  The aim of this paper is to study two-weight norm inequalities for fractional
maximal functions and fractional Bergman operator defined on the upper-half
space. Namely, we characterize those pairs of weights for which these maximal
operators satisfy strong and weak type inequalities. Our characterizations are
in terms of Sawyer and B\'ekoll\'e-Bonami type conditions. We also obtain a
$\Phi$-bump characterization for these maximal functions, where $\Phi$ is a
Orlicz function. As a consequence, we obtain two-weight norm inequalities for
fractional Bergman operators. Finally, we provide some sharp weighted
inequalities for the fractional maximal functions.
