Singular integral operators with kernels supported in higher dimensional subvarieties

Abstract

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