Singular integral operators with non-smooth kernels on irregular domains

Let χ 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(χ), we give a sufficient condition on the kernel of T so that T is of weak type (1,1), hence bounded on L_p(χ) for 1 < p ≤ 2; our condition is weaker than the usual Hörmander integral condition. ii) Assuming that T is a bounded linear operator on L₂(Ω) where Ω is a measurable subset of χ, we give a sufficient condition on the kernel of T so that T is of weak type (1,1), hence bounded on L_p(Ω) for 1 < p ≤ 2. iii) We establish sufficient conditions for the maximal truncated operator T_*, which is defined by T_*u(Greek cursive chi) = sup_ε>0|T_εu(Greek curve chi)|, to be L_p bounded, 1 < p < ∞. 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.