Abstract. 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 $\phi $ on $[0,\infty )$ that is positive, increasing, and $o(\log(x))$ for $x \rightarrow\infty $ as well as a pair of commuting invertible non-periodic measure-preserving transformations on a space $\Omega $ of finite measure, a function $f \in L\phi(L)(\Omega )$ is constructed whose associated multiparameter ergodic averages fail to converge almost everywhere in the unrestricted sense.

AMS Subject Classification (1991): 47A25, 28D05, 28D15

Keyword(s): multiparameter ergodic averages, multiparameter ergodic maximal operators

Received June 16, 2009, and in revised form September 11, 2009. (Registered under 80/2009.)