(H-p - L-p)-Type inequalities for subsequences of Norlund means of Walsh-Fourier series

We investigate the subsequence {t2n f} of Norlund 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 to the space weak-Lp for 0 < p < 1/(1+ alpha), where 0 < alpha < 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 Norlund logarithmic means and Cesaro means.