Averaging operators and related topics in harmonic analysis

这个项目是在谐波分析和近似理论，在数学学科的分析领域。这些方法在理解自然科学和工程中的现象方面有许多应用。调和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。该项目的主要目标之一是扩展谐波分析中当前的数学工具箱，以促进更深入的理论理解，这最终将有利于应用。指导研究生从事研究是该项目的一个重要教育组成部分。首席研究员将在谐波分析的几个项目工作。(i)第一个项目是关于欧氏空间子流形上某些平均的精确正则性和Lebesgue空间中相关极大算子的有界性。PI将主要考虑非卷积变量，并强调幂零群上的各种球面极大算子。(ii)在第二个项目中，PI将研究膨胀集受限时波动方程解的局部平滑问题。即使在径向函数的简化版本中也出现了新的现象。人们期望的结果取决于各种概念的尺寸的膨胀集，闵可夫斯基维度，准Assouad维度和中间尺度的尺寸。PI还将研究相关问题涉及的Lp改进界的限制扩张集的球面极大算子，为开放的情况下，扩张集是不Assouad正则的。(iii)PI将在稀疏支配中进行各种关于端点估计的项目，专注于真正的多尺度现象。原子分解技术和单尺度算子的Lp改进结果起着至关重要的作用。(iv)第四个项目是在近似理论和关注的表征近似空间的非线性小波逼近。PI和他的合作者将专注于近似阶数较高且近似空间不是赋范空间的有趣案例。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。

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.