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 $Φ(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 $λ$ 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 $δ=\max \left( \frac{n}{3},\frac{n-n_{s}}{2},n_{1}\right) >1,$ $\left \Vert T_λ\right \Vert _{L^{2}\rightarrow L^{2}}=O\left( λ^{-1/\left( 2δ\right) }\right) $, while in the endpoint case $δ=1$ the bound becomes $\left \Vert T_λ\right \Vert_{L^{2}\rightarrow L^{2}}=O\left( λ^{-1/2}\log λ\right) $. The decay rate is sharp, up to a power of $\log λ$ when $δ=1$. We further show that $δ$ is exactly the modified Newton distance for the phase function, thus verifies the conjecture of \citet{Greenleaf07} in this case.