  We develop basic properties of solutions to the Dirac-Hodge and Laplace
equations in upper half space endowed with the hyperbolic metric. Solutions to
the Dirac-Hodge equation are called hypermonogenic functions while solutions to
this version of Laplace's equation are called hyperbolic harmonic functions. We
introduce a Borel-Pompeiu formula and a Green's formula for hyperbolic harmonic
functions. Using a Cauchy Integral formula we are able to introduce Hardy
spaces of solutions to the Dirac-Hodge equation. We also provide new arguments
describing the conformal covariance of hypermonogenic functions and invariance
of hyperbolic harmonic functions. We introduce intertwining operators for the
Dirac-Hodge operator and hyperbolic Laplacian.
