Representations of bounded harmonic functions
The paper deals with open subsets D of \({\mathbb{R}}^ d\), \(d\geq 2\) (called Poissonian), where every bounded harmonic function on D is a Poisson integral of a bounded function on its boundary. The intersection of two Poissonian open sets is proved to be Poissonian and a sufficient condition for the union of two Poissonian sets to be Poissonian is given. The authors also give some necessary and sufficient conditions for an open set to be Poissonian, and in particular a necessary and sufficient condition for a Greenian to be Poissonian in terms of its Martin boundary.
scientific article; zbMATH DE number 4202681