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

建议编号：0200713和0200584 PI：Alexander Volberg、Fedor Nazarov和Serguei Treil将进行关于非齐次调和分析和带矩阵权的加权范数不等式的研究。在PI以前的工作中，发展了一种估计具有非倍增测度的空间上的Calderon-Zygmund算子的方法，并引入了一种新的Bellman函数方法来解决调和分析中的问题。这些技术将被应用于解决调和分析和算子理论中的几个公开问题，并探索新的方向，这些新方向以前被认为是由于缺乏技术工具而难以处理的。将特别注意揭示算子理论、调和分析和随机控制之间的新关系。调和分析的主要研究方向是：-正规算子扰动的谱理论和调和分析中的相关问题：希尔伯特变换的两个权重估计和共不变子空间的嵌入定理。-非齐次T(B)定理及其在分析能力(电强度容量)的推广中的应用；高维中曲率的作用。-调和分析中随机最优控制中的贝尔曼函数方法；带有矩阵变元的函数及其在非对易问题中的应用。调和分析通过将复杂过程表示为具有众所周知的行为的初等(正弦波、小波)的和来研究复杂过程。现代调和分析的核心部分是处理一种或另一种类型的“奇异算符”。这类运算符在科学领域中无处不在：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变量信号的新方法。这里的主要困难是，在这类问题中出现的数学对象是不可交换的：乘积取决于项的顺序，这使事情变得非常复杂。将一种源于随机最优控制的基于Bellman函数的新方法应用于调和分析。一个重要的研究方向是正规算子扰动的谱理论：这个方向的结果将在数学物理中产生重要的结果。另一个方向涉及非对易调和分析，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.