Iterated trilinear Fourier integrals with arbitrary symbols

Abstract

We prove $L^p$ 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 ${{m}}_1(\xi_1, \xi_2)$ and ${m}_2(\xi_2, \xi_3)$, such that the singular set of $m_1$ lies in the hyperplane $ \xi_1=\xi_2$ and that of $m_2$ lies in the hyperplane $\xi_2=\xi_3$. While previous work \cite{MTT2} requires that the multipliers satisfy $\chi_{\xi_1 <\xi_2} \cdot \chi_{\xi_2<\xi_3}$, our results allow for the case of the arbitrary multipliers, which have common singularities.