Research in harmonic analysis and partial differential equations

该奖项由《2009年美国复苏和再投资法案》(公法111-5)资助。该协会将从事调和分析和偏微分方程(PDE)的研究。在偏微分方程中，重点研究薛定谔演化(SE)的动力学性质。正在进行的研究的一个主题是对SE的分散估计。他还将致力于非线性SE的数学问题，其动机是光纤通信系统中的数值研究。在调和分析中，他专注于欧几里得空间中以勒贝格范数不等式为中心的问题。特别是，他建议继续研究与分形测度相关的限制估计，以及它们在偏微分方程组和几何测度论中的应用。他还建议继续研究广义Radon变换(GRT)的映射性质--欧氏空间的低维子流形上的一大类平均算子。通过应用为Kakeya问题开发的技术，他在某些情况下获得了有趣的结果。GRT映射性质的结果在傅里叶限制现象和多维傅里叶级数可和理论中有着重要的应用。调和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。非线性随机共振的研究直接受到光纤通信系统中的工程问题的推动，所采用的方法有可能在一系列应用中得到应用。GRT映射特性的研究在工程上有着广泛的应用。例如，应用于患者身体密度函数的X射线变换(这是一种特定的GRT)本质上是通过磁共振成像获得的数据。傅立叶限制的研究、多维傅立叶级数的可和理论、色散估计是研究一大类偏微分方程不可替代的工具。拟议的研究将有助于对这些问题的总体理解。计划中的研究还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。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI will undertake research in harmonic analysis, and partial differential equations (PDE). In PDE, the focus is on the dynamical properties of Schrodinger evolution (SE). One subject of on-going research is dispersive estimates for SE. He will also work on mathematical problems on non-linear SE motivated by the numerical studies in fiber optic communication systems. In harmonic analysis, he focuses on problems in Euclidean spaces centered around Lebesgue norm inequalities. In particular, he proposes to continue his investigations on restriction estimates relative to fractal measures, and on their applications in PDE and geometric measure theory. He also proposes to continue his research on the mapping properties of generalized Radon transforms (GRT) -- a huge class of averaging operators over lower dimensional submanifolds of Euclidean spaces. By applying the techniques developed for Kakeya problems, he obtained interesting results in some cases. The results on the mapping properties of GRT have important applications in Fourier restriction phenomenon and in general in the summability theory of multi-dimensional Fourier series.Harmonic 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. The proposed research on nonlinear SE are directly motivated by the engineering problems in fiber optic communication systems, and the methods used are likely to be useful in a range of applications. The study of the mapping properties of GRT has various applications in engineering. For example, the X-ray transform (which is a particular GRT) applied to the density function of a patients body is essentially the data obtained by magnetic resonance imaging. The study of Fourier restriction, summability theory of multi-dimensional Fourier series, and dispersive estimates are irreplaceable tools in the study of a wide class of PDE. The proposed research would make a contribution to the general understanding of these problems. The planned research is also related to certain discrete problems of interest in combinatorics and number theory, which in most cases remain wide open.