Multi-parameter singular radon transforms I: The L 2 theory

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The purpose of this paper is to study the L 2 boundedness of operators of the form f ↦ ψ(x) ∫ f (γ t (x))K(t)dt, where γ t (x) is a C ∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ N × ℝ n , satisfying γ 0(x) ≡ x, ψ is a C ∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ n , and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ N . 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 L 2. The case when K is a Calderon-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderon- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of L p boundedness, while the third paper deals with the special case when γ is real analytic.