Let be a space of homogeneous type. The aims of this paper are as follows: i) Assuming that is a bounded linear operator on we give a sufficient condition on the kernel of so that is of weak type (1,1), hence bounded on for ; our condition is weaker than the usual Hörmander integral condition. ii) Assuming that is a bounded linear operator on where is a measurable subset of , we give a sufficient condition on the kernel of so that is of weak type (1,1), hence bounded on for . iii) We establish sufficient conditions for the maximal truncated operator , which is defined by = sup, to be bounded, . Applications include weak (1,1) estimates of certain Riesz transforms and boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.
Cite this article
Xuan Thinh Duong, Alan G.R. McIntosh, Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoam. 15 (1999), no. 2, pp. 233–265DOI 10.4171/RMI/255