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Calderón-Zygmund theory with noncommuting kernels via $H_1^c$

Antonio Ismael Cano-Mármol, Éric Ricard

TL;DR

The paper develops a semicommutative Calderón-Zygmund framework for noncommuting kernels by constructing a new atom-based predual $ ext{H}_1^c(oldsymbol{ ext{A}})$ to the BMO-space $ ext{BMO}^r(oldsymbol{ ext{R}}, ext{M})$ and proving the duality $ ext{H}_1^c(oldsymbol{ ext{A}})^* = ext{BMO}^r(oldsymbol{ ext{R}}, ext{M})$. It then proves endpoint boundedness for Calderón-Zygmund operators with operator-valued kernels under a Hörmander condition, via atom decompositions and a kernel-approximation argument, establishing boundedness from $ ext{H}_1^c(oldsymbol{ ext{A}})$ to $L_1(oldsymbol{ ext{R}} imes ext{M})$. The framework extends classical CZ theory to the semicommutative noncommutative setting and yields interpolation consequences and endpoint estimates, connecting to Mei's operator-valued Hardy spaces. The worked example shows compatibility with Mei's theory and recovers known vector-valued H_1–BMO results, highlighting applicability to noncommutative harmonic analysis with noncommuting kernels.

Abstract

We study an alternative definition of the $H_1$-space associated to a semicommutative von Neumann algebra $L_\infty(\mathbb{R}) \overline{\otimes} \mathcal{M}$, first studied by Mei. We identify a "new" description for atoms in $H_1$. We then explain how they can be used to study $H_1^c$-$L_1$ endpoint estimates for Calderón-Zygmund operators with noncommuting kernels.

Calderón-Zygmund theory with noncommuting kernels via $H_1^c$

TL;DR

The paper develops a semicommutative Calderón-Zygmund framework for noncommuting kernels by constructing a new atom-based predual to the BMO-space and proving the duality . It then proves endpoint boundedness for Calderón-Zygmund operators with operator-valued kernels under a Hörmander condition, via atom decompositions and a kernel-approximation argument, establishing boundedness from to . The framework extends classical CZ theory to the semicommutative noncommutative setting and yields interpolation consequences and endpoint estimates, connecting to Mei's operator-valued Hardy spaces. The worked example shows compatibility with Mei's theory and recovers known vector-valued H_1–BMO results, highlighting applicability to noncommutative harmonic analysis with noncommuting kernels.

Abstract

We study an alternative definition of the -space associated to a semicommutative von Neumann algebra , first studied by Mei. We identify a "new" description for atoms in . We then explain how they can be used to study - endpoint estimates for Calderón-Zygmund operators with noncommuting kernels.
Paper Structure (7 sections, 21 theorems, 144 equations)

This paper contains 7 sections, 21 theorems, 144 equations.

Table of Contents

  1. Column/row Hilbert-valued $L_p$-spaces
  2. Noncommutative spaces $L_p(\mathcal{M};L^c_2(\Omega))$
  3. Duality between Hardy spaces and $\mathrm{BMO}$-spaces
  4. Calderón-Zygmund operators with operator-valued kernels
  5. An example
  6. Vector-valued Hardy spaces
  7. Proof of Lemma \ref{['lem:garnett']}

Key Result

Corollary 1.1

Let $H$ and $K$ be two Hilbert spaces, and let $T : H \longrightarrow K$ be a bounded linear operator. Then $\mathrm{Id}_{L_p(\mathcal{M})} \otimes T$ admits a unique continuous extension (resp. weak$^*$ continuous) $\widetilde{T}$ from $L_p(\mathcal{M};H^c)$ into $L_p(\mathcal{M};K^c)$ for $1\leq p

Theorems & Definitions (43)

  • Corollary 1.1
  • Lemma 1.2
  • Proposition 1.3
  • proof
  • Lemma 1.4
  • proof
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • ...and 33 more