调和分析在一致L^p预解估计与薛定谔方程定量唯一延拓性中的应用

The classical uniform L^p Sobolev estimate established by Kenig, Ruiz and Sogge has played an important role in many studies (such as unique continuation problem) concerning Schrodinger operators, recently, its generalization on manifolds has received enough attention by many mathematicians. The applicant mainly focus on the sharp uniform L^p resolvent estimates on manifolds as well as quantitative unique continuation problems for Schrodinger equations. Partial results have been achieved towards these problems. In this project, we would like to further study them by combining basic ideas in Harmonic analysis and spectral theory of Schrodinger operators. For the resolvent estimates, we shall transfer the problem to the study of the corresponding "discrete restriction estimates" on manifolds, and then rely on L^p estimates of oscillatory integral operators in Harmonic analysis. The aim is to reveal the inner connections between resolvent estimates and geometric properties (such as the curvature condition) of the manifold; For the quantitative unique continuation problem of the Schrodinger equation, unlike the traditional method by using "Carleman estimates", we are inspired by the Uncertainty Principle (such as the Hardy Uncertainty Principle, Nazarov Uncertainty Principle) in Harmonic analysis. We devote ourself to establishing such kind of Uncertainty Principle for nonnegative selfadjoint operators, then we take advantage of the long time propagation property of the solution. We hope that our approach can provide new insight in the study of this type of problems.

由kenig，Ruiz，Sogge建立的经典一致Sobolev估计已在与薛定谔算子相关的许多问题(如唯一延拓性)中起着重要作用，近年来其在流形上的推广也得到数学家的充分重视。申请人关注流形上的最优一致L^p预解式估计以及薛定谔方程的有关定量唯一延拓性，已在前期取得部分结果。本项目拟在此基础上充分运用调和分析中的基本想法与薛定谔算子的谱理论来研究上述问题。对于预解式估计，拟通过将问题转化到流形上的“离散限制性估计”，并结合调和分析中有关振荡积分算子的L^p估计，由此揭示其与流形自身的几何性态（例如曲率）之间的内在联系；对于薛定谔方程解的定量唯一延拓性问题，不同于传统的Carleman估计方法，我们受调和分析中不确定性原理（如Hardy不确定性原理，Nazarov不确定性原理）的启发，拟对一般的非负自伴算子建立相应的不确定性原理，并利用解的长时间传播估计，希望为这类问题的研究开拓新的思路。

本项目研究了高阶薛定谔算子的一致预解估计与薛定谔方程的定量唯一延拓性问题。我们分别证明：1）带位势薛定谔方程的（时间）两点型定量唯一延拓性质；2）一类抛物方程的能稳区域的刻画问题，我们对于分数阶热方程以及Hermite热方程的能稳区域分别给出了充分必要条件；3） 对一类高阶、满足非退化条件的椭圆算子建立了端点的一致L^p-L^q型的限制-弱型Sobolev估计；4）对于任意正实数s，研究了分数薛定谔算子的散射理论，证明了波算子的存在性和渐近完备性，以及有限衰减性质的特征函数；5）在 H^p(R^n)空间中，得到了在各个方向受限光滑条件的上的 Mikhlin-Hormander 型乘子定理。上述成果较好的帮助人们认识和理解了薛定谔方程的定量唯一延拓性问题，丰富了高阶薛定谔方程的散射理论，加深了人们对薛定谔方程的唯一性与有关控制问题的理解，同时对调和分析自身的问题研究也提供了某些参考。