Holomorphic Singular Integrals in Several Complex Variables and Applications

这一研究项目寻求开发数学工具，通过收集容易到达且不需要破坏网站的较小数据集，允许从位于难以到达的地方的大数据集中提取信息，而对于这些数据，实际到达将需要扰乱研究地点。例如，我们寻找的信息可能是树的核心(树皮下的木材)的温度。或者，比如说，位于地下深处的土壤温度。在这两个例子中，执行直接测量将需要扰乱研究对象(钻探树木；钻探地面)，这是昂贵和破坏性的。相反，利用这个项目中开发的基于数学的方法，它足以测量树皮上的温度或地球表面的温度(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“大姐”)，该积分的输出将是树中心的温度(或地下深处的土壤温度)。数学中处理这些问题的部分被称为“调和分析”和“奇异积分”；所采用的方法被称为“积分表示公式”。这些方法甚至适用于树皮非常粗糙的树(“类分形”)而不是光滑树皮(“非光滑区域的积分公式”)。这个项目汇集了一般分析领域不同部分的技术和问题，主要侧重于一个和多个复变量的复函数理论，以及欧几里德空间上的实调和分析。其中一个主要目标是建立n维复欧氏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2n维实空间中非光滑区域上的奇异积分的Calderon-Zygmund理论相结合。去掉光滑性假设，算子和它们作用的区域之间的几何相互作用变得突出起来：Lanzani和她的合作者最近的进展为将这一理论推广到C2范畴以下并使其与过去30年来在几何测度论、偏微分方程组和拟共形映射中发展的真实分析技术平起平坐奠定了基础。一个新的组成部分建立在主要研究者和她的合作者M.Pramanik最近的工作基础上，并试图通过探索其Schwartz核的适当对称形式来研究上述全纯奇异积分算子。其他问题包括在复杂欧几里得空间的d-bar复合体的背景下调查div-curl不等式。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.