A generalized maximum principle for convolution operators in bounded regions (Q2510964)

In [\textit{J. Reißinger}, J. Math. Anal. Appl. 178, No.~1, 176--195 (1993; Zbl 0795.46020)], the author studied a particular type of Dirichlet problems for convolution operators in bounded regions \(\Omega\) of \(\mathbb R^N\) and proved, in the case of rectangular regions, some results about the existence, uniqueness and best approximation of the solutions for kernels in certain Sobolev spaces. He also pointed out that these results hold for more general regions in the case of dimensions 1 and 2, but that the relative proof cannot be easily generalized to higher dimensions. In the present paper, these results are proved for sufficiently general sets in any arbitrary dimension \(N\). Another aspect of these researches is the existence of a kind of maximum principle for the so called \(k\)-harmonic functions \(f \in L^p\) defined by \(T_kf=0\), where \(T_k\) is a convolution operator whose kernel \(k\) works inside \(\Omega\).