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D. Ornstein proved that the ergodic maximal function satisfies a reverse weak type inequality, and from this he deduced that the integrability of the maximal function      f ∗  {f^ * }       implies that      f f       belongs to      L log + L L{\log ^ + }L      . Weighted analogues of these results are proved.

Weighted reverse weak type inequalities for the ergodic maximal function and the classes 𝐿𝑙𝑜𝑔⁺𝐿