Transference of Weak Type Bounds of Multiparameter Ergodic and Geometric Maximal Operators

Let U ₁,...,U _d be a non-periodic collection of commuting measure pre- serving transformations on a probability space ΩΣμ): Also let Γ be a nonempty subset of Z ^d ₊ and B the associated collection of rectangular parallelepipeds in R ^d with sides par- allel to the axes and dimensions of the form n ₁ x...x n _d with (n ₁,...,n _d) ∈ Γ. The associated multiparameter geometric and ergodic maximal operators M _B and M _Γ are deFIned respectively on L ¹(R ^d) and L ¹(Ω) by {equation presented} Given a Young function φ, it is shown that M _B satisfies the weak type estimate {equation presented} for a pair of positive constants CB, cB if and only if MΓ satisfies a corresponding weak type estimate {equation presented} for a pair of positive constants CΓ,cΓ. Applications of this transference principle regarding the a.e. convergence of multiparameter ergodic averages associated to rare bases are given.