Singular integral operators with kernels supported in higher dimensional subvarieties

In this paper, we introduce a class of singular Radon transforms on R-n with kernels supported in a subvariety in R-n x R-n determined by a polynomial mapping from R-n x R-n into R-n. The class of considered operators is related to the composition of homogeneous singular integral operators. We prove that the operators are bounded on L-p provided that the kernels are rough in L(log L)(2)(Sn-1 x Sn-1). The condition L(log L)(2)(Sn-1 x Sn-1) is observed to be optimal in the sense that the power 2 can not be replaced by a smaller number.