## Publicly Accessible Penn Dissertations

2019

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Philip T. Gressman

#### Abstract

In this thesis, we study the following multilinear oscillatory integral introduced by Christ, Li, Tao and Thiele \cite{CLTT}

I_{\lambda}(f_1,...f_n)=\int_{\mathbb{R}^m}e^{i\lambda P(x)}\prod_{j=1}^n f_j(\pi_j(x))\eta(x)dx,

where $P:\mathbb{R}^m\to\mathbb{R}$ is a real-valued measurable function, $\eta$ is a compactly supported smooth cutoff function. Each $\pi_j$ is a surjective linear transformation from $\mathbb{R}^m$ to $\mathbb{R}^{k_j}$, where $1\le k_j\le m-1$.

Each $f_j:\mathbb{R}^{k_j}\rightarrow \mathbb{C}$ is a locally integrable function with respect to Lebesgue measure on $\mathbb{R}^{k_j}$.

In Chapter 2, we first introduce the nondegeneracy degree along with the nondegeneracy norm defined in \cite{CLTT} to characterize the nondegeneracy condition of the phase function. In the same chapter, we will summarize some powerful tools that can help to simplify the problem and introduce the idea of a special geometric structure called separation".

There are three results in this thesis. The first proves trilinear oscillatory integrals with nondegenerate polynomial phase always have the decay property. The second one extends the one-dimensional case whose phase function has large nondegeneracy degree. The third result deals with the case where every linear mapping preserves the direct sum decomposition.

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