Singular integral operators with non-smooth kernels on irregular domains

Abstract

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 ≤ 2 ; our condition is weaker than the usual Hörmander 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 ≤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 < \infty . 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.