限制在一致嵌入子流形上的谱投影估计
Spectral and Schrödinger estimates restricted to submanifolds
AI总结：
本文研究了具有非正截面曲率和有界几何流形上的一致嵌入子流形，估计了$\log$尺度谱投影算子的$L^2(M)\to L^q(Σ)$范数，推广了X. Chen的结果至非紧情形，并在偶渐近双曲曲面上证明了非困陷测地线上的尖锐谱投影估计。
摘要：
设$M$为具有非正截面曲率和有界几何的流形，$Σ$为$M$的一致嵌入子流形。我们估计$\log$尺度谱投影算子从$L^2(M)$到$L^q(Σ)$的范数。这是X. Chen的结果向非紧情形的推广。我们还在具有有界几何且曲率向下夹紧于0的偶渐近双曲曲面上，证明了限制在非困陷测地线上的任意小尺度谱窗的尖锐谱投影估计。
英文摘要：
Let $M$ be a complete Riemannian manifold with nonpositive sectional curvature and smooth bounded geometry. We prove logarithmic spectral projection estimates restricted to uniformly embedded submanifolds. On surfaces with pinched negative curvature, we also prove frequency-independent $L^2\to L^3$ Schrödinger restriction estimates on compact geodesic segments for times $cλ^{-1}\logλ$. If such a surface is even asymptotically hyperbolic, the $L^2\to L^q$ spectral estimates on complete nontrapped geodesics, for $q>2$, gain the square root of the window width. The high-frequency Schrödinger estimate holds for all times on these surfaces, both on compact segments and on complete nontrapped geodesics. Under a regularity assumption at the spectral threshold, the all-time estimate holds on the absolutely continuous spectral subspace.
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