# Mapping properties of weakly singular periodic volume potentials in Roumieu classes

## TL;DR

This paper investigates the mapping properties of weakly singular periodic volume potentials, demonstrating their bilinear and continuous dependence on densities and kernels within Roumieu classes, extending prior non-periodic results.

## Contribution

It establishes the bilinear and continuous mapping of periodic volume potentials into Roumieu classes, extending previous non-periodic potential results to the periodic setting.

## Key findings

- Volume potentials map into Roumieu classes of analytic functions.
- The mapping is bilinear and continuous.
- Results extend non-periodic potential analysis to periodic cases.

## Abstract

The analysis of the dependence of integral operators on perturbations plays an important role in the study of inverse problems and of perturbed boundary value problems. In this paper we focus on the mapping properties of the volume potentials with weakly singular periodic kernels. Our main result is to prove that the map which takes a density function and a periodic kernel to a (suitable restriction of the) volume potential is bilinear and continuous with values in a Roumieu class of analytic functions. Such result extends to the periodic case some previous results obtained by the authors for non periodic potentials and it is motivated by the study of perturbation problems for the solutions of boundary value problems for elliptic differential equations in periodic domains.