$(H_p-L_p)$ type inequalities for subsequences of Nörlund means of Walsh-Fourier series

We investigate the subsequence $\{t_{2^n}f\}$ of Nörlund means with respect to the Walsh system generated by non-increasing and convex sequences. In particular, we prove that a big class of such summability methods are not bounded from the martingale Hardy spaces $H_p$ to the space $weak-L_p$ for $0<p<1\mathrm{/}(1+\alpha )$, where $0<\alpha <1$. Moreover, some new related inequalities are derived. As application, some well-known and new results are pointed out for well-known summability methods, especially for Nörlund logarithmic means and Cesàro means.