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 L_2(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 L_2(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(x) = sup ε>0 | T_ε u(x) | , 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.