Let 1 < p < ∞. We show the boundedness of operator-valued commutators [πa, Mb] on the noncommutative <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L_{p}(L_{\infty}({\mathbb R}){\overline \otimes}{\cal M})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>L</mi> <mrow> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow> <mover> <mo>⊗</mo> <mo accent="false">¯</mo> </mover> </mrow> <mrow> <mi mathvariant="script">M</mi> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> for any von Neumann algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\cal M}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">M</mi> </mrow> </math></EquationSource> </InlineEquation>, where πa is the d-adic martingale paraproduct with symbol a ∈ BMOd(ℝ) and Mb is the noncommutative left multiplication operator with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$b \in {{\rm BMO}_{\cal M}^{d}}(\mathbb R)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>b</mi> <mo>∈</mo> <mrow> <msubsup> <mrow> <mi mathvariant="normal">BMO</mi> </mrow> <mrow> <mi mathvariant="script">M</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msubsup> </mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>.

Boundedness of operator-valued commutators involving martingale paraproducts

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发表日期：

2026年1月21日

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Zhenguo Wei,
Hao Zhang

摘要

Let 1 < p < ∞. We show the boundedness of operator-valued commutators [π_a, M_b] on the noncommutative $L_{p}(L_{\infty}({\mathbb R}){\overline \otimes}{\cal M})$ $L_{p} (L_{\infty} (R) \bar{\otimes} M)$ for any von Neumann algebra ${\cal M}$ $M$ , where π_a is the d-adic martingale paraproduct with symbol a ∈ BMO^d(ℝ) and M_b is the noncommutative left multiplication operator with $b \in {{\rm BMO}_{\cal M}^{d}}(\mathbb R)$ $b \in {BMO}_{M}^{d} (R)$ .