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arXiv · 1304.3943

Lacunary Fourier and Walsh-Fourier series near L^1

Francesco Di Plinio
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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.

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Francesco Di Plinio. 2013-12-04. Lacunary Fourier and Walsh-Fourier series near L^1. https://doi.org/10.1007/s13348-013-0094-3

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