# Mapping properties for operator-valued pseudodifferential operators on toroidal Besov spaces

## TL;DR

This paper establishes the continuity of operator-valued pseudodifferential operators on toroidal Besov spaces, broadening understanding of their mapping properties without restrictions on the Banach spaces involved.

## Contribution

It proves continuity properties for operator-valued pseudodifferential operators on vector-valued toroidal Besov spaces without assumptions on the Banach spaces.

## Key findings

- Operators are continuous on toroidal Besov spaces.
- Symbols with limited smoothness still yield bounded operators.
- Kernel representation links to dyadic decomposition in Besov spaces.

## Abstract

In this paper, we consider pseudodifferential operators on the torus with operator-valued symbols and prove continuity properties on vector-valued toroidal Besov spaces, without assumptions on the underlying Banach spaces. The symbols are of limited smoothness with respect to $x$ and satisfy a finite number of estimates on the discrete derivatives. The proof of the main result is based on a description of the operator as a convolution operator with a kernel representation which is related to the dyadic decomposition appearing in the definition of the Besov space.