具齐次多项式相位的(2+1)维振荡积分算子的衰减率
Decay rates for (2+1)-dimensional oscillatory integral operators with homogeneous polynomial phases
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中文总结 AI 辅助
本文研究具齐次多项式相位的(2+1)维振荡积分算子，证明其衰减率，验证Greenleaf07的相关猜想，确定衰减率的最优性。
中文摘要 AI 辅助
考虑振荡积分算子$T_{\lambda}f(y)=\int_{\mathbb{R}^{2}}e^{i\lambda S\left( x_{1},x_{2},y\right) }\Phi(x_{1},x_{2},y)f(x_{1},x_{2})dx_{1}dx_{2}$，其中$\Phi(x_{1},x_{2},y)\in C_{0}^{\infty}\left( \mathbb{R}^{3}\right)$，$S\left( x_{1},x_{2},y\right) \in C_{0}^{\infty}\left( \mathbb{R}^{3}\right)$为实值函数，$\lambda$为大实数。我们证明：若$S\left( x_{1},x_{2},y\right) =y^{n_{1}}h_{n-n_{1}}\left(x_{1},x_{2}\right) +\cdots+y^{n_{s}}h_{n-n_{s}}\left( x_{1},x_{2}\right)$是次数为$n$的齐次多项式，满足$0<n_{1}<n_{2}<\cdots <n_{s}<n$，且$h_{n-n_{1}}\left( x_{1},x_{2}\right)$与$h_{n-n_{s}}\left( x_{1},x_{2}\right)$在复数域上分解为线性项时无重因子（即非退化），则当$\delta=\max \left( \frac{n}{3},\frac{n-n_{s}}{2},n_{1}\right) >1$时，算子范数满足$\left\Vert T_{\lambda}\right\Vert _{L^{2}\rightarrow L^{2}}=O\left( \lambda^{-1/\left( 2\delta \right) }\right)$；在端点情形$\delta=1$时，该范数界为$\left\Vert T_{\lambda}\right\Vert_{L^{2}\rightarrow L^{2}}=O\left( \lambda^{-1/2}\log \lambda \right)$，且该衰减率是最优的（仅在$\delta=1$时差一个$\log \lambda$的幂次）。我们进一步证明$\delta$恰好是该相位函数的修正牛顿距离，从而在该情形下验证了Greenleaf07提出的猜想。
英文摘要
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.
