Mathematical Sciences: Geometric Inequalities in Fourier Analysis

这项工作将几何问题与傅里叶分析结合起来，共同努力利用流形的对称结构来确定卷积、傅里叶变换、分数积分、泊松积分及其产生器等自然算子的基本解析不等式。这些是描述流形上的函数空间分析的数学工具。研究的重点将是理解由经典李群及其相关对称空间定义的全局流形的分析，特别是在实数，复数或四元数上定义的双曲几何的1阶非紧对称空间。利用Cayley变换和立体投影等共形变换，证明了这些非紧致域上的问题在与旋转和酉群以及边界流形（如Heisenberg群）相关的紧致域上具有等价的实现。这项工作的主要工具是利用流形的底层对称结构来证明尖锐的分数阶积分不等式。这些结果为微分算子和其他与可微性相关的无穷小产生器提供了尖锐的Sobolev估计。这些不等式在微分几何、非线性偏微分方程、数学物理和概率论等问题中有广泛的应用。此外，还可以与弦理论中与共形度量相关的变分问题建立联系。

This work will combine geometric problems with Fourier analysis in a concerted effort to exploit the symmetry structure of manifolds in determining that basic analytic inequalities for natural operators such as convolution, Fourier transforms, fractional integration, Poisson integrals and their generators. These are the mathematical instruments which characterize analysis on function spaces over a manifold. The thrust of the research will be to understand analysis on global manifolds defined by the classical Lie groups and their associated symmetric spaces, especially the rank one non-compact symmetric spaces with hyperbolic geometry, defined over real, complex or quaterionic numbers. Using conformal transformations such as the Cayley transform and stereographic projection, one shows that problems on these non-compact domains have equivalent realizations on compact domains associated with the rotation and unitary groups as well as boundary manifolds such as the Heisenberg group. The main tool in this work is the use of the underlying symmetry structure of the manifolds to prove sharp fractional integral inequalities. Such results provide as immediate consequences sharp Sobolev estimates for differential operators and other infinitesimal generators associated with differentiability. These inequalities have wide application to problems in differential geometry, non- linear partial differential equations, mathematical physics and probability. In addition, connections may be made to variational problems in string theory associated with conformal metrics.