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Restrictions of Békollé--Bonami weights and Bloch functions

Alberto Dayan, Adrián Llinares, Karl-Mikael Perfekt

TL;DR

This work extends Wolff’s restriction phenomenon to Békollé–Bonami weights with bounded hyperbolic oscillation on the unit disk, proving an analogue: a restricted weight $w$ on a subset $ oq$ extends to a global $Wei B_p$ with bounded hyperbolic oscillation if and only if a power $w^q$ lies in the restricted class $B_{p, oq}$ for some $q>1$. Central to the approach is a dyadic factorization: any $wei B_{p, oq}$ with $p>1$ can be written as $w=w_1 w_2^{1-p}$ with $w_1,w_2ei B_{1, oq}$ and both of bounded hyperbolic oscillation, enabling control of restricted and global properties via Rubio de Francia’s scheme. The paper also develops a Bloch-space restriction framework by translating Bloch functions into Bloch martingales on the dyadic tree, yielding necessary trace conditions on interpolating sequences and a sharp counterexample showing the limits of these conditions. Together, these results connect weighted analytic function theory in the disk with probabilistic martingale methods, extending classical restriction theorems to a broader, more irregular weight class and clarifying the trace problem for Bloch functions on interpolating sequences.

Abstract

We characterize the restrictions of Békollé--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.

Restrictions of Békollé--Bonami weights and Bloch functions

TL;DR

This work extends Wolff’s restriction phenomenon to Békollé–Bonami weights with bounded hyperbolic oscillation on the unit disk, proving an analogue: a restricted weight on a subset extends to a global with bounded hyperbolic oscillation if and only if a power lies in the restricted class for some . Central to the approach is a dyadic factorization: any with can be written as with and both of bounded hyperbolic oscillation, enabling control of restricted and global properties via Rubio de Francia’s scheme. The paper also develops a Bloch-space restriction framework by translating Bloch functions into Bloch martingales on the dyadic tree, yielding necessary trace conditions on interpolating sequences and a sharp counterexample showing the limits of these conditions. Together, these results connect weighted analytic function theory in the disk with probabilistic martingale methods, extending classical restriction theorems to a broader, more irregular weight class and clarifying the trace problem for Bloch functions on interpolating sequences.

Abstract

We characterize the restrictions of Békollé--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 -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.
Paper Structure (5 sections, 15 theorems, 70 equations, 4 figures)

This paper contains 5 sections, 15 theorems, 70 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Weights of bounded hyperbolic oscillation
  3. Factorization of restricted weights
  4. Proof of Theorem \ref{['theo:main1']}
  5. Restrictions of Bloch functions and martingales

Key Result

Theorem 1.1

Let $w \geq 0$ be a weight on a measurable subset $\Omega \subset \mathbb{T}$, and let $1 \leq p < \infty$. Then the following are equivalent.

Figures (4)

  • Figure 1: Example of $S(I)$, $T(I)$ and their centre.
  • Figure 2: Example of $S(I)$ and $S(J_1) \cup S(J_2)$.
  • Figure 3: The first generations of the random walk.
  • Figure 4: The first generations of the Kahane martingale.

Theorems & Definitions (26)

  • Theorem 1.1: Wolff, as cited in Garcia-CuervaRubiodeFrancia
  • Theorem 1.2: Sundberg sundberg
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • ...and 16 more