( H p − L p ) $(H_{p}-L_{p})$ -Type inequalities for subsequences of Nörlund means of Walsh–Fourier series

Abstract

Abstract 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 w e a k − L p $\mathit{weak-}L_{p} $ for 0 < p < 1 / ( 1 + α ) $0< p<1/(1+\alpha ) $ , where 0 < α < 1 $0<\alpha <1$ . 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 Nörlund logarithmic means and Cesàro means.

Keywords

• Walsh system
• Nörlund means
• Cesàro means
• Nörlund logarithmic means
• Martingale Hardy space
• Convergence