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

Abstract

Abstract We prove that there exists a martingale f ∈ H p $f\in 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 w e a k − L p $weak-L_{p} $ for 0 < p < 1 $0< p<1 $ . We also prove that for any f ∈ L p $f\in L_{p}$ , p ≥ 1 $p\geq 1 $ , L 2 n f $L_{2^{n}}f$ converge to f at any Lebesgue point x. Moreover, some new related inequalities are derived.

Keywords

• Walsh system
• Logarithmic means
• Partial sums
• Fejér means
• Martingale Hardy space
• Convergence