Sharp estimates for trilinear oscillatory integral forms
This dissertation consists of 4 chapters. In Chapter 1, we will briefly introduce some background of the trilinear oscillatory integrals and the motivations for their study. We also outline some key ideas in their proofs, as well as the major novelty of this dissertation. Chapter 2 serves as analytic preparation for the proof of our main theorem. We first extend a result of H ̈ormander to a trilinear setting, which can be viewed as a junior version of the main result. Secondly, we establish a trilinear analogue of Phong-Stein’s van der Corput Lemma, which is our major analytic ingredient in the proof of our main theorem. Chapter 3 is the heart of this dissertation. An algorithm of resolution of singularities in R2 is presented in detail. The content of this chapter is self-contained and the readers who are merely interested in this algorithm may jump in this chapter directly. Chapter 4 is designed to prove the main theorem. By using the algorithm in Chapter 3, a small neighborhood of a singular point is decomposed into finitely many curved triangular regions. In each of these regions, we can employ the analytic tools developed in Chapter 2 to obtain optimal control for the trilinear oscillatory integrals.
Oscillatory integrals, sharp estimates, Newton polyhedron, trilinear forms, resolution of singularities
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign