The One-dimensional Martingale Hardy Spaces and Partial Sums and Fej\'er Means with respect to Walsh system

TL;DR

This paper establishes new inequalities for partial sums and Fejér means in Walsh system Hardy spaces, demonstrating their optimality and providing both known and novel applications.

Contribution

It introduces and proves the best possible inequalities for Walsh system partial sums and Fejér means in Hardy spaces, extending existing results.

Findings

New $(H_p,L_p)$ inequalities for Walsh partial sums and Fejér means

Optimality of the derived inequalities

Applications to known and new results in harmonic analysis

Abstract

In this paper we prove and discuss some new $\left( H_p,L_{p}\right)$ type inequalities for partial Sums and Fej\'er means with respect to Walsh system. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.