Marcinkiewicz integral on hardy spaces

In this paper we prove that the Marcinkiewicz integral (mu)(omega) is an operator of type (H 1,L 1) and of type (H 1,(infinity),L 1,(infinity)). As a corollary of the results above, we obtain again the the weak type (1,1) boundedness of (mu)(omega), but the smoothness condition assumed on (omega) is weaker than Steinʹs condition.