Multi-parameter singular Radon transforms II: The [formula omitted] theory

Abstract

The purpose of this paper is to study the Lp boundedness of operators of the formf↦ψ(x)∫f(γt(x))K(t)dt, where γt(x) is a C∞ function defined on a neighborhood of the origin in (t,x)∈RN×Rn, satisfying γ0(x)≡x, ψ is a C∞ cut-off function supported on a small neighborhood of 0∈Rn, and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0∈RN. We also study associated maximal operators. The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on Lp (1<p<∞). The case when K is a Calderón–Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their work to the case when K is (for instance) given by a “product kernel”. Even when K is a Calderón–Zygmund kernel, our methods yield some new results. This is the second paper in a three part series. The first paper deals with the case p=2, while the third paper deals with the special case when γ is real analytic.