Lacunary Fourier and Walsh-Fourier series near L^1

TL;DR

This paper establishes almost everywhere convergence of lacunary Fourier and Walsh-Fourier series for functions in a specific Orlicz class, using less restrictive integrability conditions than previous results.

Contribution

It introduces a new convergence theorem under weaker integrability assumptions and provides a self-contained proof for the Walsh-Fourier case avoiding Antonov's lemma.

Findings

Almost everywhere convergence for lacunary Fourier series in LloglogLloglogloglogL

Weaker integrability conditions than previous theorems by Lie and Do-Lacey

Novel weak-L^p bounds for Walsh-Carleson operator

Abstract

We prove that, for functions in the Orlicz class LloglogLloglogloglogL, lacunary subsequences of the Fourier and the Walsh-Fourier series converge almost everywhere. Our integrability condition is less stringent than the homologous assumption in the almost everywhere convergence theorems of Lie (Fourier case) and Do-Lacey (Walsh-Fourier case), where the quadruple logarithmic term is replaced by a triple logarithm. Our proof of the Walsh-Fourier case is self-contained and, in antithesis to Do and Lacey's argument, avoids the use of Antonov's lemma, arguing directly via novel weak-L^p bounds for the Walsh-Carleson operator.