Decay Properties Of Multilinear Oscillatory Integrals

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