We prove almost everywhere convergence of continuous-time quadratic averages with respect to two commuting $\mathbb {R}$ -actions, coming from a single jointly measurable measure-preserving $\mathbb {R}^2$ -action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.

中文翻译：

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某些连续时间双遍历平均值的逐点收敛

我们证明了关于两个通勤$\mathbb {R}$ -actions的连续时间二次平均的几乎处处收敛 ，来自一个概率空间上的单个联合可测量保测 $\mathbb {R}^2$ -action . 证明的关键成分来自最近关于多线性奇异积分的工作；更具体地说，来自对三角形希尔伯特变换的曲线模型的研究。