Collaborative Research: Multidimensional and Non-Homogeneous Harmonic Analysis: Bellman Functions, Perturbations of Normal Operators and Two Weight Estimates of Singular Integrals

合作研究：多维非齐次调和分析：贝尔曼函数、正规算子的扰动和奇异积分的两种权重估计

• 批准号: 0200584
• 负责人: Serguei Treil
• 金额: $ 15.35万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200584&HistoricalAwards=false
• 关键词: Collaborative; Research; Multidimensional; Non; Homogeneous

Proposal Numbers: 0200713 and 0200584PIs: Alexander Volberg, Fedor Nazarov, and Serguei TreilABSTRACTResearch will be conducted on non-homogeneous harmonic analysis and on weighted norm inequalities withmatrix weights. In previous work by the PIs a techniquefor estimating Calderon-Zygmund operators on spaces withnon-doubling measures was developed, and a novel method of Bellman functions was introduced for problemsin Harmonic Analysis. These techniques will be applied to solve several open problems in harmonic analysis and operator theory and to investigate new directions,which previously were deemed untractable because of the lack of technical tools. Special attention will be paid to uncovering new relations between Operator Theory, Harmonic Analysis and Stochastic Control. Amongthe main directions of the proposed research are:- Spectral theory for perturbations of normal operators and related problems in Harmonic Analysis: two weight estimates for Hilbert Transform and embedding theorems for the co-invariant subspaces.- Non-homogeneous T(b) theorems and their applications to generalizations of analytic capacity (electric intensity capacity); the role of curvature in higher dimensions.- Bellman function method in stochastic optimal controland in harmonic analysis; functions with matrix argumentsand their applications to non-commutative problems.Harmonic analysis investigates complex processes by representing them as a sum of elementary ones (sinusoidal waves, wavelets) with well understood behavior. A centralpart of modern harmonic analysis deals with "singularoperators" of one type or another. Such operators are pervasive in the scientific landscape: they turn up in mathematical physics, probability, engineering, image processing, etc. A new way to treat multivariate signalswill be discussed. The main difficulty here is that the mathematical objects arising in such problems are non-commutative: the product depends on the order of terms,and that complicates things immensely. A new method basedon Bellman functions, which originating in the stochastic optimal control, will be exploited in harmonic analysis. One important direction of research is the spectral theory for the perturbation of normal operators: results in this direction would have important consequences in mathematical physics. Another direction deals with non-commutative harmonic analysis, i.e. with treating multivariatesignals.

提案编号：0200713和0200584PIS：Alexander Volberg，Fedor Nazarov和Serguei Treilabstractresearch将进行非均匀的谐波分析，并使用amtrix重量进行加权规范不等式。 在PIS的先前工作中，开发了使用NONNON倍增措施在空间上估算Calderon-Zygmund操作员的技术，并引入了一种新颖的Bellman功能方法，以解决谐波分析中的问题。这些技术将用于解决谐波分析和操作者理论中的几个开放问题，并研究新的方向，因为缺乏技术工具，这些方向以前被认为是无法提取的。将特别注意揭示操作员理论，谐波分析和随机控制之间的新关系。 拟议研究的主要方向包括： - 谐波分析中正常操作员扰动以及相关问题的光谱理论：希尔伯特转化的两个权重估计值，并嵌入了共同存在的子空间定理的定理。曲率在较高维度中的作用。-钟形功能方法在随机最佳控制和谐波分析中的作用；具有矩阵参数的功能及其在非共同问题上的应用。谐波分析通过将它们表示为具有广泛理解的行为的基本过程（正弦波，小波）来研究复杂的过程。 现代谐波分析的中心部分涉及一种或另一种类型的“单一动物”。 此类操作员在科学景观中普遍存在：他们在数学物理学，概率，工程，图像处理等方面出现了。讨论一种治疗多元信号的新方法。这里的主要困难是在此类问题中产生的数学对象是非交通的：产品取决于术语的顺序，并且使事情变得非常复杂。源自随机最佳控制的基于方法的贝尔曼功能将在谐波分析中被利用。研究的一个重要方向是正常运算符扰动的光谱理论：朝这个方向朝着数学物理学带来重要的后果。 另一个方向涉及非共同谐波分析，即处理多叶牙的信号。