded property, and some heat kernel
inequalities are presented for a class of O-U type semigroups with jump. These
inequalities and semigroup properties are indeed equivalent, and thus sharp,
for the Gaussian case. As an application of the log-Harnack inequality, the HWI
inequality is established for the Gaussian case. Perturbations with linear
growth are also investigated.
