Lacunary Fourier and Walsh-Fourier series near L^1

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.

Final version accepted on Coll. Math

Cited by in corpus (5)

• On weighted norm inequalities for the Carleson and Walsh-Carleson operators
• Weak-L^p bounds for the Carleson and Walsh-Carleson operators
• Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type
• Uniform boundedness of the Fourier partial sum operators on the weighted spaces of local fields
• Endpoint bounds for the bilinear Hilbert transform