Restrictions of B\'ekoll\'e--Bonami weights and Bloch functions

TL;DR

This paper characterizes how Békollé--Bonami weights with bounded hyperbolic oscillation behave when restricted to subsets of the unit disc, extending Wolff's restriction theorem to this context and connecting it with Bloch functions and dyadic martingales.

Contribution

It provides a new restriction theorem for Békollé--Bonami weights and Bloch functions, bridging complex analysis and probabilistic martingale techniques.

Findings

Established an analogue of Wolff's restriction theorem for Békollé--Bonami weights.

Connected the restriction problem for Bloch functions with dyadic martingale analysis.

Extended understanding of weight restrictions in complex analysis.

Abstract

We characterize the restrictions of B\'ekoll\'e--Bonami weights of bounded hyperbolic oscillation, to subsets of the unit disc, thus proving an analogue of Wolff's restriction theorem for Muckenhoupt weights. Sundberg proved a discrete version of Wolff's original theorem, by characterizing the trace of $BMO$-functions onto interpolating sequences. We consider an analogous question in our setting, by studying the trace of Bloch functions. Through Makarov's probabilistic approach to the Bloch space, our question can be recast as a restriction problem for dyadic martingales with uniformly bounded increments.