Iterated trilinear Fourier integrals with arbitrary symbols

Abstract

We prove Lp estimates for trilinear multiplier operators with singular symbols. These operators arise in the study of iterated trilinear Fourier integrals, which are trilinear variants of the bilinear Hilbert transform. Specifically, we consider trilinear operators determined by multipliers that are products of two functions m1(ξ1,ξ2) and m2(ξ2,ξ3), such that the singular set of m1 lies in the hyperplane ξ1=ξ2 and that of m2 lies in the hyperplane ξ2=ξ3. While previous work [8] requires that the multipliers satisfy χξ1<ξ2⋅χξ2<ξ3, our results allow for the case of the arbitrary multipliers, which have common singularities.