A Generalized Maximum Principle for Convolution Operators in Bounded Regions

Abstract

Dealing with the technically motivated concept of convolution operators in bounded regions of \(\mathbb{R}^{N}\) with an underlying nearby boundary condition we extend a formerly proved result about the existence and uniqueness of suitable solutions for dimension \(N\leq 2\) to arbitrary dimensions \(N\). Thus, a first substantial result in a sufficiently generalized form, beyond the very specific case of rectangular regions, is established in this field. The result can also be seen as a generalized maximum principle for so called \(k\)-harmonic functions where \(k\) is the kernel of the given convolution operator.