# Weighted boundedness of maximal functions and fractional Bergman operators

## TL;DR

This paper characterizes two-weight inequalities for fractional maximal functions and Bergman operators on the upper-half space, providing conditions and sharp bounds for their boundedness in weighted spaces.

## Contribution

It offers new Sawyer and Békollé-Bonami type conditions and a Φ-bump characterization for fractional maximal functions and Bergman operators.

## Key findings

- Characterization of weight pairs for strong and weak inequalities
- Introduction of a Φ-bump condition for these operators
- Derivation of sharp weighted inequalities

## Abstract

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.