Multi-parameter singular Radon transforms III: Real analytic surfaces

Abstract

The goal of this paper is to study operators of the form,Tf(x)=ψ(x)∫f(γt(x))K(t)dt, where γ is a real analytic function defined on a neighborhood of the origin in (t,x)∈RN×Rn, satisfying γ0(x)≡x, ψ is a cutoff function supported near 0∈Rn, and K is a “multi-parameter singular kernel” supported near 0∈RN. A main example is when K is a “product kernel.” We also study maximal operators of the form,Mf(x)=ψ(x)sup0<δ1,…,δN≪1∫|t|<1|f(γδ1t1,…,δNtN(x))|dt. We show that M is bounded on Lp (1<p⩽∞). We give conditions on γ under which T is bounded on Lp (1<p<∞); these conditions hold automatically when K is a Calderón–Zygmund kernel. This is the final paper in a three part series. The first two papers consider the more general case when γ is C∞.