Singular integral operators with kernels supported in higher dimensional subvarieties

In this paper, we introduce a class of singular Radon transforms on Rⁿ with kernels supported in a subvariety in Rⁿ× Rⁿ determined by a polynomial mapping from Rⁿ× Rⁿ into Rⁿ. 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(S^n-1× S^n-1). The condition L(log L) ²(S^n-1× S^n-1) is observed to be optimal in the sense that the power 2 can not be replaced by a smaller number.