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
• 负责人: Alexander Volberg
• 金额: --
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0501067&HistoricalAwards=false
• 关键词: Non; Homogeneous; Harmonic; Analysis; two

ABSTRACT.PI's propose to concentrate their efforts on several classical problems inAnalysis and Spectral Theory that remained unsolved for the last 20--50years, due to the lack of appropriate technical tools. Among the problemsare:* bilipschitz equivalence for higher dimensional analogues of analyticcapacity; * two weight estimates for the Hilbert Transform;* well-posedness of the inverse scattering problem for the discreteSchrodinger operator,i.e., uniqueness of the inverse nonlinear Fourier transform;* selected problems of noncommutative harmonic analysisAlthough the problems span several different areas of analysis andmathematical physics, our recent research revealed striking connectionsbetween the proposed problems. To put it briefly, they all are unified bythe fact that in all of them the same type of singular kernels (usually theCauchy kernel) appears. Also, the problems share the same difficulty, thekernel got "spoiled'' by multiplication by virtually arbitrary functions(weights). Recent developments in the non-homogeneous harmonic analysis,which treats exactly this type of situations, made successful solution ofthe proposed problems plausible.Harmonic analysis investigates complex processes by representing them as asum of elementary ones (sinusoidal waves, wavelets) with well understoodbehavior. A central part of modern harmonic analysis deals with "singularintegral operators" of one type or another. Such operators are pervasive inthe scientific landscape: they turn up in mathematical physics, probability,engineering, image processing, etc. While the theory of singular integraloperators is now well developed (starting with works of Calderon and Zygmundand continued by numerous researchers after them), it deals with theoperators defined on a nice "smooth" set, like the usual Euclidean space.However, in many problems one needs to investigate such operators on a "bad"set, like surfaces with singularities and even on more pathological sets.The non-homogeneous harmonic analysis was introduced by the PI's to dealexactly with such situations: recent solution by X. Tolsa of the famoussubbaditivity problem for the analytic capacity is one of the mostimpressive applications of this PI's theory of nonhomogeneous analysis. PI'spropose to attack several classical problems, where the framework of thenon-homogeneous harmonic analysis appear naturally.

摘要。PI建议将他们的精力集中在分析和光谱理论中的几个经典问题上，这些问题由于缺乏适当的技术工具，在过去的20- 50年里一直没有得到解决。这些问题包括：*解析能力的高维类似物的bilipschitz等价；* Hilbert变换的两个权值估计；*离散chrodinger算子逆散射问题的适定性，即。，非线性傅里叶逆变换的唯一性；虽然这些问题跨越了分析和数学物理的几个不同领域，但我们最近的研究揭示了所提出问题之间惊人的联系。简而言之，它们都是统一的，因为它们都有相同类型的奇异核（通常是柯希核）出现。而且，这些问题有相同的困难，内核被几乎任意函数（权重）的乘法“破坏”了。非齐次谐波分析的最新发展，正是处理这类情况，使所提出的问题的成功解决成为可能。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。现代谐波分析的一个核心部分是处理一类或另一类的“奇异积分算子”。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。虽然奇异积分算子的理论现在发展得很好（从Calderon和zygmund的工作开始，并由许多研究者在他们之后继续），但它处理的是在一个漂亮的“光滑”集合上定义的算子，就像通常的欧几里得空间一样。然而，在许多问题中，人们需要在“坏”集合上研究这样的算子，比如有奇点的曲面，甚至在更多的病态集合上。PI引入了非齐次谐波分析来精确地处理这种情况：X. Tolsa最近解决了著名的解析能力次性问题，这是PI的非齐次分析理论最令人印象深刻的应用之一。PI提出了几个经典问题，其中非齐次谐波分析的框架自然出现。