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

论文编号：Alexander Volberg， Fedor Nazarov， and Serguei treilabstract将研究非齐次谐波分析和矩阵权重加权范数不等式。在前人的工作中，提出了一种在非加倍测度空间上估计Calderon-Zygmund算子的方法，并引入了一种新的Bellman函数方法用于调和分析问题。这些技术将用于解决谐波分析和算符理论中的几个开放问题，并探索新的方向，这些问题以前由于缺乏技术工具而被认为是难以处理的。将特别注意揭示算子理论、谐波分析和随机控制之间的新关系。主要研究方向有：正规算子摄动的谱理论及谐波分析中的相关问题；希尔伯特变换的两个权估计；协不变子空间的嵌入定理。-非齐次T(b)定理及其在解析能力（电强度容量）推广中的应用；曲率在高维中的作用。-谐波分析中随机最优控制中的Bellman函数法；具有矩阵参数的函数及其在非交换问题中的应用。谐波分析通过将复杂过程表示为具有良好理解行为的基本过程（正弦波，小波）的总和来研究复杂过程。现代谐波分析的核心部分是处理一种或另一种类型的“奇异算子”。这样的运算符在科学领域无处不在：它们出现在数学物理、概率、工程、图像处理等领域。本文将讨论一种处理多元信号的新方法。这里的主要困难在于，这些问题中出现的数学对象是非交换的：乘积取决于项的顺序，这使事情变得非常复杂。本文提出了一种基于Bellman函数的谐波分析方法，该方法起源于随机最优控制。一个重要的研究方向是正则算子摄动的谱理论：这个方向的结果将在数学物理中产生重要的影响。另一个方向处理非交换谐波分析，即处理多变量信号。

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.