Lacunary Fourier and Walsh-Fourier series near L1

We prove the following theorem: given a lacunary sequence of integers {n(j)}, the subsequences F(nj)f and W(nj)f of respectively the Fourier and the Walsh-Fourier series of f : T -> C converge almost everywhere to f whenever integral(T) vertical bar f(x)vertical bar log log(e(e) + vertical bar f(x)vertical bar) log log log log (e(eee) + vertical bar f(x)vertical bar) dx < infinity (1). 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-L-p bound for the lacunary Walsh-Carleson operator parallel to sup(nj) vertical bar W(nj)f vertical bar parallel to(p,infinity) <= K log(e + p')parallel to f parallel to(p) for all 1 < p <= 2.