Holomorphic Singular Integrals in Several Complex Variables and Applications

该研究项目旨在开发数学工具，通过收集容易到达且不需要破坏研究地点的小型数据集，从而可以从位于难以到达的地方的大型数据集中提取信息，而实际到达这些数据集需要破坏研究地点。例如，我们寻找的信息可能是树核（树皮下的木头）的温度。或者说，地下深处土壤的温度。在这两个例子中，进行直接测量都需要破坏研究对象（钻采油树和钻地面），这既昂贵又具有破坏性。相反，利用这个项目开发的基于数学的方法，测量树皮或地球表面的温度就足够了（两者都不需要钻探）。然后将容易收集的数据绘制成一个积分（微积分中研究的积分的“姐姐”），这个积分的输出将是树的核心温度（或地下深处土壤的温度）。处理这些问题的数学部分被称为“调和分析”和“奇异积分”；所采用的方法称为“积分表示公式”。这些方法甚至适用于树皮非常粗糙（“分形”）而不是光滑树皮（“非光滑域的积分公式”）的树木。该项目汇集了一般分析领域不同部分的技术和问题，主要侧重于一个和几个复变量的复函数理论，以及欧几里德空间的实调和分析。主要目标之一是发展n维复欧氏空间非光滑域的全纯核类柯西奇异积分理论，成功地将环境域的复杂结构与二维实空间非光滑域上奇异积分的Calderon-Zygmund理论相融合。除去平滑假设，使算子和它们所作用的域之间的几何相互作用凸显出来：Lanzani和她的合作者最近的进展为推动这一理论远远低于c2类别奠定了基础，并使其与过去三十年来在几何测量理论、偏微分方程和拟共形映射中发展起来的真正的分析技术相一致。在首席研究员和她的合作者M. Pramanik最近的工作基础上，一个新的组成部分试图通过探索Schwartz核的合适对称形式来研究上述全纯奇异积分算子。进一步的问题包括复欧几里德空间中d-棒复形的div旋度不等式的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project seeks to develop mathematical tools that allow to draw information on large sets of data located in places that are hard to reach, and for which actual reach would require disrupting the site of study, by collecting smaller data sets that are within easy reach and do not require disrupting the site. For instance, the information we seek may be the temperature at the core of a tree (the wood underneath the bark). Or, say, the temperature of soil located deep down underground. In both examples, performing direct measurements would require disrupting the object of study (drilling the tree; drilling the ground) which is expensive and disruptive. Instead, with the mathematics-based methods developed in this project it is enough to measure temperature on the tree's bark, or on the earth's surface (and neither requires drilling). Then one plots the easily-collected data into an integral (the "big sister" of the integrals studied in calculus) and the output of this integral will be the temperature at the core of the tree (or temperature of the soil deep underground). The part of mathematics that deals with these problems is called "harmonic analysis" and "singular integrals"; the methods employed are called "integral representation formulas." These methods work even for e.g., trees that have very rough bark ("fractal-like") as opposed to smooth bark ("integral formulas for non-smooth domains").This project brings together techniques and problems from different parts of the general field of analysis, with primary emphasis on complex function theory in one and several complex variables, and on real harmonic analysis on Euclidean space. One of the main goals is to develop a theory of Cauchy-like singular integrals with holomorphic kernel and for non-smooth domains in n-dimensional complex Euclidean space that successfully blends the complex structure of the ambient domain with the Calderon-Zygmund theory for singular integrals on non-smooth domains in 2n-dimensional real space. Stripping away the smoothness assumptions brings to the fore the geometric interplay between the operators and the domains on which they act: recent advances by Lanzani and her collaborators have set the ground for pushing this theory well below the C2-category and for bringing it on par with the real analysis techniques that were developed over the past thirty years in Geometric Measure Theory, Partial Differential Equations and Quasiconformal mapping. A novel component builds upon recent work by the principal investigator and her collaborator M. Pramanik and seeks to investigate the aforementioned holomorphic singular integral operators by exploring suitable symmetrized forms of their Schwartz kernels. Further problems include the investigation of div-curl inequalities in the context of the d-bar complex in complex Euclidean space.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.