Lacunary Fourier and Walsh-Fourier series near L1

We prove the following theorem: given a lacunary sequence of integers {nj}, the subsequences Fnj f and Wnj f of respectively the Fourier and the Walsh-Fourier series of f: T → ℂ converge almost everywhere to (Formula presented.) Our integrability condition (1) is less stringent than the homologous assumption in the almost everywhere convergence theorems of Lie [14] (Fourier case) and Do and Lacey [6] (Walsh-Fourier case), where a triple-log term appears in place of the quadruple-log term of (1). Our proof of the Walsh-Fourier case is self-contained and, in antithesis to [6], avoids the use of Antonov's lemma [1, 19], relying instead on the novel weak-Lp bound for the lacunary Walsh-Carleson operator (Formula presented.). © 2013 Universitat de Barcelona.