Research in harmonic analysis and partial differential equations

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI将进行谐波分析和偏微分方程（PDE）的研究。在偏微分方程中，重点是薛定谔演化（SE）的动力学性质。正在进行的研究的一个主题是分散估计SE。他还将致力于光纤通信系统数值研究所激发的非线性SE的数学问题。在调和分析中，他专注于以勒贝格范数不等式为中心的欧几里得空间中的问题。特别是，他建议继续他的调查限制估计相对于分形措施，并就其应用在偏微分方程和几何测度理论。他还建议继续研究广义拉东变换（GRT）的映射特性-一个巨大的类平均运营商在低维子流形的欧几里德空间。通过应用技术开发的Kakeya问题，他获得了有趣的结果在某些情况下。GRT映射性质的结果在Fourier限制现象和一般的多维Fourier级数的可和性理论中有重要的应用，调和分析在自然科学和工程中有着广泛的应用。 它是目前广泛应用于应用程序中的一系列强大而多样的工具的基础，并为未来的进一步应用提供了希望。拟议的研究涉及的基础问题，最终可能有助于支持这种未来的应用。提出的非线性SE的研究直接受到光纤通信系统中的工程问题的启发，所使用的方法可能在一系列应用中是有用的。广义相对论映射性质的研究在工程中有着广泛的应用。例如，应用于患者身体的密度函数的X射线变换（其是特定的GRT）本质上是通过磁共振成像获得的数据。傅里叶限制、多维傅里叶级数的可和性理论以及色散估计的研究是研究一类广泛的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普遍理解。计划中的研究还涉及组合学和数论中某些感兴趣的离散问题，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.