In this paper, we investigate the boundedness of a square function operator from ergodic theory acting on noncommutative $L_p$-spaces. The main result is a weak type $(1,1)$ estimate of this operator. We also show the $(L_{\infty },\mathrm{BMO})$ estimate, and thus all the strong type $(L_p,L_p)$ estimates by interpolation. The main new difficulty lies in the fact that the kernel of this square function operator does not enjoy any regularity, while the Lipschitz regularity assumption is crucial in showing such endpoint estimates for the noncommutative Calderón-Zygmund singular integrals.

中文翻译：

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遍历理论的平方函数的非可交换弱类型（1,1）估计

在本文中，我们从遍历于非交换性的遍历理论研究平方函数算子的有界性 ${\mathrm{大号}}_p$-空间。主要结果是弱类型$（\mathrm{1个}，\mathrm{1个}）$此运算符的估计值。我们还展示了$（{\mathrm{大号}}_{\infty }，\mathrm{BMO}）$ 估计，因此所有强类型 $（{\mathrm{大号}}_p，{\mathrm{大号}}_p）$通过插值估算。主要的新困难在于以下事实：该平方函数算子的核不享有任何正则性，而Lipschitz正则性假设对于显示非交换Calderón-Zygmund奇异积分的此类端点估计至关重要。