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

• Main Authors: Persson, L.-E (Author), Schipp, F. (Author), Tephnadze, G. (Author), Weisz, F. (Author)
• Format: Article
• Language: English
• Published: Birkhauser 2022
• Subjects: Almost everywhere convergence; Carleson–Hunt theorem; Fourier analysis; Kolmogorov theorem; Vilenkin group; Vilenkin system; Vilenkin–Fourier series
• Online Access: View Fulltext in Publisher
• ISBN: 10695869 (ISSN)
• DOI: 10.1007/s00041-022-09938-2

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∈ Lp(Gm) 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(Gm) such that the partial sums with respect to Vilenkin systems diverge everywhere. © 2022, The Author(s).