A Dirichlet Problem for Convolution Operators in Bounded Regions

We study boundary value problems for convolution operators in bounded subregions Ω of RN. Instead of the topological boundary we introduce "boundaries" which are subsets of Ω of positive N-dimensional measure. For a boundary adapted to the convolution kernel we prove uniqueness for the corresponding Dirichlet problem. For general boundaries the existence of a unique best approximating solution is connected with the inverse of some trace operator. In case of convolution kernels of Nikol′skij-Sobolev type we prove uniqueness and existence results for both, exact solutions and best approximating solutions.