Geometric Inequalities in Fourier Analysis

摘要本研究计划的目的是发展对流形几何分析的广泛概念理解，以流形傅立叶分析中的尖锐不等式作为其范例，作为编码流形固有不变量、对称性和潜在几何结构的几何信息的作用。我们的直接重点是在等周不等式，临界Sobolev嵌入，对数Sobolev和levy - gromov泛函，以及结合Hardy-Littlewood-Sobolev不等式，stein - weiss积分，Birman-Schwinger核和Riesz势的框架的多线性分数积分，特别是在李群，对称空间，复流形和包括双曲空间在内的射射几何，SL（2，R）和Heisenberg群的设置中获得新的和更清晰的结果。尖锐嵌入估计是建立pde解的存在性和正则性以及控制流形上的振荡行为的关键工具。该程序反映了保形变形、流体力学、几何概率、统计力学、弦理论和湍流等问题的自然相互作用和刺激。计算方法是获得对几何对称复杂性更清晰理解的重要工具。描述物理现象的数学模型和描述动态过程的微分方程的分析需要一个丰富多样的分析框架。科学家们说，“数学法则支撑着我们宇宙的结构。”这项研究计划的目的是发展一个支持性的数学结构，帮助我们理解不确定性、熵和无序、几何对称、重整化和缩放参数、时空模型的非同质性、表面积和体积之间的等周比较以及双曲几何的精细细节之间的关系。我们希望获得广泛和严格的数学工具的理解，支持对自然基本定律的定量描述，并开发和探索数学结构，模拟从流体动力学到弦理论的物理现象，特别是通过使用对称模式和固有常数来编码有关潜在理论框架的几何信息。

AbstractThe objective of this research program is to develop a broad conceptualunderstanding of geometric analysis on manifolds which takes as itsparadigm the role of sharp inequalities in Fourier analysis on themanifold as encoding geometric information about intrinsic invariants,symmetries and the underlying geometric structure of the manifold. Our direct focus is to obtain new and sharper results on isoperimetricinequalities, critical Sobolev embedding, log Sobolev and Levy-Gromovfunctionals, and multilinear fractional integrals that incorporate theframework of the Hardy-Littlewood-Sobolev inequality, Stein-Weissintegrals, Birman-Schwinger kernels and Riesz potentials, especially inthe setting of Lie groups, symmetric spaces, complex manifolds and affinegeometry including hyperbolic space, SL(2,R) and the Heisenberg group.Sharp embedding estimates are a critical tool to establish existence andregularity for solutions to pde's, and to control oscillatory behavior on a manifold. This program reflects a natural interplay and stimulus withproblems in conformal deformation, fluid dynamics, geometric probability,statistical mechanics, string theory and turbulence. Computational methodsare essential tools to obtain a clearer understanding of the complexity ofthe geometric symmetry.Mathematical models that characterize physical phenomena and the analysisof differential equations that describe dynamical processes require a richand diverse analytic framework. Scientists say that "mathematical lawsunderpin the fabric of our universe." The aim of this research program is to develop a supporting mathematical structure that helps us to understand relations that connect uncertainty, entropy and disorder, geometric symmetry, renormalization and scaling arguments, non-homogeneity of space-time models, isoperimetric comparison between surface area and volume, and the fine detail of hyperbolic geometry. We want to obtain broad and rigorous understanding ofmathematical tools that support a quantitative description of the fundamentallaws of nature, and to develop and explore mathematical structures that modelphysical phenomena ranging from fluid dynamics to string theory, especiallyby using patterns of symmetry and intrinsic constants that encode geometricinformation about the underlying theoretical framework.