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 H p ( X d ) to L p ( X d ) 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 ∈ L 1 ( X d ) converge a.e. to f. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces H p ( X d ) 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.