Multi-parameter singular Radon transforms II: The theory.

The purpose of this paper is to study the boundedness of operators of the form where is a function defined on a neighborhood of the origin in , satisfying , ψ is a cut-off function supported on a small neighborhood of , and K is a “multi-parameter singular kernel” supported on a small neighborhood of . 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 ( ). 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 , while the third paper deals with the special case when γ is real analytic. [ABSTRACT FROM AUTHOR]