Some new weak $(H_p-L_p)$ type inequality for weighted maximal operators of Walsh-Fourier series

TL;DR

This paper introduces new weighted maximal operators for Walsh-Fourier series partial sums, proving their boundedness from Hardy spaces to weak Lebesgue spaces for certain weights and establishing the sharpness of these results.

Contribution

It presents novel weighted maximal operators for Walsh-Fourier series and demonstrates their boundedness and optimality in the context of Hardy and Lebesgue spaces.

Findings

Boundedness of new operators from $H_p$ to weak-$L_p$ for $0<p<1$

Identification of optimal weights for boundedness

Proof of sharpness of the main results

Abstract

In this paper we introduce some new weighted maximal operators of the partial sums of the Walsh-Fourier series. We prove that for some "optimal" weights these new operators indeed are bounded from the martingale Hardy space $H_{p}$ to the Lebesgue space $\text{weak}-L_{p},$ for $0<p<1.$ Moreover, we also prove sharpness of this result.