We obtain sharp $L^p$ bounds for oscillatory integral operators with generic homogeneous polynomial phases in several variables. The phases considered in this paper satisfy the rank one condition which is an important notion introduced by Greenleaf, Pramanik and Tang. Under certain additional assumptions, we can establish sharp damping estimates with critical exponents to prove endpoint $L^p$ estimates.

中文翻译：

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具有齐次多项式相位的振荡积分算子的锐界

我们为在几个变量中具有通用齐次多项式相位的振荡积分算子获得了尖锐的 $L^p$ 边界。本文考虑的相满足一级条件，这是 Greenleaf、Pramanik 和 Tang 引入的一个重要概念。在某些额外的假设下，我们可以使用临界指数建立尖锐的阻尼估计，以证明端点 $L^p$ 估计。