Dyadic Martingale Transforms and Weighted Walsh-Carleson Operators

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.