Abstract
We prove that there exists a martingale fin H_{p} such that the subsequence {L_{2^{n}}f } of Nörlund logarithmic means with respect to the Walsh system are not bounded from the martingale Hardy spaces H_{p} to the space weak-L_{p} for 0< p<1 . We also prove that for any fin L_{p}, pgeq 1 , L_{2^{n}}f converge to f at any Lebesgue point x. Moreover, some new related inequalities are derived.
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