Abstract
A variant of the ergodic maximal function is studied. This maximal function reduces to the usual one for $f \geqq 0$, but the cancellation between the positive and negative parts of $f$ causes interesting behavior. In particular the maximal function can be in $L^1$ even when the function is not in $L \log^+ L$. A relation between this maximal function and the ergodic Hilbert transform is studied.
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