It is shown that the maximal operator of the Marcinkiewicz means of a temperedŽ . quently, is of weak type 1, 1 , where p -1. As a consequence we obtain a 0 generalization for Fourier transforms of a summability result due to Marcinkiewicz Ž 2 . and Zhizhiashvili, more exactly, the Marcinkiewicz means of a function f g L R 1 converge a.e. to the function in question. Moreover, we prove that the Ž 2 . Marcinkiewicz means are uniformly bounded on the spaces H R and so they