Characterizations for arbitrary B\'ekoll\'e-Bonami weights

TL;DR

This paper provides a comprehensive characterization of various conditions related to Békollé-Bonami weights in the unit disc, introducing new testable side conditions that unify several inequalities and conditions.

Contribution

It introduces new necessary and sufficient side conditions for characterizing Békollé-Bonami weights, linking multiple inequalities and conditions in a unified framework.

Findings

New side conditions are simple and testable.

Characterizations unify reverse Hölder, Fujii-Wilson, and Békollé-Bonami conditions.

Side conditions relate to bounded hyperbolic oscillation.

Abstract

We precisely characterize the relationships between the reverse H\"older inequality, the Fujii-Wilson condition, the B\'ekoll\'e-Bonami $\mathrm{B}_p$ condition, the $\mathrm{B}_\infty$ condition, and the reverse Jensen inequality, for arbitrary weights in the unit disc. This is achieved by introducing new side conditions that turn out to be necessary and sufficient. The side conditions are simple and testable, and can be interpreted as integral versions of the much stronger condition of bounded hyperbolic oscillation, which has been considered earlier in the literature.