On the Convergence of Lacunary Walsh-Fourier Series

TL;DR

This paper proves almost everywhere convergence of lacunary Walsh-Fourier series for functions in a specific Orlicz space, using advanced harmonic analysis techniques and a new multi-frequency decomposition.

Contribution

It introduces a novel multi-frequency Calderon-Zygmund decomposition and extends convergence results to functions in L loglog L (logloglog L) spaces for lacunary Walsh-Fourier series.

Findings

Convergence holds for functions in L loglog L (logloglog L).

The proof combines Walsh phase plane analysis and a new multi-frequency decomposition.

An improved Hausdorff-Young inequality for lacunary sequences is established.

Abstract

We study the Walsh-Fourier series of S_{n_j}f, along a lacunary subsequence of integers {n_j}. Under a suitable integrability condition, we show that the sequence converges to f a.e. Integral condition is only slightly larger than what the sharp integrability condition would be, by a result of Konyagin. The condition is: f is in L loglog L (logloglog L). The method of proof uses four ingredients, (1) analysis on the Walsh Phase Plane, (2) the new multi-frequency Calderon-Zygmund Decomposition of Nazarov-Oberlin-Thiele, (3) a classical inequality of Zygmund, giving an improvement in the Hausdorff-Young inequality for lacunary subsequences of integers, and (4) the extrapolation method of Carro-Martin, which generalizes the work of Antonov and Arias-de-Reyna.