Dyadic Martingale Transforms and Weighted Walsh-Carleson Operators

TL;DR

This paper investigates weighted Walsh--Carleson maximal operators linked to dyadic martingale transforms, establishing weak type estimates, divergence criteria, and applications to various Walsh--Fourier summability methods.

Contribution

It introduces new weak type (1,1) estimates for maximal operators under specific weight conditions and connects divergence criteria to explicit ratio conditions.

Findings

Proved weak type (1,1) estimates for weighted Walsh--Carleson maximal operators.

Established divergence criteria based on weight behavior near the top dyadic scale.

Extended results to matrix transforms and various Walsh--Fourier summability methods.

Abstract

We study weighted Walsh--Carleson maximal operators arising from dyadic martingale transforms associated with Walsh--Fourier partial sums. For weights satisfying a uniform dyadic variation condition and a uniform bound at the top dyadic scale, we prove weak type~$(1,1)$ estimates for the corresponding maximal operators along subsequences. We also give divergence criteria in terms of the behavior of the weights near the top dyadic scale and, under suitable admissibility assumptions, relate these criteria to explicit ratio conditions. As applications, we obtain results on matrix transforms of Walsh--Fourier partial sums, including de la Vall\'ee Poussin means, Ces\`aro means with varying parameters, N\"orlund logarithmic means, and general N\"orlund means. In particular, we prove a Walsh--Paley analogue of the Leindler--Tandori theorem and establish everywhere divergence results for several summability methods.