  We study a counterpart of the classical Poisson integral for a family of
weighted Laplace differential equations in Euclidean half space, solutions of
which are known as generalized axially symmetric potentials. These potentials
appear naturally in the study of hyperbolic Brownian motion with drift. We
determine the optimal class of tempered distributions which by means of the
so-called $\mathscr{S}'$-convolution can be extended to generalized axially
symmetric potentials. In the process, the associated Dirichlet boundary value
problem is solved, and we obtain sharp order relations for the asymptotic
growth of these extensions.
