Singular Integrals in Geometric Analysis

几何分析(GA)是研究微分几何(DG)和偏微分方程组(PDE)界面的数学分支。这一领域既包括PDE方法在DG中的使用，也包括DG结果在PDE中的动机。在遗传算法的研究中，通常需要找到满足一定曲率条件的度量存在的完全可积性条件。这样的发现往往涉及到在调和分析中或多或少存在的各种奇异积分。拟议的研究方案将通过以下三个面向p-Laplace的目标阐明这一问题： 1.p-格林函数及其几何性质； 2.广义扭转刚度的Minkowski问题； 3.Morrey-Riesz积分及其在某些几何偏微分方程组中的应用。 这一研究项目的创新之处在于利用新的HA工具和技术(特别是Riesz势和Morrey能力加上Choquite积分的有效结合)来解决遗传算法所产生的一些关于奇异积分的基本问题。这一研究计划在一系列不同的主题中创建了一个综合体，不仅对基于容量或曲率的分析和几何方面的研究人员有用，而且对某些领域的物理学家和工程师也很感兴趣。

Geometric Analysis (GA) is a branch of mathematics treating the interface of Differential Geometry (DG) and Partial Differential Equations (PDE). This field includes both the use of PDE-methods in DG and the motivation of DG-results in PDE. In the study of GA, it is naturally and frequently needed to find complete integrability conditions for the existence of metrics with certain curvature conditions. Such a finding often involves various kinds of singular integrals that exist more or less in Harmonic Analysis (HA). The proposed research program will shed light upon this issue through the following three p-Laplace oriented objectives: 1. p-Green's functions and their geometric properties; 2. Minkowski's problem for generalized torsional rigidity; 3. Morrey-Riesz's integrals with applications to some geometric PDE. The novelty of this research program is to utilize new HA tools and techniques (especially, an effective incorporation of Riesz potentials and Morrey capacities plus Choquet integrals) to settle some fundamental problems on singular integrals arising from GA. This research program, creating a synthesis among a host of distinct topics, will not only be useful to researchers in both analysis and geometry based on either capacity or curvature, but also be of interest to both physicists and engineers in certain areas.