Role of structure in oscillatory integral estimates
Although oscillatory integrals intrinsically depend only on measure-theoretic properties of the phase and amplitude functions, in practice most techniques for deriving estimates invoke structural information either implicitly or explicitly. The main focus of this thesis is to study the role of certain structural assumptions, which are of a geometric nature and arise naturally in applications. In particular, we will derive relations with sublevel set estimates, which are often easier to understand, manipulate and exploit. In the first instance, we study one-dimensional oscillatory integrals. Under our assumptions, we establish a relation to sublevel set estimates which is essentially sharp. As a consequence of our analysis we show that one-dimensional oscillatory integral estimates behave well under composition with functions possessing well-understood structure, key examples of this are given by polynomials, but also, for instance, real power functions with exponent greater than one. We then proceed to address convex phase functions in general dimensions, as was previously considered in seminal work of Bruna, Nagel & Wainger. We again derive a sharp relation between such oscillatory integrals and their sublevel sets. The assumption of finite line type seen in the work of Bruna, Nagel & Wainger is replaced with a naturally appearing geometric assumption which holds in many situations where the phase is not of finite type. A key aspect of our argument is the aforementioned idea that oscillatory integral estimates depend only on measure-theoretic properties, which we exploit to consider rearrangements of the phase with additional structure. In contrast with Bruna, Nagel & Wainger, our hypotheses and the constants appearing in the resulting estimates are preserved under scaling. In another part of the thesis, we consider the separate but related problem of understanding conditions under which uniform sublevel set estimates hold. We establish a framework in which sublevel set estimates are related to lower bounds on L^p means, suggesting further structural considerations. Within this framework, we note that in some settings, such as the set of functions on the unit cube with their Laplacian bounded below by a fixed positive constant previously considered by Carbery, Christ & Wright, uniform sublevel set estimates only narrowly fail. In the course of these proofs we establish some extensions of existing results for mean value inequalities which may be of independent interest.