  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.
