Iterated trilinear Fourier integrals with arbitrary symbols

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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 MTT2 requires that the multipliers satisfy χξ₁ <ξ₂ · χξ₂<ξ₃, our results allow for the case of the arbitrary multipliers, which have common singularities.