Holomorphic Singular Integral techniques for Non-Smooth domains and applications

这个数学研究项目涉及复数和调和分析中的所谓积分公式的研究。积分公式是通过收集容易到达的非常小的样本来恢复位于难以到达的地方的大型数据集的信息的重要工具。例如，积分公式可以用来恢复固体(比方说一棵树，甚至一颗行星)内部的温度，而不必探测身体上的洞(也就是在树上的例子中，不必在树干上钻孔)。取而代之的是，人们测量表面水平的温度(比如树皮上的温度)，并将这些值绘制在积分公式中：输出将是内部温度的值。这个项目的一个新奇之处在于，它允许处理外表面非常粗糙(而不是非常光滑)的物体。PI和她的合作者将把一般分析领域不同部分的技术和问题结合在一起，特别是一个和几个复变量的实调和复函数理论。其中一个主要目标是建立欧几里德复空间中非光滑区域的具有全纯核的Cauchy-like奇异积分理论，它成功地将环境区域的复结构与实空间中非光滑区域上的奇异积分的Calderon-Zygmund理论相结合。要做到这一点，需要接受底层复杂结构施加的额外刚性，特别是统称为(或与之关联)伪凸性的几何特性。一些复变量的应用包括某些正交投影算子的正则性，例如在非光滑区域的新背景下的Bergman和Szego投影。

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.