Multi-parameter singular Radon transforms II: The LpLp theory

The purpose of this paper is to study the LpLp boundedness of operators of the formf↦ψ(x)∫f(γt(x))K(t)dt, where γt(x)γt(x) is a C∞C∞ function defined on a neighborhood of the origin in (t,x)∈RN×Rn(t,x)∈RN×Rn, satisfying γ0(x)≡xγ0(x)≡x, ψ   is a C∞C∞ cut-off function supported on a small neighborhood of 0∈Rn0∈Rn, and K   is a “multi-parameter singular kernel” supported on a small neighborhood of 0∈RN0∈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 LpLp (1