# The class $B_\infty$

## TL;DR

This paper investigates the properties of the Békollé-Bonami weight class $B_
abla$, showing that under certain restrictions, these weights regain some classical properties and applying this to analyze spectra of specific integral operators.

## Contribution

The authors demonstrate that restricting $B_
abla$ weights to those nearly constant on top halves restores classical properties and apply this to spectral analysis of integral operators.

## Key findings

- Restricted $B_
abla$ weights exhibit classical properties.
- Application to spectra of integral operators.
- Identification of conditions for weight regularity.

## Abstract

We explore properties of the class of B\'ekoll\'e-Bonami weights $B_\infty$ introduced by the authors in a previous work. Although B\'ekoll\'e-Bonami weights are known to be ill-behaved because they do not satisfy a reverse H\"older property, we prove than when restricting to a class of weights that are "nearly constant on top halves", one recovers some of the classical properties of Muckenhoupt weights. We also provide an application of this result to the study of the spectra of certain integral operators.