Multi-parameter singular Radon transforms III: Real analytic surfaces

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(t,x)∈RN×Rn, satisfying γ0(x)≡xγ0(x)≡x, ψ   is a cutoff function supported near 0∈Rn0∈Rn, and K   is a “multi-parameter singular kernel” supported near 0∈RN0∈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 MM is bounded on LpLp (1