Averaging operators and related topics in harmonic analysis

谐波分析中的平均运算符和相关主题

• 批准号: 2348797
• 负责人: Andreas Seeger
• 金额: $ 33.5万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-09-01 至 2027-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348797&HistoricalAwards=false
• 关键词: Averaging; operators; related; topics; harmonic

This project is in harmonic analysis and approximation theory, areas within the mathematical discipline of analysis. The methods have found many applications in understanding phenomena in the natural sciences and engineering. Harmonic analysis seeks to provide efficient mathematical tools for these disciplines and contributes to the unification of seemingly unrelated areas. One of the main objectives of this project is to expand the current mathematical toolbox in harmonic analysis to contribute towards a deeper theoretical understanding which will ultimately be beneficial for applications. The mentoring of graduate students in research is an important educational component of the project. The principal investigator will work on several projects in harmonic analysis. (i) The first project is concerned with the precise regularity properties of certain averages over sub-manifolds of Euclidean space and the boundedness of associated maximal operators in Lebesgue spaces. The PI will consider mainly non-convolution variants and emphasize various classes of spherical maximal operators on nilpotent groups. (ii) In a second project the PI will study versions of the local smoothing problem for solutions of the wave equation when the dilation set is restricted. New phenomena show up even in the simplified version for radial functions. One expects that the outcomes depend on various notions of dimensions of the dilation sets, the Minkowski dimension, the quasi-Assouad dimension and intermediate scales of dimensions. The PI will also study the related problem concerns the Lp improving bounds for spherical maximal operators with restricted dilation sets, for the open case when the dilation set is not Assouad regular. (iii) The PI will pursue various projects on endpoint estimates in sparse domination, focusing on true multiscale phenomena. Atomic decompositions techniques and sharp Lp improving results for single-scaled operators play a crucial role. (iv) A fourth project is in approximation theory and concerns the characterization of approximation spaces for nonlinear wavelet approximation. The PI and his collaborators will focus on the interesting cases when the order of approximation is high, and the approximation spaces will not be normed spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目正在谐波分析和近似理论中，这是分析的数学学科中的领域。这些方法在理解自然科学和工程中的现象中发现了许多应用。谐波分析旨在为这些学科提供有效的数学工具，并有助于统一看似无关的领域。该项目的主要目的之一是在谐波分析中扩展当前的数学工具箱，以有助于更深入的理论理解，最终将对应用程序有益。研究生在研究中的指导是该项目的重要教育组成部分。首席研究人员将在谐波分析中从事多个项目。 （i）第一个项目与欧几里得空间的子序列相比某些平均值的确切规律性属性以及Lebesgue空间中相关最大运算符的界限。 PI将主要考虑非卷积变体，并强调各种在nilpotent组上的球形最大算子。 （ii）在第二个项目中，PI将在限制扩张集时研究局部平滑问题的局部平滑问题。新现象即使在简化的径向功能中也出现。人们期望结果取决于扩张集的各种概念，minkowski维度，准空间维度和尺寸的中间尺度。 PI还将研究相关的问题涉及LP通过限制扩张集的球形最大运算符的界限，而当扩张集不是Assouad常规的情况下。 （iii）PI将在稀疏统治的端点估计中追求各种项目，重点是真正的多尺度现象。原子分解技术和尖锐的LP改善单级操作员的结果起着至关重要的作用。 （iv）第四个项目是在近似理论中，涉及非线性小波近似近似空间的表征。当近似值较高时，PI和他的合作者将重点关注有趣的案例，并且近似空间将不会成为规范的空间。该奖项反映了NSF的法定任务，并且认为值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来获得支持。