  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).
