Topics in Fourier Analysis

傅立叶分析主题

• 批准号: 0652890
• 负责人: Andreas Seeger
• 金额: $ 30万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652890&HistoricalAwards=false
• 关键词: Topics; Fourier; Analysis

1. We shall conduct research on various problems in harmonic analysis. The project will focus on local smoothing properties of solutions for wave equations. Results on local smoothing follow from a crucial inequality involving decompositions of cone multipliers which was originally formulated by Wolff. This inequality and several variants of it have a wide range of applications; one of them concerns the Sobolev regularity for averaging operators along curves and boundedness of associated maximal operators. It is of interest to determine the range of these inequalities. Other parts of the proposal deal with Fourier restriction and extension problems, with variational Carleson theorems, with singular maximal functions and with the wave equation on the Heisenberg group.2. A major part of this project is concerned with the regularity properties of some basic linear partial differential operators of mathematical physics. Quantitative results for these operators are applicable to problems in nonlinear equations with a wide range of applications in the physical sciences. They also yield applications within mathematics, namely to problems on expansions in eigenfunctions of the Laplacian on compact manifolds and the regularity of averaging operators. The study of these averaging operators is motivated by problems in the theory of medical imaging.

1. 我们将对谐波分析中的各种问题进行研究。本项目将重点研究波动方程解的局部平滑特性。局部平滑的结果来自于一个关键的不等式，该不等式涉及锥乘子的分解，该不等式最初由Wolff提出。这个不等式和它的几个变体有广泛的应用；其中一个是关于沿曲线平均算子的Sobolev规律性和相关极大算子的有界性。确定这些不平等的范围是很有趣的。提案的其他部分处理傅里叶限制和扩展问题，变分Carleson定理，奇异极大函数和海森堡群上的波动方程。本课题主要研究数学物理中一些基本线性偏微分算子的正则性。这些算子的定量结果适用于在物理科学中有着广泛应用的非线性方程问题。它们在数学中也有应用，即在紧流形上拉普拉斯特征函数的展开式问题和平均算子的规律性问题。这些平均算子的研究是由医学成像理论中的问题所激发的。