标题： 带有临界奇性的薛定谔方程的均匀预解式和斯特里哈茨估计 显示英文标题

标题： Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities

摘要： 本文研究了具有实值势的薛定谔方程的全局色散性质，这些势表现出临界奇异性，我们的势类比反平方型势更一般，并包括几种各向异性势。 我们首先证明了加权预解式估计，这些估计在能量上是统一的，并且适用于莫雷-卡姆帕纳托空间中的大量权重函数。 还研究了洛伦兹空间中的统一索博列夫不等式。 证明采用了迭代预解式恒等式和经典的乘子技术。 作为应用，推导出了包括端点情况在内的完整的全局时间斯特里哈茨估计集。 在斯特里哈茨估计的证明中，我们开发了一个关于扰动的一般准则，确保可以从预解式估计中恢复齐次和非齐次端点估计。 最后，我们还通过使用莫尔理论的一个基本版本，研究了具有临界奇异性长程排斥势的统一预解式估计。 显示英文摘要

摘要： This paper deals with global dispersive properties of Schr\"odinger equations with real-valued potentials exhibiting critical singularities, where our class of potentials is more general than inverse-square type potentials and includes several anisotropic potentials. We first prove weighted resolvent estimates, which are uniform with respect to the energy, with a large class of weight functions in Morrey-Campanato spaces. Uniform Sobolev inequalities in Lorentz spaces are also studied. The proof employs the iterated resolvent identity and a classical multiplier technique. As an application, the full set of global-in-time Strichartz estimates including the endpoint case is derived. In the proof of Strichartz estimates, we develop a general criterion on perturbations ensuring that both homogeneous and inhomogeneous endpoint estimates can be recovered form resolvent estimates. Finally, we also investigate uniform resolvent estimates for long range repulsive potentials with critical singularities by using an elementary version of the Mourre theory.