$L^2$-Theory for nonlocal operators on domains

TL;DR

This thesis develops an $L^2$-theory framework for nonlocal Lévy-type operators on bounded domains, advancing the mathematical understanding of their properties and applications in analysis.

Contribution

It introduces a comprehensive $L^2$-theory for nonlocal operators on bounded domains, bridging the gap between nonlocal and local operator analysis.

Findings

Established $L^2$-boundedness for nonlocal operators

Derived regularity results for solutions on bounded domains

Connected nonlocal operator theory with classical PDE analysis

Abstract

This thesis explores the $L^2$-Theory for nonlocal operators of L\'evy type on bounded domains, as well as their local counterparts. The research was completed at Bielefeld University in Germany and has since garnered significant attention in the field of analysis of nonlocal operators. To enhance its visibility, we believe it is convenient to make it available on arXiv. Interested readers are strongly encouraged to use the following referencing format: "Guy Fabrice Foghem Gounoue. $L^2$-Theory for nonlocal operators on domains. PhD thesis, Bielefeld University, \url{https://doi.org/10.4119/unibi/2946033}, 2020."