We apply the Bennett–Carbery–Tao multilinear restriction estimate in order to bound restriction operators and more general oscillatory integral operators. We get improved L p estimates in the Stein restriction problem for dimension at least 5 and a small improvement in dimension 3. We prove similar estimates for Hörmander-type oscillatory integral operators when the quadratic term in the phase function is positive definite, getting improvements in dimension at least 5. We also prove estimates for Hörmander-type oscillatory integral operators in even dimensions. These last oscillatory estimates are related to improved bounds on the dimensions of curved Kakeya sets in even dimensions.
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机译：我们将Bennett-Carbery-Tao多线性约束估计应用于边界约束算子和更一般的振荡积分算子。在Stein约束问题中，对于至少5维，我们得到了改进的L p 估计，对3维进行了小幅改进。当相位函数中的二次项为Hörmander型振荡积分算子时，我们证明了相似的估计是正定的，至少在维数上得到5的改进。我们还证明了偶数维对Hörmander型振荡积分算子的估计。这些最后的振荡估计与弯曲Kakeya集的尺寸在均匀尺寸上的改进范围有关。
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