Limit properties of Poisson kernels of tube domains

Abstract:

If certain local boundary conditions hold near $P \in \partial \Gamma$, the Poisson kernel belonging to a proper cone $\Gamma \subset {{\mathbf {R}}^n}$ converges to a tight $C_0^\ast$ limit as its parameter converges admissibly to P in $\Gamma$. This limit can be identified with a lower-dimensional Poisson kernel. The result always works for polytopic and “rounded” cones; for these, a result on the decrease at infinity is obtained which in fact implies convergence almost everywhere in the appropriate sense of the Poisson integral to certain of its boundary values.