An Analogy of the Carleson–Hunt Theorem with Respect to Vilenkin Systems

Abstract

In this paper we discuss and prove an analogy of the Carleson–Hunt theorem with respect to Vilenkin systems. In particular, we use the theory of martingales and give a new and shorter proof of the almost everywhere convergence of Vilenkin–Fourier series of f ∈ L _p(G_m)for p > 1 in case the Vilenkin system is bounded. Moreover, we also prove sharpness by stating an analogy of the Kolmogorov theorem for p = 1 and construct a function f ∈ L₁(G_m) such that the partial sums with respect to Vilenkin systems diverge everywhere.