Endpoint Maximal Theorems

端点极大定理

• 批准号: 1201314
• 负责人: Brian Street
• 金额: $ 9万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-09-15 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201314&HistoricalAwards=false
• 关键词: Endpoint; Maximal; Theorems

This project aims to study endpoint maximal inequalities for families of convolution operators with smooth measures on manifolds under anisotropic dilation structures, making use of local smoothing estimates in the context of an advanced Calderon-Zygmund argument. The (properly scaled) convolution in three dimensions with the arc-length measure on the twisted cubic is the ultimate model case. In addition, LaVictoire and his collaborators hope to apply these methods to obtain new estimates for singular averaging operators with canonical folding relations in nilpotent groups, and also to complete a related project on nonstandard ergodic averages derived from sparse sets in discrete virtually nilpotent groups. Finally, LaVictoire plans to develop and test an innovative approach to supplementary mathematics instruction inspired by the website Project Euler.Methods and techniques from the mathematics field of harmonic analysis have often led to important applications to the natural sciences and engineering. Harmonic analysis, in its broadest sense, concerns the study of functions by means of calculus-type integrals that lead to the discovery or recovery of important information which can be very hard to detect from the function's technical definition. This project looks at various processes that distort a given function (the data) in a geometrically patterned way similar e.g., to a photograph taken with a shaking camera, or to a deep-space image warped by a massive gravitational field. LaVictoire will investigate the important problem of when can the original data be satisfactorily reconstructed from the distorted data; such problems can be studied using recent advances in several areas of harmonic analysis.

该项目旨在研究各向异性膨胀结构下流形上具有光滑测度的卷积算子族的端点极大不等式，在高级Calderon-Zygmund参数的背景下使用局部光滑估计。在扭曲的三次曲面上的弧长测量的三维（适当缩放的）卷积是最终的模型情况。此外，LaVictoire和他的合作者希望应用这些方法来获得新的估计奇异平均运营商与规范折叠关系的幂零群，也完成了相关的项目非标准遍历平均来自稀疏集离散虚拟幂零群。最后，LaVictoire计划开发和测试一种创新的方法，以补充数学教学的启发网站项目Euler.方法和技术从数学领域的谐波分析往往导致重要的应用到自然科学和工程。调和分析，在其最广泛的意义上，涉及通过微积分类型的积分，导致发现或恢复的重要信息，可以很难从功能的技术定义检测功能的研究。这个项目着眼于以几何模式的方式扭曲给定函数（数据）的各种过程，例如，一张用震动的照相机拍摄的照片，或者一张被巨大的引力场扭曲的深空图像。LaVictoire将调查的重要问题时，可以令人满意地重建原始数据从失真的数据，这些问题可以研究使用最近的进展，在几个领域的谐波分析。