Some new results and inequalities for subsequences of Nörlund logarithmic means of Walsh–Fourier series

Abstract

We prove that there exists a martingale f ∈ H_p such that the subsequence {L<sub>2n</sub 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 f ∈ L_p, p ≥ 1, L_2n f converge to f at any Lebesgue point x. Moreover, some new related inequalities are derived.