  Dimension-independent Harnack inequalities are derived for a class of
subordinate semigroups. In particular, for a diffusion satisfying the
Bakry-Emery curvature condition, the subordinate semigroup with power $\alpha$
satisfies a dimension-free Harnack inequality provided $\alpha \in(1/2, 1)$,
and it satisfies the log-Harnack inequality for all $\alpha \in (0,1).$ Some
infinite-dimensional examples are also presented.
