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

提案编号：0200713和0200584 PI：亚历山大沃尔伯格，Fedor Nazarov和Serguei Treil摘要研究将进行非齐次调和分析和加权范数不等式与矩阵权重。 在以前的工作中，PI发展了一种在非倍测度空间上估计Calderon-Zygmund算子的技术，并为调和分析问题引入了一种新的Bellman函数方法。这些技术将被应用于解决几个开放的问题，谐波分析和算子理论，并探讨新的方向，以前被认为是难以驾驭的，因为缺乏技术工具。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。 其中的主要方向拟议的研究是：-谱理论的扰动正常运营商和相关问题的调和分析：两个重量估计希尔伯特变换和嵌入定理的共同不变的子空间。非齐次T（B）定理及其在解析电容（电强度电容）推广中的应用;曲率在高维中的作用。随机最优控制中的贝尔曼函数方法和调和分析中的贝尔曼函数方法;具有矩阵参数的函数及其在非交换问题中的应用。调和分析通过将复杂过程表示为具有良好行为的基本过程（正弦波，小波）的和来研究复杂过程。 现代调和分析的一个中心部分涉及一种或另一种类型的“奇异算子”。 这种算子在科学领域中是普遍存在的：它们出现在数学物理、概率论、工程、图像处理等领域。这里的主要困难是，在这些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。本文提出了一种基于Bellman函数的谐波分析新方法，该方法起源于随机最优控制理论。一个重要的研究方向是谱理论的扰动正常运营商：结果在这个方向将有重要的后果，在数学物理。 另一个方向涉及非交换谐波分析，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.