Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series

A generalization of Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series is investigated with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the Marcinkiewicz-θ-means of a tempered distribution is bounded from Hp(Xd) to Lp(Xd) for all d/(d+α)<p?∞ and, consequently, is of weak type (1,1), where 0<α?1 is depending only on θ and X=R or X=T. As a consequence we obtain a generalization of a summability result due to Marcinkiewicz and Zhizhiashvili for d-dimensional Fourier transforms and Fourier series, more exactly, the Marcinkiewicz-θ-means of a function f∈L₁(Xd) converge a.e. to f. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces Hp(Xd) and so they converge in norm (d/(d+α)<p<∞). Similar results are shown for conjugate functions. Some special cases of the Marcinkiewicz-θ-summation are considered, such as the Fejér, Cesàro, Weierstrass, Picar, Bessel, de La Vallée-Poussin, Rogosinski and Riesz summations.