Lacunary Fourier and Walsh–Fourier series near

• Autores: Francesco Di Plinio
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 65, Fasc. 2, 2014, págs. 219-232
• Idioma: inglés
• DOI: 10.1007/s13348-013-0094-3
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• Resumen

  • 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