On some pointwise inequalities

In a recent paper (Studia Math. 138 (2000) 285-291) we proved pointwise estimates relating some classical maximal and singular integral operators. Here we show that, in a sense, there are more flexible inequalities which not only imply the previously known results but also give something new. In particular, they hold for the multilinear Calderón-Zygmund operators. This result gives a new approach to a recent work by Grafakos and Torres, and unifies some classical inequalities by Cotlar and Coifman and Fefferman.