Holomorphic Singular Integrals in Several Complex Variables and Applications

多复变量中的全纯奇异积分及其应用

• 批准号: 1901978
• 负责人: Loredana Lanzani
• 金额: $ 21.44万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901978&HistoricalAwards=false
• 关键词: Holomorphic; Singular; Integrals; Several; Complex

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.

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较小数据集，来获取位于难以到达的地方的大型数据集的信息，并且实际到达需要破坏研究地点。例如，我们寻求的信息可能是树的核心（树皮下的木材）的温度。或者说，地下深处土壤的温度。在这两个例子中，执行直接测量将需要破坏研究对象（钻树;钻地），这是昂贵的和破坏性的。 相反，在这个项目中开发的基于南极学的方法足以测量树皮或地球表面的温度（两者都不需要钻探）。然后将容易收集的数据绘制成积分（微积分中研究的积分的“大姐”），该积分的输出将是树的核心温度（或地下深处土壤的温度）。数学中处理这些问题的部分称为“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例如，该项目汇集了来自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真实的调和分析。其中一个主要目标是发展一个理论的柯西类奇异积分与全纯核和非光滑域在n维复欧几里德空间，成功地融合了复杂的结构的环境域与Calderon-Zygmund理论的奇异积分的非光滑域在2n维真实的空间。去掉光滑性假设，算子和它们作用的域之间的几何相互作用就凸显出来了：Lanzani和她的合作者最近的进展为将这一理论推到C2范畴之下奠定了基础，并使其与过去30年在几何测度论、偏微分方程和拟共形映射中发展起来的真实的分析技术相提并论。一个新的组成部分建立在最近的工作，由主要研究者和她的合作者M。Pramanik和旨在调查上述全纯奇异积分算子探索合适的对称化形式的施瓦茨内核。进一步的问题包括调查的div-curl不平等的背景下，d-酒吧复杂的复杂的欧几里德space.This奖项反映了NSF的法定使命，并已被认为是值得支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。