Multi-Parameter singular radon transforms

Abstract

The purpose of this announcement is to describe a development given in a series of forthcoming papers by the authors that concern operators of the form f → ψ (x) ∫ f(Υt(x))K (t) dt, where Υt (x) = Υ (t, x) is a C∞ function defined on a neighborhood of the origin in (t, x) ε ℝN × ℝn satisfying Υ0 (x) ≡ x, K (t) is a "multi-parameter singular kernel" supported near t = 0, and ψ is a cutoff function supported near x = 0. This note concerns the case when K is a "product kernel." The goal is to give conditions on Υ such that the above operator is bounded on Lp for 1 < p < ∞. Associated maximal functions are also discussed. The "single-parameter" case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger. The theory here extends these results to the multi-parameter context and also deals eifectively with the case when Υ is real-analytic. © International Press 2011.