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Weighted norm estimates of noncommutative Calderón-Zygmund operators

Wenfei Fan, Yong Jiao, Lian Wu, Dejian Zhou

TL;DR

This work develops the weighted endpoint theory for operator-valued Calderón-Zygmund operators in a noncommutative setting. It proves weighted weak-type bounds for noncommutative maximal CZ operators, establishes weighted square-function estimates, and derives a weighted $H_1-L_1$ inequality under $A_1$ weights and regularity conditions weaker than Lipschitz, leveraging a refined noncommutative CZ decomposition and a new weighted atomic theory for martingales. The approach combines lacunary reductions, kernel regularity in $\mathcal{H}_{2r_w'}$, and weighted atomic decompositions to extend unweighted and commutative results to the weighted noncommutative context, with applications to operator-valued harmonic analysis. These results broaden the scope of weighted inequalities in noncommutative harmonic analysis and pave the way for weighted martingale inequalities in operator-valued settings.

Abstract

This paper is devoted to studying weighted endpoint estimates of operator-valued singular integrals. Our main results include weighted weak-type $(1,1)$ estimate of noncommutative maximal Calderón-Zygmund operators, corresponding version of square functions and a weighted $H_1- L_1$ type inequality. All these results are obtained under the condition that the weight belonging to the Muchenhoupt $A_1$ class and certain regularity assumptions imposed on kernels which are weaker than the Lipschitz condition.

Weighted norm estimates of noncommutative Calderón-Zygmund operators

TL;DR

This work develops the weighted endpoint theory for operator-valued Calderón-Zygmund operators in a noncommutative setting. It proves weighted weak-type bounds for noncommutative maximal CZ operators, establishes weighted square-function estimates, and derives a weighted inequality under weights and regularity conditions weaker than Lipschitz, leveraging a refined noncommutative CZ decomposition and a new weighted atomic theory for martingales. The approach combines lacunary reductions, kernel regularity in , and weighted atomic decompositions to extend unweighted and commutative results to the weighted noncommutative context, with applications to operator-valued harmonic analysis. These results broaden the scope of weighted inequalities in noncommutative harmonic analysis and pave the way for weighted martingale inequalities in operator-valued settings.

Abstract

This paper is devoted to studying weighted endpoint estimates of operator-valued singular integrals. Our main results include weighted weak-type estimate of noncommutative maximal Calderón-Zygmund operators, corresponding version of square functions and a weighted type inequality. All these results are obtained under the condition that the weight belonging to the Muchenhoupt class and certain regularity assumptions imposed on kernels which are weaker than the Lipschitz condition.
Paper Structure (14 sections, 17 theorems, 220 equations)

This paper contains 14 sections, 17 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Noncommutative $L_p$ spaces
  4. Vector-valued noncommutative $L_p$ spaces
  5. Weights
  6. Noncommutative weighted $L_p$ spaces
  7. Noncommuative (martingale) Hardy spaces
  8. Noncommutative Calderón-Zygmund decomposition
  9. Weighted estimate for maximal singular integral operators
  10. Weighted estimate for square functions
  11. Weighted estimate in Hardy spaces
  12. Atomic decomposition of weighted Hardy spaces
  13. Proof of Theorem \ref{['HL1']}
  14. Proof of Remark \ref{['indspaces']}

Key Result

Theorem 1.3

Let $T$ be a Calderón-Zygmund operator associated to a kernel $K$ satisfying sizec. Given a weight ${w}\in A_1$, suppose that $K\in \mathcal{H}_{2r_w'}$, where $r_w$ comes from Lemma weightprop(i) and $r_w'$ denotes the conjugate number of $r_w$. If there exists some $p_0>1$ such that $(T_\varepsilo or equivalently, we have

Theorems & Definitions (44)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • ...and 34 more