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Abstract 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. 

An inequality of Brascamp-Lieb-Luttinger and of Rogers states that among subsets of Euclidean space R d \mathbb {R}^d of specified Lebesgue measures, (tuples of) balls centered at the origin are maximizers of certain functionals defined by multidimensional integrals. For d > 1 d>1 , this inequality only applies to functionals invariant under a diagonal action of Sl ⁡ ( d ) \operatorname {Sl}(d) . We investigate functionals of this type, and their maximizers, in perhaps the simplest situation in which Sl ⁡ ( d ) \operatorname {Sl}(d) invariance does not hold. Assuming a more limited symmetry encompassing dilations but not rotations, we show under natural hypotheses that maximizers exist, and, moreover, that there exist distinguished maximizers whose structure reflects this limited symmetry. For small perturbations of the Sl ⁡ ( d ) \operatorname {Sl}(d) –invariant framework we show that these distinguished maximizers are strongly convex sets with infinitely differentiable boundaries. It is shown that in the absence of partial symmetry, maximizers fail to exist for certain arbitrarily small perturbations of Sl ⁡ ( d ) \operatorname {Sl}(d) –invariant structures.