Singular integral operators with non-smooth kernels on irregular domains

Let chi be a space of homogeneous type. The aims of this paper are as follows. i) Assuming that T is a bounded linear operator on L-2(chi), we give a sufficient condition on the kernel of T so that T is of weak type (1, 1), hence bounded on L-p(chi) for 1 < p less than or equal to 2; our condition is weaker than the usual Hormander integral condition, ii) Assuming that T is a bounded linear operator on L-2(Omega) where Omega is a measurable subset of chi, we give a sufficient condition on the kernel of T so that T is of weak type (1, 1), hence bounded on L-p(Omega) for 1 < p less than or equal to 2. iii) We establish sufficient conditions for the maximal truncated operator T-*, which is defined by T(*)u(x) = sup(epsilon>0) \T(epsilon)u(x)\, to be L-p bounded, 1 < p < infinity. Applications include weak (1, 1) estimates of certain Riesz transforms, and L-p boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.