Date of Award

2022

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mathematics

First Advisor

Philip T. Gressman

Abstract

In this thesis, we study bilinear oscillatory integral operators of the form \[ I_\lambda(f_1, f_2) = \int_M e^{i\lambda \Phi(x)} f_1(x^1) f_2(x^2) a(x) d\sigma(x) \] where $x^1:= (x_1, \dots, x_d)$, $x^2:= (x_{d+1}, \dots, x_{2d})$, and $\rho, \Phi, a$ are smooth functions on an open box $B_1$ with $a$ compactly supported, $\partial_i \rho$ nonvanishing on $B_1$ for each $i$, and $M := \{ x \in B_1 \,|\, \rho(x) = 0\}$. Under an additional determinant condition that has similarities to both a mixed Hessian condition on $\Phi$ and a Phong-Stein rotational curvature condition on $\rho$, we prove that this operator has optimal $L^2$ decay, namely that \[ |I_\lambda(f_1, f_2)| \leq C|\lambda|^{-\frac{d-1}2} ||f_1||_{L^2(\mathbb{R}^d)} ||f_2||_{L^2(\mathbb{R}^d)} \] The proof uses a frequency space decomposition which is a higher-dimensional analogue of one developed in earlier work with Gressman, and applies this to the functions $f_1$ and $f_2$ to generate a kernel which captures the oscillatory behavior of the phase and can be analyzed using stationary phase arguments, among others. The constant $C$ in the bound depends continuously on parameters based on $a, \Phi, \rho$, the dimension $d$, and the size of the support of the integrand, and so the result is stable under small perturbations of these objects.

We then study two specific bilinear operators which have polynomial phase, and show how the results of the main theorem can be leveraged to prove decay even when the determinant condition in the hypothesis does not hold. We also use these examples to show that the decay of the operator is affected by the precise way in which the determinant condition fails.

Included in

Mathematics Commons

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