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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.
