Weak type inequalities for ergodic strong maximal operators

Fava's weak type L log L estimate for strong two-parameter ergodic maximal operators associated to pairs of commuting non-periodic measure-preserving transformations is shown to be sharp. Moreover, given a function Φ on [0,∞) that is positive, increasing, and o(log(x)) for x → ∞ as well as a pair of commuting invertible non-periodic measure-preserving transformations on a space ω of finite measure, a function f ε LΦ(L)(ω) is constructed whose associated multiparameter ergodic averages fail to converge almost everywhere in the unrestricted sense.