  We study boundary value problems for some differential operators on Euclidean
space and the Heisenberg group which are invariant under the conformal group of
a Euclidean subspace resp. Heisenberg subgroup. These operators are shown to be
self-adjoint in certain Sobolev type spaces and the related boundary value
problems are proven to have unique solutions in these spaces. We further find
the corresponding Poisson transforms explicitly in terms of their integral
kernels and show that they are isometric between Sobolev spaces and extend to
bounded operators between certain $L^p$-spaces.
  The conformal invariance of the differential operators allows us to apply
unitary representation theory of reductive Lie groups, in particular recently
developed methods for restriction problems.
