Decay rates for (2+1)-dimensional oscillatory integral operators with homogeneous polynomial phases

Abstract

Consider the oscillatory integral operators \begin{equation} T_λf(y)=\int_{\mathbb{R}^{2}}e^{iλS\left( x_{1},x_{2}% ,y\right) }Φ(x_{1},x_{2},y)f(x_{1},x_{2})dx_{1}dx_{2},\nonumber \end{equation} where $\Phi (x_1,x_2,y)\in C_0^{\infty }\left({\mathbb{R}}^3\right)$, $S\left(x_1,x_2,y\right)\in C_0^{\infty }\left({\mathbb{R}}^3\right)$ is real valued, and $\lambda$ is a large real number. We prove that, if $S\left(x_1,x_2,y\right)=y^{n_1}{h}_{n-n_1}\left(x_1,x_2\right)+\cdots +y^{n_s}{h}_{n-n_s}\left(x_1,x_2\right)$ is a homogeneous polynomial of degree $n,$ where $0<n_1<n_2<\cdots <n_s<n$, and $h_{n-n_1}\left(x_1,x_2\right)$ and $h_{n-n_s}\left(x_1,x_2\right)$ are non-degenerate in the sense that there are no multiple factors when they are factored into linear terms over complex numbers, then for $\delta =\max \left(\frac{n}{3},\frac{n-n_s}{2},n_1\right)>1,$ ${\Vert T_{\lambda }\Vert }_{L^2\to L^2}=O\left({\lambda }^{-1/\left(2\delta \right)}\right)$, while in the endpoint case $\delta =1$ the bound becomes ${\Vert T_{\lambda }\Vert }_{L^2\to L^2}=O\left({\lambda }^{-1/2}\log \lambda \right)$. The decay rate is sharp, up to a power of $\log \lambda$ when $\delta =1$. We further show that $\delta$ is exactly the modified Newton distance for the phase function, thus verifies the conjecture of \citet{Greenleaf07} in this case.