Schauder theorems for a class of (pseudo-)differential operators on finite- and infinite-dimensional state spaces

We prove maximal regularity results in Hölder and Zygmund spaces for linear stationary and evolution equations driven by a class of differential and pseudo-differential operators (Formula presented.), both in finite and in infinite dimension. The assumptions are given in terms of the semigroup generated by (Formula presented.). 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 fractional powers, in infinite dimension.