Weighted boundedness of maximal functions and fractional Bergman operators

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 $\mathrm{\Phi }$-bump characterization for these maximal functions, where $\mathrm{\Phi }$ 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.