Properties at Averaging Operators, and Applications to Fourier Analysis

PI将进行调和分析和偏微分方程组的基础研究，重点研究以勒贝格范数不等式为中心的欧氏空间中的调和分析问题。正在进行的研究(部分与M.Christian联合)的一个主题是广义Radon变换的映射性质，广义Radon变换是低维子流形上的一大类平均算子。在某些情况下，PI通过应用Bourain、Wolff等人开发的技术来解决Kakeya问题，获得了满意的结果。除了在特殊情况下，广义Radon变换仍然知之甚少，Kakeya相关的方法可以提供更多的东西。了解广义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。其他相关的问题，包括傅里叶限制现象和几何测度论中的问题，如子流形到欧氏空间的填充，也将被研究。傅立叶分析在自然科学和工程中一直得到广泛的应用，它是目前广泛应用于应用中的一系列强大而多样的工具的基础，并提供了未来进一步应用的前景。拟议的研究涉及的基础性问题可能最终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其相关的振荡积分问题是研究一类知识不完备的偏微分方程组不可替代的工具。拟议的研究将有助于对这些问题的总体理解。计划中的研究与组合学和数论中感兴趣的某些离散问题有关，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.