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On the compactness of the bi-commutator

Henri Martikainen, Tuomas Oikari

TL;DR

The paper advances the theory of bi-parameter operator compactness by analyzing the bi-commutator [T1,[b,T2]] for two non-degenerate Calderón-Zygmund operators on bi-parameter spaces. It develops a two-pronged strategy: a direct off-diagonal compactness result in the p1<q1, p2<q2 regime combined with an extrapolation framework to cover diagonal and near-diagonal cases, and a thorough study of product and rectangular BMO/VMO spaces to characterize conditions on the symbol b. The authors prove sufficiency for compactness under product-VMO-type conditions in off-diagonal regimes and derive a full extrapolation-based mechanism to extend these results to diagonal configurations, provided at least one index is strict. They also establish necessary conditions via bi-parameter rectangular weak factorization and Uchiyama-type arguments, showing that compactness forces b to lie in appropriate VMO-type spaces, with special considerations when beta parameters reach or exceed unity. Overall, the work blends new bi-parameter approximation results with extrapolation principles to yield sharp sufficient and necessary criteria for compactness of bi-commutators, enriching the understanding of multi-parameter harmonic analysis.

Abstract

We prove compactness results and characterizations for the bi-commutator $[T_1,[b, T_2]]$ of a symbol $b$ and two non-degenerate Calderón-Zygmund singular integral operators $T_1, T_2$. Our strategy for proving sufficient conditions for compactness is to first establish them in the mixed-norm $L^{p_1}L^{p_2}\to L^{q_1}L^{q_2}$ off-diagonal case with $p_i < q_i$, and then extend these to other exponents, including the diagonal $p_i = q_i$, with a new extrapolation argument. In particular, the natural product $\operatorname{VMO}$ condition is obtained as a sufficient condition in the diagonal. A full characterization is obtained, both in terms of a vanishing mean oscillation type condition and in terms of the approximability of the symbol, whenever the inequality $p_i \le q_i$ is strict for at least one index. The extrapolation scheme for proving sufficiency requires us to prove new approximation results in relevant bi-parameter function spaces that are of independent interest. The necessity results are obtained by carefully combining recent rectangular approximate weak factorization methods with a classical idea of Uchiyama.

On the compactness of the bi-commutator

TL;DR

The paper advances the theory of bi-parameter operator compactness by analyzing the bi-commutator [T1,[b,T2]] for two non-degenerate Calderón-Zygmund operators on bi-parameter spaces. It develops a two-pronged strategy: a direct off-diagonal compactness result in the p1<q1, p2<q2 regime combined with an extrapolation framework to cover diagonal and near-diagonal cases, and a thorough study of product and rectangular BMO/VMO spaces to characterize conditions on the symbol b. The authors prove sufficiency for compactness under product-VMO-type conditions in off-diagonal regimes and derive a full extrapolation-based mechanism to extend these results to diagonal configurations, provided at least one index is strict. They also establish necessary conditions via bi-parameter rectangular weak factorization and Uchiyama-type arguments, showing that compactness forces b to lie in appropriate VMO-type spaces, with special considerations when beta parameters reach or exceed unity. Overall, the work blends new bi-parameter approximation results with extrapolation principles to yield sharp sufficient and necessary criteria for compactness of bi-commutators, enriching the understanding of multi-parameter harmonic analysis.

Abstract

We prove compactness results and characterizations for the bi-commutator of a symbol and two non-degenerate Calderón-Zygmund singular integral operators . Our strategy for proving sufficient conditions for compactness is to first establish them in the mixed-norm off-diagonal case with , and then extend these to other exponents, including the diagonal , with a new extrapolation argument. In particular, the natural product condition is obtained as a sufficient condition in the diagonal. A full characterization is obtained, both in terms of a vanishing mean oscillation type condition and in terms of the approximability of the symbol, whenever the inequality is strict for at least one index. The extrapolation scheme for proving sufficiency requires us to prove new approximation results in relevant bi-parameter function spaces that are of independent interest. The necessity results are obtained by carefully combining recent rectangular approximate weak factorization methods with a classical idea of Uchiyama.
Paper Structure (13 sections, 20 theorems, 178 equations)

This paper contains 13 sections, 20 theorems, 178 equations.

Table of Contents

  1. Introduction and main results
  2. Notation and definitions
  3. On relevant function spaces and their properties
  4. Product $BMO/VMO$
  5. Hölder spaces and rectangular fractional $\operatorname{BMO}$/$\operatorname{VMO}$
  6. Approximation in the relevant bi-parameter spaces
  7. Approximation in $\dot{C}^{0,\alpha}$-valued Hölder spaces
  8. Approximation in $\operatorname{BMO}$-valued Hölder spaces
  9. Sufficient conditions for compactness
  10. The case $p_i < q_i$, $i=1,2$
  11. Compactness extrapolation principle for bi-commutators
  12. Necessary conditions for compactness
  13. The cases where $\beta_i\geq 1$ for $i=1$ or $i=2$

Key Result

Theorem 1.1

Let $b\in L^2_{\operatorname{loc}}(\mathbb{R}^d)$ and $T_i$ be non-degenerate CZOs on $\mathbb{R}^{d_i}$, $i=1,2$. Then, the compactness of the bi-commutator has sufficient and necessary conditions according to the following table

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.4
  • proof
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • proof
  • Definition 2.11
  • ...and 36 more