Resumen de Lacunary Fourier and Walsh–Fourier series near L^1

Francesco Di Plinio

• We prove the following theorem: given a lacunary sequence of integers \{n_j\}, the subsequences \mathsf{F }_{n_j} f and \mathsf{W }_{n_j} f of respectively the Fourier and the Walsh–Fourier series of f: \mathbb T \rightarrow \mathbb{C } converge almost everywhere to f whenever \begin{aligned} \int \limits _{\mathbb{T }} |f(x)| \log \log (\mathrm{e}^\mathrm{e}+|f(x)|)\log \log \log \log \left( \mathrm{e}^{\mathrm{e}^{\mathrm{e}^\mathrm{e}}}+ |f(x)|\right) \mathrm{d}x <\infty \quad (1). \end{aligned} 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 \begin{aligned} \big \Vert \sup _{n_j} |\mathsf{W }_{n_j} f|\big \Vert _{p,\infty } \le K \log ( \mathrm{e}+ p') \Vert f\Vert _p \qquad \forall 1