Collaborative Research: Calderon-Zygmund Operators in Highly Irregular Environments, and Applications
合作研究:高度不规则环境中的 Calderon-Zygmund 算子及其应用
基本信息
- 批准号:1600065
- 负责人:
- 金额:$ 39万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2016
- 资助国家:美国
- 起止时间:2016-06-01 至 2020-05-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Calderon-Zygmund operators are mathematical objects that play an important role in the understanding of many physical phenomena, ranging from heat transfer to turbulence in dynamical systems. The classical theory of these operators was designed to work on smooth functions. However, nature often provides us with very irregular media with which to engage. This creates the need for a very low-regularity form of the theory of singular integrals, which the principal investigators on this project have constructed. A consequence of the low-regularity theory is that through the action of Calderon-Zygmund operators on a set in a Euclidean space of a very high dimension, one can sometimes conclude that the set itself is of a much lower dimension than the ambient space, an important piece of information from the perspective of data science. To refine this approach to data analysis is one of the main goals of this project. This project considers several problems in nonhomogeneous harmonic analysis, geometric measure theory, and spectral theory. The common theme uniting the problems is the behavior of singular operators with very good (Calderon-Zygmund) kernels in very bad environments (e.g., on sets with no a priori structure, in spaces with matrix weights). Specifically, the project will pursue the following avenues of research: (1) the David-Semmes problem to characterize the rectifiability of sets and measures in high-dimensional Euclidean space in terms of the boundedness of the corresponding Riesz transforms; (2) the geometry of reflection-less measures; (3) the geometric characterization of higher-dimensional analogues of positive analytic capacity; (4) two-weight estimates for very simple singular operators in the non-Hilbert setting; and (5) sharp estimates for classical operators with matrix weights. Singular integral operators with respect to bad measures and very irregular sets appear naturally in many problems of analysis. One of the reasons for their increasing interest in recent years has been the study of analytic capacity. While the theory for the two-dimensional case (i.e., the Cauchy transform on the complex plane) and the theory of analytic capacity that emerged as its by-product are now very well understood, the analogous theory in higher dimensions has not been fully developed. The main roadblock here is the lack of geometric tools in higher dimensions. Additionally, in higher dimensions, nonhomogeneous situations arise more often than in the plane and more often one might expect. For example, boundary value problems in (otherwise smooth) domains with cusps lead to nonhomogeneous problems, because, unlike what happens in the two-dimensional setting, surface measure on the boundary of such a domain is non-doubling. This becomes an even more vexing problem if one wants to consider harmonic measure estimates for domains on whose boundaries "surface measure" is practically arbitrary. This is an important issue that the project seeks to confront.
