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Two inequalities for commutators of singular integral operators satisfying Hörmander conditions of Young type

Yuru Li, Jiawei Tan, Qingying Xue

TL;DR

This work addresses weighted endpoint estimates for general commutators $T_{oldsymbol{b}}$ of singular integral operators under Hörmander conditions of Young type. The authors develop a robust pointwise sparse domination strategy and leverage dyadic analysis to derive Fefferman-Stein inequalities with arbitrary weights and a quantitative Coifman-Fefferman inequality, while decoupling the ε-parameter from the commutator length $m$. They extend the framework to non-convolution kernels via $L^A$-Hörmander conditions and establish broad applications to $L^r$-Hörmander operators, ω-CZ operators with Dini conditions, Calderón commutators, homogeneous singular integrals, and Fourier multipliers, including sharp constants and weight dependencies. Collectively, the results unify and sharpen previous work (e.g., Pérez, Rivera-Ríos) and provide a versatile toolkit for quantitative weighted endpoint estimates in non-convolution harmonic analysis contexts.

Abstract

In this paper, we systematically study the Fefferman-Stein inequality and Coifman-Fefferman inequality for the general commutators of singular integral operators that satisfy Hörmander conditions of Young type. Specifically, we first establish the pointwise sparse domination for these operators. Then, relying on the dyadic analysis, the Fefferman-Stein inequality with respect to arbitrary weights and the quantitative weighted Coifman-Fefferman inequality are demonstrated. We decouple the relationship between the number of commutators and the index $\varepsilon$, which essentially improved the results of Pérez and Rivera-Ríos (Israel J. Math., 2017). As applications, it is shown that all the aforementioned results can be applied to a wide range of operators, such as singular integral operators satisfying the $L^r$-Hörmander operators, $ω$-Calderón-Zygmund operators with $ω$ satisfying a Dini condition, Calderón commutators, homogeneous singular integral operators and Fourier multipliers.

Two inequalities for commutators of singular integral operators satisfying Hörmander conditions of Young type

TL;DR

This work addresses weighted endpoint estimates for general commutators of singular integral operators under Hörmander conditions of Young type. The authors develop a robust pointwise sparse domination strategy and leverage dyadic analysis to derive Fefferman-Stein inequalities with arbitrary weights and a quantitative Coifman-Fefferman inequality, while decoupling the ε-parameter from the commutator length . They extend the framework to non-convolution kernels via -Hörmander conditions and establish broad applications to -Hörmander operators, ω-CZ operators with Dini conditions, Calderón commutators, homogeneous singular integrals, and Fourier multipliers, including sharp constants and weight dependencies. Collectively, the results unify and sharpen previous work (e.g., Pérez, Rivera-Ríos) and provide a versatile toolkit for quantitative weighted endpoint estimates in non-convolution harmonic analysis contexts.

Abstract

In this paper, we systematically study the Fefferman-Stein inequality and Coifman-Fefferman inequality for the general commutators of singular integral operators that satisfy Hörmander conditions of Young type. Specifically, we first establish the pointwise sparse domination for these operators. Then, relying on the dyadic analysis, the Fefferman-Stein inequality with respect to arbitrary weights and the quantitative weighted Coifman-Fefferman inequality are demonstrated. We decouple the relationship between the number of commutators and the index , which essentially improved the results of Pérez and Rivera-Ríos (Israel J. Math., 2017). As applications, it is shown that all the aforementioned results can be applied to a wide range of operators, such as singular integral operators satisfying the -Hörmander operators, -Calderón-Zygmund operators with satisfying a Dini condition, Calderón commutators, homogeneous singular integral operators and Fourier multipliers.
Paper Structure (21 sections, 30 theorems, 235 equations)

This paper contains 21 sections, 30 theorems, 235 equations.

Table of Contents

  1. Introduction and main results
  2. Definitions and notations
  3. The general commutators
  4. Historical background
  5. Motivation
  6. Main results
  7. Preliminary
  8. $\omega$-Calderón-Zygmund operators
  9. Sparse family
  10. Sharp maximal functions
  11. The classical weights
  12. Young functions and Orlicz maximal functions
  13. Proofs of Theorem \ref{['thm5.1']}
  14. Proofs of Theorem \ref{['thm5.2']}
  15. Proofs of Theorem \ref{['thm5.3']}, Corollaries \ref{['cor5.1']} and \ref{['cor5.2']}
  16. ...and 6 more sections

Key Result

Theorem 1.6

Let $m\in\mathbb{N}$ and $A$ be a Young function. Assume that $T$ is an $\bar{A}$-Hörmander operators. If $A\in \mathcal{Y}(p_0,p_1)~(1\leq p_0\leq p_1<\infty)$, then for any locally integrable functions $\vec{b}=(b_1,\ldots,b_m)$ on $\mathbb{R}^n$ and bounded function $f$ with compact support, ther where $C_T=c_{n,p_0,p_1}\max\{c_{A,p_0},c_{A,p_1}\}(H_{\bar{A}}+\|T\|_{L^2\rightarrow L^2})$ and

Theorems & Definitions (58)

  • Definition 1.1: $L^r$-Hörmander condition, kur
  • Definition 1.2: hor+guo
  • Definition 1.3: $L^A$-Hörmander condition of non-convolution type
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 48 more