Bloch functions and Bekoll\'e-Bonami weights

TL;DR

This paper explores the relationship between Bekollé-Bonami weights, Bloch functions, and harmonic analysis, providing new characterizations, counterexamples, and applications to spectral properties of Cesàro operators.

Contribution

It introduces new distance formulas for Bekollé-Bonami weights, characterizes weights with powers also in the class, and offers a counterexample related to Bloch space approximation.

Findings

Characterization of weights with all powers in Bekollé-Bonami class

Counterexample to a conjecture on Bloch space closure

Insights into spectral properties of Cesàro operators

Abstract

We study analogues of well-known relationships between Muckenhoupt weights and $BMO$ in the setting of Bekoll\'e-Bonami weights. For Bekoll\'e-Bonami weights of bounded hyperbolic oscillation, we provide distance formulas of Garnett and Jones-type, in the context of $BMO$ on the unit disc and hyperbolic Lipschitz functions. This leads to a characterization of all weights in this class, for which any power of the weight is a Bekoll\'e-Bonami weight, which in particular reveals an intimate connection between Bekoll\'e-Bonami weights and Bloch functions. On the open problem of characterizing the closure of bounded analytic functions in the Bloch space, we provide a counter-example to a related recent conjecture. This shed light into the difficulty of preserving harmonicity in approximation problems in norms equivalent to the Bloch norm. Finally, we apply our results to study certain spectral properties of Cesar\'o operators.