Weak type (1,1) bound criterion for singular integral with rough kernel and its applications

In this paper, a weak type (1,1) bound criterion is established for singular integral operator with rough kernel. As some applications of this criterion, we prove some important operators with rough kernel in harmonic analysis, such as Calder'on commutator, higher order Calder'on commutator, general Calder'on commutator, Calder'on commutator of Bajsanski-Coifman type and general singular integral of Muckenhoupt type, are all of weak type (1,1).