On Rearrangements of Vilenkin-Fourier Series which Preserve almost Everywhere Convergence

Abstract

ABSTRACT. It is known that the partial sums of Vilenkin-Fourier series of Lq functions (q > 1) converge a.e. In this paper we establish the L result for a class of rearrangements of the Vilenkin-Fourier series, and the Lq result (1 < q < 2) for a subclass of rearrangements.In the case of the Walsh-Fourier series, these classes include the Kaczmarz rearrangement studied by L. A. 2Balashov. The L result for the Kaczmarz rearrangement was first proved by K. H. Moon. The techniques of proof involve a modification of the Carleson-Hunt method and estimates on maximal functions of the Hardy-Littlewood type that arise from these rearrangements.