Properties at Averaging Operators, and Applications to Fourier Analysis

PI将进行调和分析和偏微分方程的基础研究，重点研究以勒贝格范数不等式为中心的欧几里得空间中的调和分析问题。一个正在进行的研究课题（部分与M. Christ合作）是广义Radon变换的映射性质——广义Radon变换是一大类低维子流形上的平均算子。在某些情况下，PI通过应用Bourgain， Wolff和其他人为Kakeya问题开发的技术获得了令人满意的结果。除了在特殊情况下，广义Radon变换的理解仍然很少，而与kakya相关的方法可以提供更多的东西。理解广义Radon变换是分析多维傅里叶级数和积分、更一般的振荡积分以及线性和非线性波动方程的可和性的重要组成部分。其他相关问题，包括傅里叶限制现象和几何测量理论中的问题，如子流形在欧几里德空间中的填充，也将被研究。傅里叶分析在自然科学和工程中有着广泛的应用。它是目前在应用程序中广泛使用的强大而多样的工具阵列的基础，并提供了未来进一步应用程序的承诺。拟议的研究涉及基础问题，这些问题最终可能有助于支持这种未来的应用。多维傅立叶级数的可和性理论及相关的振荡积分问题是研究一类广泛的偏微分方程的不可替代的工具，而这些偏微分方程目前的知识状态是不完整的。拟议的研究将有助于对这些问题的普遍理解。计划中的研究与组合学和数论中某些离散问题有关，这些问题在大多数情况下仍然是开放的。

The PI will conduct basic research on Harmonic Analysis and Partial Differential Equations, focusing on problems in harmonic analysis in Euclidean spaces centered around Lebesgue norm inequalities. One subject of ongoing research (partly joint with M. Christ) is the mapping properties of Generalized Radon Transforms -- a vast class of averaging operators over lower dimensional submanifolds. Satisfactory results in some cases have been obtained by the PI, via the application of techniques developed by Bourgain, Wolff, and others for the Kakeya problem. Generalized Radon transforms are still poorly understood, except in special cases, and Kakeya-related methods have a great deal more to offer. Understanding generalized Radon transforms is an important ingredient in the analysis of summability of multi-dimensional Fourier nseries and integrals, of more general oscillatory integrals, and of linear and nonlinear wave equations. Other related problems, including the Fourier restriction phenomenon and issues in Geometric Measure Theory such as the packing of submanifolds into Euclidean space, will also be investigated.Fourier analysis has always found wide applications in natural sciences and engineering.It underlies a powerful and diverse array of tools currently widely used in applications, and offers the promise of further applications in the future. The proposed research deals with foundational issues which may ultimately help to underpin such future applications. Summability theory of multi-dimensional Fourier series and related oscillatory integral problems are irreplaceable tools in the study of a wide class of PDEs, in which the current state of knowledge is incomplete. The proposed research will contribute to the general understanding of these problems. The planned research is related to certain discrete problems of interest in Combinatorics and Number Theory, which in most cases remain wide open.