Singular Brascamp-Lieb Forms and Multilinear Fourier Multipliers with Rough or Oscillatory Multipliers

Abstract

In this thesis, we study the boundedness of several multilinear operators including singular Brascamp-Lieb forms and certain multilinear Fourier multipliers with rough or oscillatory multipliers. Several important multilinear singular integral operators in harmonic analysis such as the Coifman-Meyer multipliers, the bilinear Hilbert transform, and twisted paraproducts all fall within the class of singular Brascamp-Lieb forms. The boundedness of a singular Brascamp-Lieb form is invariant under certain linear changes of variables. Given specific dimension data, we classify singular Brascamp-Lieb forms up to equivalence and characterize their boundedness in this setting. Typically, for a given singular Brascamp-Lieb form, one imposes the Mihlin's condition on its multiplier, which is the Fourier transform of the singular kernel. This Mihlin's condition can be generalized to Hormander's condition, which allows for fractional regularity. Naturally, this raises the question: what is the minimal regularity required of the multiplier to ensure boundedness? We address this question for multipliers that may exhibit Lipschitz-type singularities. Furthermore, the linear projections appearing in singular Brascamp-Lieb forms can be replaced by nonlinear maps. This line of research traces back to the 1970s, when singular Radon transforms were first studied. We also explore multilinear generalizations of the singular Radon transform, where the associated multipliers may exhibit oscillatory behavior. A crucial step in establishing the boundedness of such multilinear oscillatory multipliers is to prove a suitable smoothing inequality. We provide partial answers regarding which classes of multilinear oscillatory multipliers admit such smoothing effects.