流形上的振荡积分算子及相关的Kakeya和Nikodym问题
Oscillatory integral operators on manifolds and related Kakeya and Nikodym problems
AI总结：
本文研究了黎曼流形上的Carleson-Sjölin算子，在常截面曲率和Sogge混沌曲率条件下通过不同方法获得了改进的$L^p$界，并作为副产品得到了Nikodym极大函数的新界。
摘要：
我们考虑黎曼流形上的Carleson-Sjölin算子，这些算子自然产生于流形上Bochner-Riesz问题的研究。它们是Hörmander型振荡积分算子的特例。我们在两种情况下获得了Carleson-Sjölin算子改进的$L^p$界：底层流形具有常截面曲率的情况，以及流形满足Sogge混沌曲率条件的情况。\n这两个结果依赖于截然不同的方法：为了证明前一个结果，我们证明了在黎曼流形上，距离函数满足Bourgain条件当且仅当该流形具有常截面曲率。为了获得第二个结果，我们向Hörmander型振荡积分算子引入了“接触阶”的概念，证明了如果一个Hörmander型振荡积分算子具有有限接触阶，那么它总是具有比“最坏情况”（遵循Bourgain和Guth，以及Guth、Hickman和Iliopoulou的精神）更好的$L^p$界，并最终验证了对于满足Sogge混沌曲率条件的黎曼流形，其距离函数总是具有有限接触阶。\n作为副产品，我们获得了常截面曲率流形上Nikodym极大函数的新界。
英文摘要：
We consider Carleson-Sjölin operators on Riemannian manifolds that arise naturally from the study of Bochner-Riesz problems on manifolds. They are special cases of Hörmander-type oscillatory integral operators. We obtain improved $L^p$ bounds of Carleson-Sjölin operators in two cases: The case where the underlying manifold has constant sectional curvature and the case where the manifold satisfies Sogge's chaotic curvature condition. The two results rely on very different methods: To prove the former result, we show that on a Riemannian manifold, the distance function satisfies Bourgain's condition if and only if the manifold has constant sectional curvature. To obtain the second result, we introduce the notion of "contact orders" to Hörmander-type oscillatory integral operators, prove that if a Hörmander-type oscillatory integral operator is of a finite contact order, then it always has better $L^p$ bounds than "worst cases" (in spirit of Bourgain and Guth, and Guth, Hickman and Iliopoulou), and eventually verify that for Riemannian manifolds that satisfy Sogge's chaotic curvature condition, their distance functions alway have finite contact orders. As byproducts, we obtain new bounds for Nikodym maximal functions on manifolds of constant sectional curvatures.
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