Some new results for subsequences of N\"orlund logarithmic means of Walsh-Fourier series

TL;DR

This paper investigates the behavior of N"orlund logarithmic means of Walsh-Fourier series, showing unboundedness in certain spaces for some functions and convergence properties for others, along with new inequalities.

Contribution

It establishes the unboundedness of specific subsequences of N"orlund means in weak-Lp spaces and proves convergence at Lebesgue points for functions in Lp, introducing new related inequalities.

Findings

Existence of functions with unbounded subsequences of N"orlund means in weak-Lp spaces for 0<p<1.

Convergence of N"orlund means to the original function at Lebesgue points for p≥1.

Derivation of new inequalities related to N"orlund logarithmic means.

Abstract

We prove that there exists a martingale $f\in H_p $ such that the subsequence $\{L_{2^n}f \}$ of N\"orlund logarithmic means with respect to the Walsh system are not bounded in the Lebesgue space $weak-L_p $ for $0<p<1 $. Moreover, we prove that for any $f\in L_p(G),$ $p\geq 1, $ $L_{2^n}f$ converge to $f$ at any Lebesgue point $x$. Some new related inequalities are derived.