Singular integral operators with non-smooth kernels on irregular domains

Abstract

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 Lp(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 Lp(Ω) for 1 0|Tεu(Greek curve chi)|, to be Lp bounded, 1 < p < ∞. Applications include weak (1, 1) estimates of certain Riesz transforms, and Lp boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.