Weak Type Inequalities for Maximal Operators Associated to Double Ergodic Sums

Given an approach region G{cyrillic} ∈ ℤ²₊ and a pair U, V of commuting nonperiodic measure preserving transformations on a probability space (Ω,∑, μ), it is shown that either the associated multiparameter ergodic averages of any function in L¹(Ω) converge a.e. or that, given a positive increasing function φ on [0, ∞) that is o(log x) as x → ∞, there exists a function g ∈ Lφ(L) (Ω) whose associated multiparameter ergodic averages fail to converge a.e.