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 defined on the upper-half plane. 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 Békollé–Bonami-type conditions. We also obtain a Φ -bump characterization for these maximal functions, where Φ 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.