卡尔德隆-齐格蒙德算子是数学对象,在理解许多物理现象(从传热到动力系统中的湍流)方面发挥着重要作用。 这些算子的经典理论被设计用于处理平滑函数。然而,大自然经常为我们提供非常不规则的媒介来参与。这就需要一种非常低正则性形式的奇异积分理论,该项目的主要研究人员已经构建了这种形式。低正则性理论的一个结果是,通过卡尔德隆-齐格蒙德算子对高维欧几里得空间中的集合的作用,有时可以得出结论,该集合本身的维数比周围空间低得多,从数据科学的角度来看,这是一条重要的信息。完善这种数据分析方法是该项目的主要目标之一。该项目考虑了非齐次谐波分析、几何测度理论和谱理论中的几个问题。统一这些问题的共同主题是在非常恶劣的环境中(例如,在没有先验结构的集合上、在具有矩阵权重的空间中)具有非常好的(Calderon-Zygmund)核的奇异算子的行为。具体来说,该项目将寻求以下研究途径:(1)David-Semmes问题,根据相应的Riesz变换的有界性来表征高维欧几里德空间中集合和测度的可修正性; (2)无反射措施的几何形状; (3) 正分析能力的高维类似物的几何表征; (4) 非希尔伯特设置中非常简单的奇异算子的二重估计; (5) 对具有矩阵权重的经典算子的敏锐估计。与不良测度和非常不规则集有关的奇异积分算子自然地出现在许多分析问题中。近年来他们越来越感兴趣的原因之一是对分析能力的研究。虽然二维情况的理论(即复平面上的柯西变换)和作为其副产品出现的分析能力理论现在已经得到很好的理解,但更高维度的类似理论尚未得到充分发展。这里的主要障碍是缺乏更高维度的几何工具。此外,在更高的维度中,非均匀情况比在平面中出现得更频繁,而且比人们预期的更频繁。例如,具有尖点的(否则平滑的)域中的边界值问题会导致非齐次问题,因为与二维设置中发生的情况不同,此类域边界上的表面测量是非重重的。如果想要考虑其边界“表面测量”实际上是任意的域的调和测量估计,这将成为一个更加令人烦恼的问题。这是该项目寻求面对的一个重要问题。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Alexander Volberg其他文献
NONCOMMUTATIVE BOHNENBLUST–HILLE INEQUALITY IN THE HEISENBERG–WEYL AND GELL-MANN BASES WITH APPLICATIONS TO FAST LEARNING
海森堡-韦尔和盖尔曼基中的非交换 Bohnenblust-Hille 不等式及其在快速学习中的应用
- DOI:
- 发表时间:
2023 - 期刊:
- 影响因子:0
- 作者:
Joseph Slote;Alexander Volberg;Haonan Zhang - 通讯作者:
Haonan Zhang
Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension-one Hausdorff measure ✩
如果谐波测度相对于同维一豪斯多夫测度绝对连续,则它是可校正的 ✩
- DOI:
- 发表时间:
- 期刊:
- 影响因子:0
- 作者:
C. Acad;Sci;Ser. I Paris;Jonas Azzam;Steve Hofmann;J. M. Martell;S. Mayboroda;Mihalis Mourgoglou;X. Tolsa;Alexander Volberg - 通讯作者:
Alexander Volberg
Alexander Volberg的其他文献
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{{ truncateString('Alexander Volberg', 18)}}的其他基金
Collaborative Research: Non-homogeneous Harmonic Analysis, Spectral Theory, and Weighted Norm Estimates
合作研究:非齐次谐波分析、谱理论和加权范数估计
- 批准号:
2154402 - 财政年份:2022
- 资助金额:
$ 39万 - 项目类别:
Standard Grant
Collaborative research: Weighted Estimates with Matrix Weights and Non-Homogeneous Harmonic Analysis
合作研究:矩阵权重加权估计和非齐次谐波分析
- 批准号:
1900268 - 财政年份:2019
- 资助金额:
$ 39万 - 项目类别:
Continuing Grant
Collaborative Research: Universality Phenomena and Some Hard Problems of Non-homogeneous Harmonic Analysis
合作研究:非齐次谐波分析的普遍性现象和一些难题
- 批准号:
1265549 - 财政年份:2013
- 资助金额:
$ 39万 - 项目类别:
Continuing Grant
Collaborative Research: Bellman function, Harmonic Analysis and Operator Theory
合作研究:贝尔曼函数、调和分析和算子理论
- 批准号:
0758552 - 财政年份:2008
- 资助金额:
$ 39万 - 项目类别:
Continuing Grant
Non-Homogeneous Harmonic Analysis, two weight estimates, and spectral problems
非齐次谐波分析、两次权重估计和谱问题
- 批准号:
0501067 - 财政年份:2005
- 资助金额:
$ 39万 - 项目类别:
Continuing Grant
Multidimensional and Non-Homogeneous Harmonic Analysis: Bellman Functions, Pertubations of Normal Operators and Two Weight Estimates of Singular Integrals
多维非齐次调和分析:贝尔曼函数、正规算子的摄动和奇异积分的两个权重估计
- 批准号:
0200713 - 财政年份:2002
- 资助金额:
$ 39万 - 项目类别:
Continuing Grant
Mathematical Sciences: Three Measures on Fractals
数学科学:分形的三种测度
- 批准号:
9302728 - 财政年份:1993
- 资助金额:
$ 39万 - 项目类别:
Standard Grant
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