Weak Type Inequalities for Ergodic Strong Maximal Operators
Acta Scientiarum Mathematicarum
Fava's weak type LlogL 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∈Lϕ(L)(Ω) is constructed whose associated multiparameter ergodic averages fail to converge almost everywhere in the unrestricted sense.
Hagelstein, Paul, Alexander M. Stokolos.
"Weak Type Inequalities for Ergodic Strong Maximal Operators."
Acta Scientiarum Mathematicarum, 76 (3-4): 427-441.