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Boundedness and compactness of Bergman projection commutators in two-weight setting

Bingyang Hu, Ji Li, Nathan A. Wagner

TL;DR

The paper develops a two-weight theory for commutators of Bergman projections in the upper half-space and extends to the unit ball, distinguishing between analytic and non-analytic symbols. For analytic symbols, it provides a full Bloom-type boundedness and compactness framework: $[b,P_\alpha]$ and $[b,P_\alpha^{+}]$ are bounded iff $b\in {\rm BMOA}_{\nu}$ and compact iff $b\in {\rm VMOA}_{\nu}$, with sharp norm equivalences. When symbols are merely measurable, the authors show that an Aleman–Pott–Reguera (APR) regularity condition on the weight is necessary for boundedness, and they prove partial sufficiency/necessity results along with a counterexample demonstrating sharpness. The methodology blends sparse domination, complex median methods, Hankel operator techniques, and a dyadic Bergman-tree framework, providing a coherent two-weight theory for Bergman projection commutators in complex function spaces. The results generalize to the ball and offer a foundation for further study of commutators with varying symbols in several complex variables.

Abstract

The goal of this paper is to study the boundedness and compactness of the Bergman projection commutators in two weighted settings via the weighted BMO and VMO spaces, respectively. The novelty of our work lies in the distinct treatment of the symbol b in the commutator, depending on whether it is analytic or not, which turns out to be quite different. In particular, we show that an additional weight condition due to Aleman, Pott, and Reguera is necessary to study the commutators when b is not analytic, while it can be relaxed when b is analytic. In the analytic setting, we completely characterize boundedness and compactness, while in the non-analytic setting, we provide a sufficient condition which generalizes the Euclidean case and is also necessary in many cases of interest. Our work initiates a study of the commutators acting on complex function spaces with different symbols.

Boundedness and compactness of Bergman projection commutators in two-weight setting

TL;DR

The paper develops a two-weight theory for commutators of Bergman projections in the upper half-space and extends to the unit ball, distinguishing between analytic and non-analytic symbols. For analytic symbols, it provides a full Bloom-type boundedness and compactness framework: and are bounded iff and compact iff , with sharp norm equivalences. When symbols are merely measurable, the authors show that an Aleman–Pott–Reguera (APR) regularity condition on the weight is necessary for boundedness, and they prove partial sufficiency/necessity results along with a counterexample demonstrating sharpness. The methodology blends sparse domination, complex median methods, Hankel operator techniques, and a dyadic Bergman-tree framework, providing a coherent two-weight theory for Bergman projection commutators in complex function spaces. The results generalize to the ball and offer a foundation for further study of commutators with varying symbols in several complex variables.

Abstract

The goal of this paper is to study the boundedness and compactness of the Bergman projection commutators in two weighted settings via the weighted BMO and VMO spaces, respectively. The novelty of our work lies in the distinct treatment of the symbol b in the commutator, depending on whether it is analytic or not, which turns out to be quite different. In particular, we show that an additional weight condition due to Aleman, Pott, and Reguera is necessary to study the commutators when b is not analytic, while it can be relaxed when b is analytic. In the analytic setting, we completely characterize boundedness and compactness, while in the non-analytic setting, we provide a sufficient condition which generalizes the Euclidean case and is also necessary in many cases of interest. Our work initiates a study of the commutators acting on complex function spaces with different symbols.

Paper Structure

This paper contains 13 sections, 19 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['mainthm001']}
  3. Some sparse domination estimates
  4. Sufficiency
  5. Necessity
  6. Proof of Theorem \ref{['mainthm002']}
  7. Sufficiency
  8. Necessity
  9. The Case of Non-Analytic Symbol
  10. Boundedness Results
  11. A Counterexample
  12. Compactness Results
  13. Unit ball analog

Key Result

Theorem 1.4

Let $\mu, \lambda \in {\mathcal{B}}_2$ and set $\nu=\mu^{\frac{1}{2}}\lambda^{-\frac{1}{2}}$. Let further, $b \in L_{ loc}^1\left(\mathbb{R}_{+}^2, dA_\alpha \right) \cap H(\mathbb{R}_{+}^2)$. Then the commutators $[b, P_\alpha]$ and $[b, P^+_\alpha]$ are bounded from $L_\alpha^2 \left(\mathbb{R}_{+

Figures (1)

  • Figure 1: Construction of $S_I$ and $Q_{\widetilde{I}}$

Theorems & Definitions (49)

  • Definition 1.1: Dyadic Békolle--Bonami ${\mathcal{B}}_2$ weights on $\mathbb{R}_+^2$
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9: Aleman--Pott--Regeura weights ($\mathcal{APR}$)
  • Definition 1.10
  • ...and 39 more