l1-summability of higher-dimensional Fourier series

It is proved that the maximal operator of the l1-Fejér means of a d-dimensional Fourier series is bounded from the periodic Hardy space Hp(T ) to L p(T ) for all d/(d+1) < p ≤ ∞ and, consequently, is of weak type (1, 1). As a consequence we obtain that the l1-Fejér means of a function f ∈ L1(T ) converge a.e. to f . Moreover, we prove that the l1-Fejér means are uniformly bounded on the spaces Hp(T ) and so they converge in norm (d/(d +1) < p < ∞). Similar results are shown for conjugate functions and for a general summability method, called θ -summability. Some special cases of the l1–θ -summation are considered, such as the Weierstrass, Picard, Bessel, Fejér, de la Vallée Poussin, Rogosinski and Riesz summations. c ⃝ 2010 Elsevier Inc. All rights reserved.