Mathematical Sciences: Geometric Inequalities in Fourier Analysis

这项工作将联合收割机几何问题与傅立叶 分析在协调一致的努力，以利用对称结构 在确定基本的分析不等式， 自然运算符，如卷积，傅立叶变换， 分数阶积分、泊松积分及其生成元。 这些数学工具描述了 流形上函数空间的分析 研究的重点将是了解分析 由经典李群定义的全局流形及其 相关对称空间，特别是秩为1的非紧空间 双曲几何的对称空间，定义在真实的上， 复数或四元数。 使用保角变换，如凯莱变换 和赤平投影，其中一个显示，这些问题 非紧域在紧域上有等价实现 域与旋转和酉群以及 像海森堡群那样的边界流形 主要 这项工作的一个工具是使用基本的对称结构 证明尖锐的分数阶积分不等式。 这样的结果提供作为直接后果尖锐索博列夫 微分算子和其他无穷小的估计 与可微性相关的生成器。 这些不平等 广泛应用于微分几何问题，非 线性偏微分方程、数学物理和 概率 此外，还可以与变化的 弦理论中与共形度规有关的问题。

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.