Weak type inequalities for the ?_1-summability of higher dimensional Fourier transforms

摘要

Under some weak conditions on θ, it was verified in [21, 17] that the maximal operator of the ? _1-θ-means of a tempered distribution is bounded from H _p(?~d) to L _p(?~d) for all d/(d + α) < p ≤ ∞, where 0 < α ≤ 1 depends only on θ. In this paper, we prove that the maximal operator is bounded from H _(d/(d+α))(?~d) to the weak L _(d/(d+α))(?~d) space. The analogous result is given for Fourier series, as well. Some special cases of the ? _1-θ-summation are considered, such as the Weierstrass, Picard, Bessel, Fejér, de La Vallée-Poussin, Rogosinski and Riesz summations.