We prove that there exists a martingale f ∈ H p such that the subsequence {L 2 n f } of Nörlund logarithmic means with respect to the Walsh system are not bounded in the Lebesgue space weak − L p for 0 < p < 1. Moreover, we prove that for any f ∈ L p (G), p ≥ 1, L 2 n f converge to f at any Lebesgue point x. Some new related inequalities are derived.
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