Mathematical Sciences: Fourier Analysis

该奖项支持的工作将集中在与奇异积分算子、曲线和曲面上的平均、极大函数、傅立叶积分算子以及紧致和非紧致流形上的傅立叶乘子相关的几个数学分析领域。数学分析中的核心问题之一是通过对仅限于曲线或曲面的函数的积分应用各种微分技术来恢复函数。虽然现在有相当多的文献关于这个主题，但曲线总是被限制在满足强曲率条件的曲线上。为了获得更一般的信息，将需要紧集上的局部极大函数估计。这项研究的主旨将是首先找到这些估计。这项工作反过来又与Calderon-Zygmund型奇异积分算子有着密切的联系。其中一个障碍是过去的论点对傅里叶变换的依赖。现在有一些证据表明，有可能取得进展，绕过这一障碍。其他工作将集中在波动方程的局部光滑化性质上。局部平滑背后的一般思想是为时间周期而不是为固定的时刻建立波动方程解的大小的测量。这些测量值是根据与方程相关的边值给出的。已经取得了一些局部光顺的结果，特别是在高维的情况下。最困难的情况是空间维度2和3。最终，我们必须建立Kakeya极大算子的变体及其p次方范数的控制。关于退化傅立叶积分算子的估计、非紧黎曼流形上的径向傅立叶乘子以及伪凸域上的Szego核的估计的工作将继续进行。调和分析中的数学研究致力于研究数学函数的精细结构及其定义的基本领域。这项工作开发的工具揭示了更经典的方法通常不能检测到的隐藏关系。

Work supported by this award will focus on several areas of mathematical analysis related to singular integral operators, averages over curves and surfaces, maximal functions, Fourier integral operators, and Fourier multipliers on compact and noncompact manifolds. One of the central problems in mathematical analysis is that of recovering a function through various differentiation techniques applied to integrals of the function restricted to curves or surfaces. Although there is now a considerable body of literature on this subject, the curves have always been confined to those satisfying strong curvature conditions. Efforts to obtain more general information will require local maximal function estimates on compact sets. The thrust of the research will be to find these estimates first. This work, in turn, has close connections with singular integral operators of Calderon-Zygmund type. One of the obstructions is the dependence of past arguments on the Fourier transform. There is now some evidence that progress is possible which will bypass this impediment. Other work will concentrate on local smoothing properties for wave equations. The general idea behind local smoothing is one of establishing measurement of the size of a solution of a wave equation for time periods rather than for a fixed instant of time. The measurements are given in terms of the boundary values associated with the equation. Some local smoothing results have been obtained, especially in higher dimensions. The most difficult cases are spatial dimensions two and three. Ultimately, one must establish variants of the Kakeya- maximal operator and control of its p-th power norm. Work on estimates of degenerate Fourier integral operators, radial Fourier multipliers on noncompact Riemannian manifolds and estimates of the Szego kernel for pseudoconvex domains will continue. Mathematical research in harmonic analysis seeks to study the fine structure of mathematical functions and their underlying domains of definition. The tools developed by this work bring out hidden relationships not ordinarily detected by more classical methods.