Characterizations for arbitrary Békollé-Bonami weights

We precisely characterize the relationships between the reverse Hölder inequality, the Fujii-Wilson condition, the Békollé-Bonami ${\mathrm{B}}_p$ condition, the ${\mathrm{B}}_{\mathrm{\infty }}$ 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.