Endpoint Maximal Theorems

本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，在先进的Calderon-Zygmund论证的背景下利用局部平滑估计。在扭曲三次体上的（适当缩放的）具有弧长测量的三维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获得幂零群中具有正则折叠关系的奇异平均算子的新估计，并完成离散虚幂零群中稀疏集衍生的非标准遍历平均的相关项目。最后，LaVictoire计划开发并测试一种受Project Euler网站启发的补充数学教学的创新方法。谐波分析的数学方法和技术经常在自然科学和工程中得到重要的应用。从广义上讲，调和分析涉及通过微积分型积分来研究函数，从而发现或恢复从函数的技术定义中很难发现的重要信息。这个项目着眼于以几何图形的方式扭曲给定函数（数据）的各种过程，类似于用抖动的相机拍摄的照片，或者被巨大的引力场扭曲的深空图像。LaVictoire将研究一个重要的问题，即什么时候可以从扭曲的数据中令人满意地重建原始数据；这些问题可以利用谐波分析几个领域的最新进展来研究。

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.