A generalization for Fourier transforms of a theorem due to Marcinkiewicz

It is shown that the maximal operator of the Marcinkiewicz means of a tempe red distribution is bounded from H-p(R-2) to L-p(R-2) for all p(0) < p <les s than or equal to> infinity and, consequently, is of weak type (1, 1), whe re p(0) < 1. As a consequence we obtain a generalization for Fourier transf orms of a summability result due to Marcinkiewicz and Zhizhiashviii, more e xactly, the Marcinkiewicz means of a function f <is an element of> L-1(R-2) converge a.e. to the function in question. Moreover, we prove that the Mar cinkiewicz means are uniformly bounded on the spaces H-p(R-2) and so they c onverge in the norm (p(0) < p < infinity). Similar results for the Riesz tr ansforms are also given. (C) 2000 Academic Press.