Holomorphic Singular Integral techniques for Non-Smooth domains and applications

非光滑域和应用的全纯奇异积分技术

• 批准号: 1503612
• 负责人: Loredana Lanzani
• 金额: $ 18万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1503612&HistoricalAwards=false
• 关键词: Holomorphic; Singular; Integral; techniques; Non

This mathematics research project deals with the study of so-called integral formulas in complex and harmonic analysis. Integral formulas are important tools for recovering information on large data sets that are located in hard-to-reach places by collecting very small samples that are within easy reach. For instance, integral formulas can be used to recover the temperature in the interior of a solid body (say a tree, or even a planet) without having to probe holes in the body (that is, in the tree example, without having to drill holes in the trunk). Instead, one measures the temperature at surface level (say on the tree's bark) and plots these values in the integral formula: the output will be the value of the temperature inside. One of the novelties of this project is that it allows to deal with objects whose outer surface is very rough (as opposed to very smooth).The PI and her collaborators will bring together techniques and problems from different parts of the general field of analysis, specifically real harmonic analysis and complex function theory in one and several complex variables. One of the main goals is to develop a theory of Cauchy-like singular integrals with holomorphic kernel and for non-smooth domains in Euclidean complex space that successfully blends the complex structure of the ambient domain with the Calderon-Zygmund theory for singular integrals on non-smooth domains in real space. Doing so requires coming to terms with the additional rigidity imposed by the underlying complex structure, in particular the geometric properties that are collectively known as (or linked to) pseudo-convexity. Applications to several complex variables include the regularity of certain orthogonal projection operators, such as the Bergman and Szego projections, in the novel context of non-smooth domains.

这个数学研究项目涉及复谐分析中所谓的积分公式的研究。积分公式是一种重要的工具，它通过收集非常小的样本来恢复位于难以到达的地方的大数据集的信息。例如，积分公式可以用来恢复固体（比如一棵树，甚至一颗行星）内部的温度，而不需要探测物体上的洞（也就是说，在树的例子中，不需要在树干上钻洞）。取而代之的是，测量表面的温度（比如树皮上的温度），并在积分公式中绘制这些值：输出将是内部温度的值。这个项目的一个新颖之处在于，它允许处理外表面非常粗糙（而不是非常光滑）的物体。PI和她的合作者将汇集来自一般分析领域不同部分的技术和问题，特别是在一个和几个复杂变量中的实谐波分析和复函数理论。本文的主要目标之一是建立欧氏复空间中具有全纯核和非光滑域的类柯西奇异积分理论，成功地将环境域的复杂结构与实际空间中非光滑域上奇异积分的Calderon-Zygmund理论相融合。这样做需要接受潜在复杂结构所施加的额外刚性，特别是统称为（或与之相关的）伪凸性的几何属性。一些复杂变量的应用包括在非光滑域的新背景下某些正交投影算子的正则性，如Bergman和Szego投影。