Iterated Trilinear Fourier Integrals With Arbitrary Symbols

Abstract

My research interests fall under the broad categories of harmonic analysis and partial differential equations, with a particular focus on multilinear harmonic analysis with time-frequency analysis techniques. I have been particularly interested in understanding the relationship between Lp estimates of multilinear singular operators and the dimension of the singularity sets of their symbols as well as applications to non-linear partial differential equations and other fields. In this paper, I 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, I consider trilinear operators determined by multipliers that are products of two functions m1 ([xi]1 , [xi]2 ) and m2 ([xi]2 , [xi]3 ), such that the singular set of m1 lies in the hyperplane [xi]1 = [xi]2 and that of m2 lies in the hyperplane [xi]2 = [xi]3 . While previous work [15] requires that the multipliers satisfy [chi][xi]1 <[xi]2 [MIDDLE DOT] [chi][xi]2 <[xi]3 , my results allow for the case of the arbitrary multipliers, which have common singularities.