Topics in Harmonic Analysis

谐波分析主题

• 批准号: 1500162
• 负责人: Andreas Seeger
• 金额: $ 42万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500162&HistoricalAwards=false
• 关键词: Topics; Harmonic; Analysis

Methods from mathematical analysis have found wide applications in understanding physical phenomena in the natural sciences and engineering. This project is concerned with topics in harmonic analysis that are designed to provide effective mathematical tools for these disciplines and that could well contribute to the unification of seemingly unrelated areas. The main goal of the project is to expand the mathematical toolbox in harmonic analysis along these lines. The project involves mentoring of graduate students and postdoctoral researchers.The principal investigator will to work on several projects in harmonic analysis. The first project is concerned with the functional calculus for selfadjoint elliptic and subelliptic partial differential operators. The research will focus on a model example in the subelliptic case, the Laplacian on the Heisenberg group. The goal is to derive new multiplier results on Lebesgue spaces, in particular for the model case of Bochner-Riesz means. In the Euclidean case such multiplier results can be based on Fourier restriction theory, which is not available on the Heisenberg group. A new approach is proposed based on the fine structure of the wave kernel on the Heisenberg group. A second project is motivated by a problem on the cost of mixing for incompressible flows that are generated by time-dependent vector fields. It turns out that this problem is related to boundedness questions on certain bilinear singular integral operators. A general theory will be developed that incorporates multilinear generalizations and is of independent interest. A third project is concerned with problems in approximation theory that involve function spaces of low order of smoothness. In many cases there are gaps between the known results for the categories of Besov and Sobolev spaces. As an example, the range for unconditional convergence of Haar wavelet expansions is well understood for Besov spaces but remains open for Sobolev spaces. One goal for this project is to determine the correct range for Sobolev spaces.

来自数学分析的方法在理解自然科学和工程中的物理现象中发现了广泛的应用。该项目涉及谐波分析中的主题，该主题旨在为这些学科提供有效的数学工具，这很可能有助于统一看似无关的领域。该项目的主要目标是在这些线路上扩展数学工具箱。该项目涉及指导研究生和博士后研究人员。第一个项目涉及自助椭圆形和亚小节部分差分算子的功能性演算。这项研究将重点介绍亚森伯格小组的Laplacian subelliptic案例中的模型示例。目的是在Lebesgue空间上得出新的乘数结果，尤其是对于Bochner-Riesz Mene的模型案例。在欧几里得的情况下，这种乘数结果可以基于傅立叶限制理论，而傅立叶限制理论在海森伯格组上不可用。根据海森伯格组的波核的精细结构提出了一种新方法。第二个项目是由一个问题的问题引起的，该问题是由时间依赖的矢量场产生的不可压缩流的混合成本。事实证明，这个问题与某些双线性单数积分运算符上的有限性问题有关。将开发一种一般理论，结合了多线性概括并且具有独立的利益。第三个项目涉及近似理论中的问题，该问题涉及较低平稳性的功能空间。在许多情况下，BESOV和Sobolev空间的类别的已知结果之间存在差距。例如，HAAR小波膨胀无条件收敛的范围在BESOV空间中得到充分了解，但仍在Sobolev空间开放。该项目的目标之一是确定Sobolev空间的正确范围。