Singular integral operators with non-smooth kernels on irregular domains

Let ? be a space of homogeneous type. The aims of this paper are as follows.i) Assuming that T is a bounded linear operator on L2(?), we give a sufficient condition on the kernel of T such that T is of weak type (1,1), hence bounded on Lp(?) for 1 < p = 2; our condition is weaker then the usual Hormander integral condition.ii) Assuming that T is a bounded linear operator on L2(O) where O is a measurable subset of ?, we give a sufficient condition on the kernel of T so that T is of weak type (1,1), hence bounded on Lp(O) for 1 < p = 2.iii) We establish sufficient conditions for the maximal truncated operator T* which is defined by T*u(x) = supe>0 |Teu(x)|, to be Lp bounded, 1 < p < 8. Applications include weak (1,1) estimates of certain Riesz transforms, and Lp boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.