Resumen de Lacunary Fourier and Walsh–Fourier series near

Francesco Di Plinio

• We prove the following theorem: given a lacunary sequence of integers {??} , the subsequences ???? and ???? of respectively the Fourier and the Walsh–Fourier series of ?:?→ℂ converge almost everywhere to ? whenever ∫?|?(?)|loglog(ee+|?(?)|)loglogloglog(eeee+|?(?)|)d?<∞(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- ?? bound for the lacunary Walsh–Carleson operator ‖‖sup??|????|‖‖?,∞≤?log(e+?′)‖?‖?∀1