A Generalization for Fourier Transforms of a Theorem due to Marcinkiewicz

It is shown that the maximal operator of the Marcinkiewicz means of a tempered distribution is bounded from HpŽR2.to LpŽR2 . for all p0 p and, consequently, is of weak type Ž1, 1., where p0 1. As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkiewicz and Zhizhiashvili, more exactly, the Marcinkiewicz means of a function f L1ŽR2 . converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz means are uniformly bounded on the spaces HpŽR2 . and so they converge in the normŽp0 p .. Similar results for the Riesz transforms are also given.