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

## TL;DR

This paper establishes maximal regularity results for a broad class of differential and pseudo-differential operators in both finite and infinite-dimensional spaces, extending classical Schauder theorems to new operator classes.

## Contribution

It provides new Schauder-type regularity theorems for operators like fractional Laplacians and Ornstein-Uhlenbeck operators in finite and infinite dimensions.

## Key findings

- Maximal regularity results in Hölder and Zygmund spaces.
- Extension of Schauder theorems to fractional and Ornstein-Uhlenbeck operators.
- Applicability to both finite and infinite-dimensional spaces.

## Abstract

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.