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Journal of Operator Theory

Volume 90, Issue 1, Summer 2023  pp. 3-24.

Noncommutative differential transforms for averaging operators

Authors:  Bang Xu
Author institution: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China and Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

Summary:  In this paper, we complete the study of mapping properties for a family of operators evaluating the difference between differentiation operators and conditional expectations acting on noncommutative $L_{p}$-spaces. To be more precise, we establish the weak type $(1,1)$ and $(L_{\infty},\mathrm{BMO})$ estimates of this difference. Consequently, in conjunction with interpolation and duality, we obtain all strong type $(p,p)$ estimates. This allows us to obtain a quick application to noncommutative differential transforms for averaging operators.

Keywords:  Calderon-Zygmund decomposition, noncommutative $L_{p}$-spaces, differential transforms, noncommutative martingales, weak $(1,1)$.

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