On the Convergence of Lacunary Walsh-Fourier Series
  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. 
Yen Do and Michael T. Lacey
math.CA