Everywhere divergent Fourier series with respect to the Walsh system and with respect to multiplicative systems

In this paper a new construction of everywhere divergent Fourier–Walsh series is presented. This construction enables one to halve the gap in the Lebesgue–Orlicz classes between the Schipp–Moon lower bound established by using Kolmogorov's construction and the Sjölin upper bound obtained by using Carleson's method. Fourier series which are everywhere divergent after a rearrangement are constructed with respect to the Walsh system (and to more general systems of characters) with the best lower bound for the Weyl factor. Some results related to an upper bound of the majorant for partial sums of series with respect to rearranged multiplicative systems are established. The results thus obtained show certain merits of harmonic analysis on the dyadic group in clarifying and overcoming fundamental difficulties in the solution of the main problems of Fourier analysis.