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Given the usual averages in ergodic theory, let n1 ≤ n2 ≤ … and . There is a strong inequality ‖Sf‖2 ≤ 25‖f‖2 and there is a weak inequality m{Sf > λ} ≤ (7000/λ)‖f‖1. Related results and questions for other variants of this square function are also discussed.
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Jones et al. (1996) studied this question.
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