Decay rates for (2+1)-dimensional oscillatory integral operators with homogeneous polynomial phases
Consider the oscillatory integral operators \begin{equation} T_{\lambda}f(y)=\int_{\mathbb{R}^{2}}e^{i\lambda S\left( x_{1},x_{2}% ,y\right) }\Phi(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 $0 1,$ $\left \Vert T_{\lambda}\right \Vert _{L^{2}\rightarrow L^{2}}=O\left( \lambda^{-1/\left( 2\delta \right) }\right) $, while in the endpoint case $\delta=1$ the bound becomes $\left \Vert T_{\lambda}\right \Vert_{L^{2}\rightarrow 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.