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Sharpness of the side condition in a characterization of Békollé-Bonami weights

Alptekin Can Goksan

TL;DR

The paper investigates the sharpness of the side condition that a weight be almost constant on the top halves of Carleson squares in the unit disc characterization of the limiting Békollé-Bonami class $B_\infty$. It proves that this side condition can be essentially dropped for radial monotone weights (with $0<w(0)<\infty$), while counterexamples show sharpness for non-monotone weights, establishing the condition as necessary in general. It further extends the $B_\infty$ characterization to encompass all twelve $A_\infty$-type conditions studied by Duoandikoetxea–Martín-Reyes–Ombrosi, and maps their interrelations on arbitrary weights on the unit disc, aided by a detailed radial-to-disc dictionary. The results yield a self-improvement property for monotone $B_p$ weights, mirroring the classical A_p theory, and provide a precise framework for when the various $A_\infty$-type properties coincide in this setting. Overall, the work unifies the disc-specific $A_\infty$ landscape under AC and clarifies the sharpness and limits of the top-half oscillation condition for $B_\infty$.

Abstract

We study the sharpness of the side condition in a recent characterization of a limiting class $B_\infty$ of Békollé-Bonami weights by Aleman, Pott and Reguera. This side condition bounds the oscillation of a weight on the top halves of Carleson squares and allows for the development of a rich theory for Békollé-Bonami weights, analogous to that of Muckenhoupt weights. First, we prove that the side condition can essentially be dropped when the weight is radial and monotonic. Then, by means of counterexamples, we show that the side condition is sharp for non-monotonic weights. In addition, we extend the characterization of the $B_\infty$ class so that it includes all twelve $A_\infty$ conditions recently studied by Duoandikoetxea, Martín-Reyes and Ombrosi, and we present a complete picture of the relationships between these twelve conditions for arbitrary weights on the unit disc. Finally, we use our results to prove an analogue of the self-improvement property of Muckenhoupt weights for monotonic Békollé-Bonami weights.

Sharpness of the side condition in a characterization of Békollé-Bonami weights

TL;DR

The paper investigates the sharpness of the side condition that a weight be almost constant on the top halves of Carleson squares in the unit disc characterization of the limiting Békollé-Bonami class . It proves that this side condition can be essentially dropped for radial monotone weights (with ), while counterexamples show sharpness for non-monotone weights, establishing the condition as necessary in general. It further extends the characterization to encompass all twelve -type conditions studied by Duoandikoetxea–Martín-Reyes–Ombrosi, and maps their interrelations on arbitrary weights on the unit disc, aided by a detailed radial-to-disc dictionary. The results yield a self-improvement property for monotone weights, mirroring the classical A_p theory, and provide a precise framework for when the various -type properties coincide in this setting. Overall, the work unifies the disc-specific landscape under AC and clarifies the sharpness and limits of the top-half oscillation condition for .

Abstract

We study the sharpness of the side condition in a recent characterization of a limiting class of Békollé-Bonami weights by Aleman, Pott and Reguera. This side condition bounds the oscillation of a weight on the top halves of Carleson squares and allows for the development of a rich theory for Békollé-Bonami weights, analogous to that of Muckenhoupt weights. First, we prove that the side condition can essentially be dropped when the weight is radial and monotonic. Then, by means of counterexamples, we show that the side condition is sharp for non-monotonic weights. In addition, we extend the characterization of the class so that it includes all twelve conditions recently studied by Duoandikoetxea, Martín-Reyes and Ombrosi, and we present a complete picture of the relationships between these twelve conditions for arbitrary weights on the unit disc. Finally, we use our results to prove an analogue of the self-improvement property of Muckenhoupt weights for monotonic Békollé-Bonami weights.
Paper Structure (8 sections, 26 theorems, 89 equations, 4 figures)

This paper contains 8 sections, 26 theorems, 89 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Preliminaries
  4. Proof of Theorem \ref{['thm_equiv_2']}
  5. Proof of Theorem \ref{['thm_mono_1']}
  6. Counterexamples
  7. Further A_infinity conditions
  8. Acknowledgements

Key Result

Theorem 1.1

Let $w$ be a weight on the unit disc which is almost constant on top halves. Then the following are equivalent:

Figures (4)

  • Figure 1: Relationships between the properties in Definitions \ref{['def_disc']} and \ref{['def_AC']}, for arbitrary weights on the unit disc. The label $B_p$ represents $\bigcap_{1<p<\infty} B_p$ in the two implications to its right and $\bigcup_{1<p<\infty} B_p$ in the three implications below it. Thick arrows indicate contributions of this paper.
  • Figure 2: Relationships between properties \ref{['P1']} - \ref{['P8']} in Definition \ref{['def_P1-P8']}, for arbitrary weights on the unit disc.
  • Figure 3: An interval $I$ of the unit circle, the associated Carleson square $Q_I$, and its top half $T_I$.
  • Figure 4: Relationships between \ref{['P4a']} and \ref{['P1']} - \ref{['P8']}, and between \ref{['P4b']} and \ref{['P1']} - \ref{['P8']}, for arbitrary weights on the unit disc.

Theorems & Definitions (73)

  • Theorem 1.1: Aleman, Pott and Reguera, 2017
  • Theorem 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • proof
  • ...and 63 more