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Let U1, . . . , Ud be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ, μ). Also let Γ be a nonempty subset of Z+ and B the associated collection of rectangular parallelepipeds in R with sides parallel to the axes and dimensions of the form n1 × · · · × nd with (n1, . . . , nd) ∈ Γ. The associated multiparameter geometric and ergodic maximal operators MB and MΓ are defined respectively on L(R) and L(Ω) by MBg(x) = sup x∈R∈B 1 |R| R |g(y)| dy and MΓ f(ω) = sup (n1,...,nd)∈Γ 1 n1 · · ·nd n1−1 ∑

Transference of weak type bounds of multiparameter ergodic and geometric maximal operators