Sharpness of the side condition in a characterization of B\'ekoll\'e-Bonami weights

TL;DR

This paper investigates the necessity of a side condition in characterizing Békollé-Bonami weights, showing it can be dropped for radial monotonic weights but is essential otherwise, and extends the theory to include various A_infinity conditions.

Contribution

It demonstrates the sharpness of the side condition for non-monotonic weights, extends the B_infinity characterization to include recent A_infinity conditions, and establishes a self-improvement property for monotonic weights.

Findings

Side condition can be dropped for radial monotonic weights.

Counterexamples show the side condition is sharp for non-monotonic weights.

Extended B_infinity characterization includes all twelve A_infinity conditions.

Abstract

We study the sharpness of the side condition in a recent characterization of a limiting class $B_\infty$ of B\'ekoll\'e-Bonami weights by Aleman, Pott and Reguera. This side condition bounds the oscillation of a weight on the top halves of Carleson squares and allows for the development of a rich theory for B\'ekoll\'e-Bonami weights, analogous to that of Muckenhoupt weights. First, we prove that the side condition can essentially be dropped when the weight is radial and monotonic. Then, by means of counterexamples, we show that the side condition is sharp for non-monotonic weights. In addition, we extend the characterization of the $B_\infty$ class so that it includes all twelve $A_\infty$ conditions recently studied by Duoandikoetxea, Mart\'in-Reyes and Ombrosi, and we present a complete picture of the relationships between these twelve conditions for arbitrary weights on the unit disc. Finally, we use our results to prove an analogue of the self-improvement property of Muckenhoupt weights for monotonic B\'ekoll\'e-Bonami weights.