Towards direct $L^2$-bounds for maximal partial sums of Walsh--Fourier series: The case of dyadic partial sums

We outline an approach to obtain direct $L^2$ estimates not requiring interpolation for so-called linearized partial sums operators associated with expansions in Walsh functions. We focus specifically on a simpler case of dyadic partial sums but also outline a second approach to proving bounds on general linearized partial sums.