Weighted Boundedness of Maximal Functions and Fractional Bergman Operators - Sehba, B.F. (Journal of Geometric Analysis, 2018)
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.
Keywords: Bergman operator; Békollè–Bonami weight; Carleson-type embedding; Dyadic grid; Maximal function; Upper-half plane