  We define H\"older classes $\Lambda_\alpha$ associated with a Markovian
semigroup and prove that, when the semigroup satisfies the $\Gamma^2 \geq 0$
condition, the Riesz transforms are bounded between the H\"older classes. As a
consequence, this bound holds in manifolds with nonnegative Ricci curvature. We
also show, without the need for extra assumptions on the semigroup, that a
version of the Morrey inequalities is equivalent to the ultracontractivity
property. This result extends the semigroup approach to the Sobolev
inequalities laid by Varopoulos. After that, we study certain families of
operators between the homogeneous H\"older classes. One of these families is
given by analytic spectral multipliers and includes the imaginary powers of the
generator, the other, by smooth multipliers analogous to those in the
Marcienkiewicz theorem. Lastly, we explore the connection between the H\"older
norm and Campanato's formula for semigroups.
