Our first result is a noncommutative form of the Jessen-Marcinkiewicz-Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of almost everywhere convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the $L_p$ -norm of the $\limsup $ of a sequence of operators as a localized version of a $\ell _\infty /c_0$ -valued $L_p$ -space. In particular, our main result gives a strong $L_1$ -estimate for the $\limsup $ —as opposed to the usual weak $L_{1,\infty }$ -estimate for the $\mathop {\mathrm {sup}}\limits $ —with interesting consequences for the free group algebra. Let $\mathcal{L} \mathbf{F} _2$ denote the free group algebra with $2$ generators, and consider the free Poisson semigroup generated by the usual length function. It is an open problem to determine the largest class inside $L_1(\mathcal{L} \mathbf{F} _2)$ for which the free Poisson semigroup converges to the initial data. Currently, the best known result is $L \log ^2 L(\mathcal{L} \mathbf{F} _2)$ . We improve this result by adding to it the operators in $L_1(\mathcal{L} \mathbf{F} _2)$ spanned by words without signs changes. Contrary to other related results in the literature, this set grows exponentially with length. The proof relies on our estimates for the noncommutative $\limsup $ together with new transference techniques. We also establish a noncommutative form of Córdoba/Feffermann/Guzmán inequality for the strong maximal: more precisely, a weak $(\Phi ,\Phi )$ inequality—as opposed to weak $(\Phi ,1)$ —for noncommutative multiparametric martingales and $\Phi (s) = s (1 + \log _+ s)^{2 + \varepsilon }$ . This logarithmic power is an $\varepsilon $ -perturbation of the expected optimal one. The proof combines a refinement of Cuculescu’s construction with a quantum probabilistic interpretation of M. de Guzmán’s original argument. The commutative form of our argument gives the simplest known proof of this classical inequality. A few interesting consequences are derived for Cuculescu’s projections.

中文翻译：

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非对易强极大值和几个方向上几乎一致的收敛

我们的第一个结果是 Jessen-Marcinkiewicz-Zygmund 定理的非交换形式，用于多参数鞅或遍历均值的最大极限。它意味着在预期 Orlicz 空间中具有初始数据的双边几乎一致收敛（几乎处处收敛的非交换模拟）。一个关键因素是引入 $L_p$ - 规范 $\limsup $ 一系列运算符作为 a 的本地化版本 $\ell _\infty /c_0$ 有价值的 $L_p$ -空间。特别是，我们的主要结果给出了强有力的 $L_1$ - 估计 $\limsup $ ——与通常的弱者相反 $L_{1,\infty }$ - 估计 $\mathop {\mathrm {sup}}\limits $ ——对自由群代数产生有趣的影响。让 $\mathcal{L} \mathbf{F} _2$ 用 $2$ 生成器，并考虑由通常的长度函数生成的自由泊松半群。确定内部最大的类是一个开放问题 $L_1(\mathcal{L} \mathbf{F} _2)$ 自由泊松半群收敛到初始数据。目前，最知名的结果是 $L \log ^2 L(\mathcal{L} \mathbf{F} _2)$ . 我们通过在其中添加运算符来改进此结果 $L_1(\mathcal{L} \mathbf{F} _2)$ 跨越了没有符号变化的单词。与文献中的其他相关结果相反，该集合随长度呈指数增长。证明依赖于我们对不可交换的估计 $\limsup $ 连同新的移情技术。我们还为强极大值建立了 Córdoba/Feffermann/Guzmán 不等式的非交换形式：更准确地说，弱 $(\Phi,\Phi)$ 不平等——相对于弱 $(\Phi ,1)$ —对于非交换多参数鞅和 $\Phi (s) = s (1 + \log _+ s)^{2 + \varepsilon }$ . 这个对数幂是 $\伐普西隆$ - 预期最优的扰动。该证明结合了对 Cuculescu 构造的改进和对 M. de Guzmán 原始论点的量子概率解释。我们论证的交换形式给出了这个经典不等式的最简单的已知证明。库库列斯库的预测得出了一些有趣的结果。