  We prove maximal regularity results in H\"older and Zygmund spaces for linear
stationary and evolution equations driven by a large class of differential and
pseudo-differential operators L, both in finite and in infinite dimension. The
assumptions are given in terms of the semigroup generated by L. We cover the
cases of fractional Laplacians and Ornstein-Uhlenbeck operators with fractional
diffusion in finite dimension, and several types of local and nonlocal
Ornstein-Uhlenbeck operators, as well as the Gross Laplacian and its negative
powers, in infinite dimension.
