We prove that there exists a martingale $f\in H_{p} $ f ∈ H p such that the subsequence $\{L_{2^{n}}f \}$ { L 2 n f } of Nörlund logarithmic means with respect to the Walsh system are not bounded from the martingale Hardy spaces $H_{p}$ H p to the space $weak-L_{p} $ w e a k − L p for $0< p<1 $ 0 < p < 1 . We also prove that for any $f\in L_{p}$ f ∈ L p , $p\geq 1 $ p ≥ 1 , $L_{2^{n}}f$ L 2 n f converge to f at any Lebesgue point x. Moreover, some new related inequalities are derived.

We investigate the subsequence $\{t_{2^{n}}f \}$ { t 2 n f } of Nörlund means with respect to the Walsh system generated by nonincreasing and convex sequences. In particular, we prove that a large class of such summability methods are not bounded from the martingale Hardy spaces $H_{p}$ H p to the space $\mathit{weak-}L_{p} $ w e a k − L p for $0< p<1/(1+\alpha ) $ 0 < p < 1 / ( 1 + α ) , where $0<\alpha <1$ 0 < α < 1 . Moreover, some new related inequalities are derived. As applications, some well-known and new results are pointed out for well-known summability methods, especially for Nörlund logarithmic means and Cesàro means.